Question: What type of Turing machine is capable of multiple actions and extends into a variety of computational paths?

Answer: A non-deterministic Turing machine

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. <hl> A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. <hl> One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. <hl> A non-deterministic Turing machine <hl> is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.

question: What type of Turing machine is capable of multiple actions and extends into a variety of computational paths?, context: A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.


Question: What is the term given to algorithms that utilize random bits?

Answer: randomized algorithms

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. <hl> Algorithms that use random bits are called randomized algorithms . <hl> A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called <hl> randomized algorithms <hl>. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.

Algorithms that use random bits are called <hl> randomized algorithms <hl> .

question: What is the term given to algorithms that utilize random bits?, context: A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.


Question: Turing machines are commonly employed to define what?

Answer: complexity classes

<hl> Many types of Turing machines are used to define complexity classes , such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. <hl> They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.

Many types of Turing machines are used to define <hl> complexity classes <hl>, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.

Many types of Turing machines are used to define <hl> complexity classes <hl> , such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines.

question: Turing machines are commonly employed to define what?, context: Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.


Question: What are two factors that directly effect how powerful a Turing machine may or may not be?

Answer: time or space

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. <hl> They are all equally powerful in principle, but when resources (such as time or space ) are bounded, some of these may be more powerful than others. <hl>

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as <hl> time or space <hl>) are bounded, some of these may be more powerful than others.

They are all equally powerful in principle, but when resources (such as <hl> time or space <hl> ) are bounded, some of these may be more powerful than others.

question: What are two factors that directly effect how powerful a Turing machine may or may not be?, context: Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.


Question: In the determination of complexity classes, what are two examples of types of Turing machines?

Answer: probabilistic Turing machines, non-deterministic Turing machines

<hl> Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines , quantum Turing machines, symmetric Turing machines and alternating Turing machines. <hl> They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, <hl> probabilistic Turing machines, non-deterministic Turing machines <hl>, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, <hl> probabilistic Turing machines, non-deterministic Turing machines <hl> , quantum Turing machines, symmetric Turing machines and alternating Turing machines.

question: In the determination of complexity classes, what are two examples of types of Turing machines?, context: Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.


Question: What is an example of a machine model that deviates from a generally accepted multi-tape Turing machine?

Answer: random access machines

<hl> Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines . <hl> Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example <hl> random access machines <hl>. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example <hl> random access machines <hl> .

question: What is an example of a machine model that deviates from a generally accepted multi-tape Turing machine?, context: Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.


Question: In considering Turing machines and alternate variables, what measurement left unaffected by conversion between machine models?

Answer: computational power

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. <hl> Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power . <hl> The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra <hl> computational power <hl>. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.

Perhaps surprisingly, each of these models can be converted to another without providing any extra <hl> computational power <hl> .

question: In considering Turing machines and alternate variables, what measurement left unaffected by conversion between machine models?, context: Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.


Question: What two resources commonly consumed by alternate models are typically known to vary?

Answer: time and memory

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. <hl> The time and memory consumption of these alternate models may vary. <hl> What all these models have in common is that the machines operate deterministically.

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The <hl> time and memory <hl> consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.

question: What two resources commonly consumed by alternate models are typically known to vary?, context: Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.


Question: What commonality do alternate machine models, such as random access machines, share with Turing machines?

Answer: the machines operate deterministically

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. <hl> What all these models have in common is that the machines operate deterministically . <hl>

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that <hl> the machines operate deterministically <hl>.

What all these models have in common is that <hl> the machines operate deterministically <hl> .

question: What commonality do alternate machine models, such as random access machines, share with Turing machines?, context: Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.


Question: What type of Turing machine can be characterized by checking multiple possibilities at the same time?

Answer: non-deterministic

However, some computational problems are easier to analyze in terms of more unusual resources. <hl> For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. <hl> The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

However, some computational problems are easier to analyze in terms of more unusual resources. For example, a <hl> non-deterministic <hl> Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

question: What type of Turing machine can be characterized by checking multiple possibilities at the same time?, context: However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.


Question: What often affects or facilitates ease of analysis in computational problems?

Answer: unusual resources

<hl> However, some computational problems are easier to analyze in terms of more unusual resources . <hl> For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

However, some computational problems are easier to analyze in terms of more <hl> unusual resources <hl>. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

However, some computational problems are easier to analyze in terms of more <hl> unusual resources <hl> .

question: What often affects or facilitates ease of analysis in computational problems?, context: However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.


Question: A non-deterministic Turing machine has the ability to capture what facet of useful analysis?

Answer: mathematical models

However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. <hl> The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems. <hl>

However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the <hl> mathematical models <hl> we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

question: A non-deterministic Turing machine has the ability to capture what facet of useful analysis?, context: However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.


