Question: What computational problem is commonly associated with prime factorization?

Answer: The integer factorization problem

<hl> The integer factorization problem is the computational problem of determining the prime factorization of a given integer. <hl> Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

<hl> The integer factorization problem <hl> is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

question: What computational problem is commonly associated with prime factorization?, context: The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.


Question: The integer factorization problem essentially seeks to determine if the value of of an input is less than what variable?

Answer: k

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. <hl> Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k . <hl> No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than <hl> k <hl>. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than <hl> k <hl> .

question: The integer factorization problem essentially seeks to determine if the value of of an input is less than what variable?, context: The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.


Question: That there currently exists no known integer factorization problem underpins what commonly used system?

Answer: modern cryptographic systems

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. <hl> Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems , such as the RSA algorithm. <hl> The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several <hl> modern cryptographic systems <hl>, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several <hl> modern cryptographic systems <hl> , such as the RSA algorithm.

question: That there currently exists no known integer factorization problem underpins what commonly used system?, context: The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.


Question: What is the most well-known algorithm associated with the integer factorization problem?

Answer: the general number field sieve

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). <hl> The best known algorithm for integer factorization is the general number field sieve , which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. <hl> However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is <hl> the general number field sieve <hl>, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

The best known algorithm for integer factorization is <hl> the general number field sieve <hl> , which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer.

question: What is the most well-known algorithm associated with the integer factorization problem?, context: The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.


Question: What is the unproven assumption generally ascribed to the value of complexity classes?

Answer: suspected to be unequal

<hl> Many known complexity classes are suspected to be unequal , but this has not been proved. <hl> For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

Many known complexity classes are <hl> suspected to be unequal <hl>, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

Many known complexity classes are <hl> suspected to be unequal <hl> , but this has not been proved.

question: What is the unproven assumption generally ascribed to the value of complexity classes?, context: Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.


Question: What is an expression that can be used to illustrate the suspected inequality of complexity classes?

Answer: P ⊆ NP ⊆ PP ⊆ PSPACE

Many known complexity classes are suspected to be unequal, but this has not been proved. <hl> For instance P ⊆ NP ⊆ PP ⊆ PSPACE , but it is possible that P = PSPACE. <hl> If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

Many known complexity classes are suspected to be unequal, but this has not been proved. For instance <hl> P ⊆ NP ⊆ PP ⊆ PSPACE <hl>, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

For instance <hl> P ⊆ NP ⊆ PP ⊆ PSPACE <hl> , but it is possible that P = PSPACE.

question: What is an expression that can be used to illustrate the suspected inequality of complexity classes?, context: Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.


Question: Where can the complexity classes RP, BPP, PP, BQP, MA, and PH be located?

Answer: between P and PSPACE

Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. <hl> Since there are many known complexity classes between P and PSPACE , such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. <hl> Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes <hl> between P and PSPACE <hl>, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

Since there are many known complexity classes <hl> between P and PSPACE <hl> , such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class.

question: Where can the complexity classes RP, BPP, PP, BQP, MA, and PH be located?, context: Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.


Question: What evidence between and among complexity classes would signify a theoretical watershed for complexity theory?

Answer: Proving that any of these classes are unequal

Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. <hl> Proving that any of these classes are unequal would be a major breakthrough in complexity theory. <hl>

Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. <hl> Proving that any of these classes are unequal <hl> would be a major breakthrough in complexity theory.

question: What evidence between and among complexity classes would signify a theoretical watershed for complexity theory?, context: Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.


Question: In what complexity class do complement problems of NP problems exist?

Answer: co-NP

<hl> Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. <hl> It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.

Along the same lines, <hl> co-NP <hl> is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.

question: In what complexity class do complement problems of NP problems exist?, context: Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.


Question: How do the yes/no answers of a complement problem of NP appear?

Answer: reversed

<hl> Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed ) <hl> of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers <hl> reversed <hl>) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers <hl> reversed <hl> )

question: How do the yes/no answers of a complement problem of NP appear?, context: Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.


Question: What is commonly believed to be the value relationship between P and co-NP

Answer: not equal

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. <hl> It is believed that NP is not equal to co-NP; however, it has not yet been proven. <hl> It has been shown that if these two complexity classes are not equal then P is not equal to NP.

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is <hl> not equal <hl> to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.

question: What is commonly believed to be the value relationship between P and co-NP, context: Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.


Question: What implication can be derived for P and NP if P and co-NP are established to be unequal?

Answer: P is not equal to NP

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. <hl> It has been shown that if these two complexity classes are not equal then P is not equal to NP . <hl>

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then <hl> P is not equal to NP <hl>.