Question: What is the most critical resource in the analysis of computational problems associated with non-deterministic Turing machines?

Answer: time

However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. <hl> The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems. <hl>

However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic <hl> time <hl> is a very important resource in analyzing computational problems.

question: What is the most critical resource in the analysis of computational problems associated with non-deterministic Turing machines?, context: However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.


Question: The time required to output an answer on a deterministic Turing machine is expressed as what?

Answer: state transitions

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. <hl> The time required by a deterministic Turing machine M on input x is the total number of state transitions , or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). <hl> A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of <hl> state transitions <hl>, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

The time required by a deterministic Turing machine M on input x is the total number of <hl> state transitions <hl> , or steps, the machine makes before it halts and outputs the answer ("yes" or "no").

question: The time required to output an answer on a deterministic Turing machine is expressed as what?, context: For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).


Question: Complexity theory classifies problems based on what primary attribute?

Answer: difficulty

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. <hl> Since complexity theory is interested in classifying problems based on their difficulty , one defines sets of problems based on some criteria. <hl> For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their <hl> difficulty <hl>, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

Since complexity theory is interested in classifying problems based on their <hl> difficulty <hl> , one defines sets of problems based on some criteria.

question: Complexity theory classifies problems based on what primary attribute?, context: For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).


Question: What is the expression used to identify any given series of problems capable of being solved within time on a deterministic Turing machine?

Answer: DTIME(f(n))

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. <hl> For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)) . <hl>

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by <hl> DTIME(f(n)) <hl>.

For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by <hl> DTIME(f(n)) <hl> .

question: What is the expression used to identify any given series of problems capable of being solved within time on a deterministic Turing machine?, context: For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).


Question: What is the most critical resource measured to in assessing the determination of a Turing machine's ability to solve any given set of problems?

Answer: time

<hl> For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. <hl> The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

For a precise definition of what it means to solve a problem using a given amount of <hl> time <hl> and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

question: What is the most critical resource measured to in assessing the determination of a Turing machine's ability to solve any given set of problems?, context: For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).


Question: Time and space are both examples of what type of resource?

Answer: complexity resources

Analogous definitions can be made for space requirements. <hl> Although time and space are the most well-known complexity resources , any complexity measure can be viewed as a computational resource. <hl> Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Analogous definitions can be made for space requirements. Although time and space are the most well-known <hl> complexity resources <hl>, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Although time and space are the most well-known <hl> complexity resources <hl> , any complexity measure can be viewed as a computational resource.

question: Time and space are both examples of what type of resource?, context: Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.


Question: A complexity resource can also be described as what other type of resource?

Answer: computational resource

Analogous definitions can be made for space requirements. <hl> Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource . <hl> Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a <hl> computational resource <hl>. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a <hl> computational resource <hl> .

question: A complexity resource can also be described as what other type of resource?, context: Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.


Question: What is typically used to broadly define complexity measures?

Answer: Blum complexity axioms

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. <hl> Complexity measures are very generally defined by the Blum complexity axioms . <hl> Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the <hl> Blum complexity axioms <hl>. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Complexity measures are very generally defined by the <hl> Blum complexity axioms <hl> .

question: What is typically used to broadly define complexity measures?, context: Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.


Question: Communication complexity is an example of what type of measure?

Answer: Complexity measures

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. <hl> Complexity measures are very generally defined by the Blum complexity axioms. <hl> Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. <hl> Complexity measures <hl> are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

question: Communication complexity is an example of what type of measure?, context: Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.


Question: Decision tree is an example of what type of measure?

Answer: Complexity measures

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. <hl> Complexity measures are very generally defined by the Blum complexity axioms. <hl> Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. <hl> Complexity measures <hl> are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

question: Decision tree is an example of what type of measure?, context: Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.


Question: What are the three primary expressions used to represent case complexity?

Answer: best, worst and average

<hl> The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. <hl> Since some inputs of size n may be faster to solve than others, we define the following complexities:

The <hl> best, worst and average <hl> case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:

question: What are the three primary expressions used to represent case complexity?, context: The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:


Question: Case complexity likelihoods provide variable probabilities of what general measure?

Answer: complexity measure

<hl> The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure ) of different inputs of the same size. <hl> Since some inputs of size n may be faster to solve than others, we define the following complexities:

The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other <hl> complexity measure <hl>) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:

The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other <hl> complexity measure <hl> ) of different inputs of the same size.

question: Case complexity likelihoods provide variable probabilities of what general measure?, context: The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:


Question: What is one common example of a critical complexity measure?

Answer: time

<hl> The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. <hl> Since some inputs of size n may be faster to solve than others, we define the following complexities:

The best, worst and average case complexity refer to three different ways of measuring the <hl> time <hl> complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:

question: What is one common example of a critical complexity measure?, context: The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:


Question: Case complexities provide three likelihoods of what differing variable that remains the same size?

Answer: inputs

<hl> The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. <hl> Since some inputs of size n may be faster to solve than others, we define the following complexities:

The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different <hl> inputs <hl> of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:

question: Case complexities provide three likelihoods of what differing variable that remains the same size?, context: The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:


Question: What provides a solution to a list of integers provided as input that ned to be sorted?

Answer: deterministic sorting algorithm quicksort

<hl> For example, consider the deterministic sorting algorithm quicksort . <hl> This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

For example, consider the <hl> deterministic sorting algorithm quicksort <hl>. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

For example, consider the <hl> deterministic sorting algorithm quicksort <hl> .

question: What provides a solution to a list of integers provided as input that ned to be sorted?, context: For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.


Question: When extensive time is required to sort integers, this represents what case complexity?

Answer: worst-case

For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. <hl> The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. <hl> If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The <hl> worst-case <hl> is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

question: When extensive time is required to sort integers, this represents what case complexity?, context: For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.


Question: What is the expression used to denote a worst case complexity as expressed by time taken?

Answer: O(n2)

For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. <hl> The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. <hl> If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time <hl> O(n2) <hl> for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

question: What is the expression used to denote a worst case complexity as expressed by time taken?, context: For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.


Question: Classification of resources is contingent on determining the upper and lower bounds of minimum time required by what?

Answer: the most efficient algorithm

<hl> To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. <hl> The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by <hl> the most efficient algorithm <hl> solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

question: Classification of resources is contingent on determining the upper and lower bounds of minimum time required by what?, context: To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).


Question: The analysis of a specific algorithm is typically assigned to what field of computational science?

Answer: analysis of algorithms

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. <hl> Analyzing a particular algorithm falls under the field of analysis of algorithms . <hl> To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of <hl> analysis of algorithms <hl>. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

Analyzing a particular algorithm falls under the field of <hl> analysis of algorithms <hl> .

question: The analysis of a specific algorithm is typically assigned to what field of computational science?, context: To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).


Question: Which bound of time is more difficult to establish?

Answer: lower bounds

<hl> To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. <hl> The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and <hl> lower bounds <hl> on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

question: Which bound of time is more difficult to establish?, context: To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).


Question: A specific algorithm demonstrating T(n) represents what measure of time complexity?

Answer: upper bound

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. <hl> To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). <hl> However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an <hl> upper bound <hl> T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

question: A specific algorithm demonstrating T(n) represents what measure of time complexity?, context: To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).


Question: What is the colloquial phrase used to convey the continuum of algorithms with unlimited availability irrespective of time?

Answer: all possible algorithms

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). <hl> However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. <hl> The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about <hl> all possible algorithms <hl> that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

question: What is the colloquial phrase used to convey the continuum of algorithms with unlimited availability irrespective of time?, context: To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).


Question: What expression is generally used to convey upper or lower bounds?

Answer: big O notation

<hl> Upper and lower bounds are usually stated using the big O notation , which hides constant factors and smaller terms. <hl> This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).

Upper and lower bounds are usually stated using the <hl> big O notation <hl>, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).

Upper and lower bounds are usually stated using the <hl> big O notation <hl> , which hides constant factors and smaller terms.

question: What expression is generally used to convey upper or lower bounds?, context: Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).


Question: What does a big O notation hide?

Answer: constant factors and smaller terms

<hl> Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms . <hl> This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).

Upper and lower bounds are usually stated using the big O notation, which hides <hl> constant factors and smaller terms <hl>. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).

Upper and lower bounds are usually stated using the big O notation, which hides <hl> constant factors and smaller terms <hl> .

question: What does a big O notation hide?, context: Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).


Question: How would one write T(n) = 7n2 + 15n + 40 in big O notation?

Answer: T(n) = O(n2)

Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 <hl> + 15n + 40, in big O notation one would write T(n) = O(n2) . <hl>

Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write <hl> T(n) = O(n2) <hl>.

+ 15n + 40, in big O notation one would write <hl> T(n) = O(n2) <hl> .

question: How would one write T(n) = 7n2 + 15n + 40 in big O notation?, context: Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).


Question: Big O notation provides autonomy to upper and lower bounds with relationship to what?

Answer: the computational model

Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. <hl> This makes the bounds independent of the specific details of the computational model used. <hl> For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).

Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of <hl> the computational model <hl> used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).

question: Big O notation provides autonomy to upper and lower bounds with relationship to what?, context: Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).


Question: What has complicated definitions that prevent classification into a framework?

Answer: complexity classes

<hl> Of course, some complexity classes have complicated definitions that do not fit into this framework. <hl> Thus, a typical complexity class has a definition like the following:

Of course, some <hl> complexity classes <hl> have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following:

question: What has complicated definitions that prevent classification into a framework?, context: Of course, some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following:


Question: Complexity classes are generally classified into what?

Answer: framework

<hl> Of course, some complexity classes have complicated definitions that do not fit into this framework . <hl> Thus, a typical complexity class has a definition like the following:

Of course, some complexity classes have complicated definitions that do not fit into this <hl> framework <hl>. Thus, a typical complexity class has a definition like the following:

Of course, some complexity classes have complicated definitions that do not fit into this <hl> framework <hl> .

question: Complexity classes are generally classified into what?, context: Of course, some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following:


Question: Difficulty in establishing a framework for complexity classes can be caused by what variable?

Answer: complicated definitions

<hl> Of course, some complexity classes have complicated definitions that do not fit into this framework. <hl> Thus, a typical complexity class has a definition like the following:

Of course, some complexity classes have <hl> complicated definitions <hl> that do not fit into this framework. Thus, a typical complexity class has a definition like the following:

question: Difficulty in establishing a framework for complexity classes can be caused by what variable?, context: Of course, some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following:


Question: Concrete bounding of computation time frequently produces complexity classes contingent upon what?

Answer: chosen machine model

<hl> But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model . <hl> For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the <hl> chosen machine model <hl>. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the <hl> chosen machine model <hl> .

question: Concrete bounding of computation time frequently produces complexity classes contingent upon what?, context: But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.


Question: A multi-tape Turing machine requires what type of time for a solution?

Answer: linear time

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. <hl> For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. <hl> If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in <hl> linear time <hl> on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

question: A multi-tape Turing machine requires what type of time for a solution?, context: But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.


Question: A language solved in quadratic time implies the use of what type of Turing machine?

Answer: single-tape Turing machines

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. <hl> For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines . <hl> If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of <hl> single-tape Turing machines <hl>. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of <hl> single-tape Turing machines <hl> .

question: A language solved in quadratic time implies the use of what type of Turing machine?, context: But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.


Question: What thesis specifies that a polynomial relationship exists within time complexities in a computational model?

Answer: Cobham-Edmonds thesis

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. <hl> If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). <hl> This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, <hl> Cobham-Edmonds thesis <hl> states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

question: What thesis specifies that a polynomial relationship exists within time complexities in a computational model?, context: But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.


Question: Decision problems capable of being solved by a deterministic Turing machine while maintaining adherence to polynomial time belong to what class?

Answer: complexity class P

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). <hl> This forms the basis for the complexity class P , which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. <hl> The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the <hl> complexity class P <hl>, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

This forms the basis for the <hl> complexity class P <hl> , which is the set of decision problems solvable by a deterministic Turing machine within polynomial time.

question: Decision problems capable of being solved by a deterministic Turing machine while maintaining adherence to polynomial time belong to what class?, context: But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.


Question: What are two examples of measurements are bound within algorithms to establish complexity classes?

Answer: time or space

<hl> Many important complexity classes can be defined by bounding the time or space used by the algorithm. <hl> Some important complexity classes of decision problems defined in this manner are the following:

Many important complexity classes can be defined by bounding the <hl> time or space <hl> used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:

question: What are two examples of measurements are bound within algorithms to establish complexity classes?, context: Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:


Question: What function is used by algorithms to define measurements like time or space?

Answer: bounding

<hl> Many important complexity classes can be defined by bounding the time or space used by the algorithm. <hl> Some important complexity classes of decision problems defined in this manner are the following:

Many important complexity classes can be defined by <hl> bounding <hl> the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:

question: What function is used by algorithms to define measurements like time or space?, context: Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:


Question: Bounding of time and space or similar measurements is often used by algorithms to define what?

Answer: complexity classes

<hl> Many important complexity classes can be defined by bounding the time or space used by the algorithm. <hl> Some important complexity classes of decision problems defined in this manner are the following:

Many important <hl> complexity classes <hl> can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:

question: Bounding of time and space or similar measurements is often used by algorithms to define what?, context: Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:


Question: What are three examples of complexity classes associated with definitions established by probabilistic Turing machines?

Answer: BPP, ZPP and RP

<hl> Other important complexity classes include BPP, ZPP and RP , which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. <hl> #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

Other important complexity classes include <hl> BPP, ZPP and RP <hl>, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

Other important complexity classes include <hl> BPP, ZPP and RP <hl> , which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines.

question: What are three examples of complexity classes associated with definitions established by probabilistic Turing machines?, context: Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.


Question: AC and NC are complexity classes typically associated with what type of circuit?

Answer: Boolean

<hl> Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. <hl> #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using <hl> Boolean <hl> circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

question: AC and NC are complexity classes typically associated with what type of circuit?, context: Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.