It has been shown that if these two complexity classes are not equal then <hl> P is not equal to NP <hl> .

question: What implication can be derived for P and NP if P and co-NP are established to be unequal?, context: Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.


Question: What variable is associated with all problems solved within logarithmic space?

Answer: L

<hl> Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes. <hl>

Similarly, it is not known if <hl> L <hl> (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.

question: What variable is associated with all problems solved within logarithmic space?, context: Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.


Question: Though unkown, what are the most commonly ascribed attributes of L in relation to P

Answer: strictly contained in P or equal to P

<hl> Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P . <hl> Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is <hl> strictly contained in P or equal to P <hl>. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is <hl> strictly contained in P or equal to P <hl> .

question: Though unkown, what are the most commonly ascribed attributes of L in relation to P, context: Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.


Question: What lies between L and P that prevents a definitive determination of the relationship between L and P?

Answer: complexity classes

<hl> Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes. <hl>

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many <hl> complexity classes <hl> between the two, such as NL and NC, and it is not known if they are distinct or equal classes.

question: What lies between L and P that prevents a definitive determination of the relationship between L and P?, context: Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.


Question: What are two complexity classes between L and P?

Answer: NL and NC

<hl> Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC , and it is not known if they are distinct or equal classes. <hl>

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as <hl> NL and NC <hl>, and it is not known if they are distinct or equal classes.

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as <hl> NL and NC <hl> , and it is not known if they are distinct or equal classes.

question: What are two complexity classes between L and P?, context: Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.


Question: What is unknown about the complexity classes between L and P that further prevents determining the value relationship between L and P?

Answer: if they are distinct or equal classes

<hl> Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes . <hl>

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known <hl> if they are distinct or equal classes <hl>.

Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known <hl> if they are distinct or equal classes <hl> .

question: What is unknown about the complexity classes between L and P that further prevents determining the value relationship between L and P?, context: Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.


Question: Problems capable of theoretical solutions but consuming unreasonable time in practical application are known as what?

Answer: intractable problems

<hl> Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems . <hl> In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.

Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as <hl> intractable problems <hl>. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.

Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as <hl> intractable problems <hl> .

question: Problems capable of theoretical solutions but consuming unreasonable time in practical application are known as what?, context: Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.


Question: Intractable problems lacking polynomial time solutions necessarily negate the practical efficacy of what type of algorithm?

Answer: exponential-time algorithms

Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. <hl> To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. <hl> For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.

Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why <hl> exponential-time algorithms <hl> might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.

question: Intractable problems lacking polynomial time solutions necessarily negate the practical efficacy of what type of algorithm?, context: Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.


Question: If NP is not equal to P, viewed through this lens, what type of problems can also be considered intractable?

Answer: NP-complete problems

Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. <hl> If NP is not the same as P, then the NP-complete problems are also intractable in this sense. <hl> To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.

Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the <hl> NP-complete problems <hl> are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.

question: If NP is not equal to P, viewed through this lens, what type of problems can also be considered intractable?, context: Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.


Question: What eponymous variation of arithmetic presents a decision problem not evidenced in P?

Answer: Presburger arithmetic

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. <hl> For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. <hl> Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in <hl> Presburger arithmetic <hl> has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

question: What eponymous variation of arithmetic presents a decision problem not evidenced in P?, context: What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.


Question: Despite the Presburger problem, and in view of intractability, what has been done to establish solutions in reasonable periods of time?

Answer: algorithms have been written

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. <hl> For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. <hl> Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet <hl> algorithms have been written <hl> that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

question: Despite the Presburger problem, and in view of intractability, what has been done to establish solutions in reasonable periods of time?, context: What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.


Question: What is an example of a problem to which effective algorithms have provided a solution in spite of the intractability associated with the breadth of sizes?

Answer: NP-complete knapsack problem

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. <hl> Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem. <hl>

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the <hl> NP-complete knapsack problem <hl> over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

question: What is an example of a problem to which effective algorithms have provided a solution in spite of the intractability associated with the breadth of sizes?, context: What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.


Question: How quickly can an algorithm solve an NP-complete knapsack problem?

Answer: in less than quadratic time

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. <hl> Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem. <hl>

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes <hl> in less than quadratic time <hl> and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

question: How quickly can an algorithm solve an NP-complete knapsack problem?, context: What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.


Question: What is the example of another problem characterized by large instances that is routinely solved by SAT handlers employing efficient algorithms?

Answer: NP-complete Boolean satisfiability problem

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. <hl> Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem . <hl>

What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the <hl> NP-complete Boolean satisfiability problem <hl>.

Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the <hl> NP-complete Boolean satisfiability problem <hl> .

question: What is the example of another problem characterized by large instances that is routinely solved by SAT handlers employing efficient algorithms?, context: What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.


Question: What tactic did researchers employ to offset the former deficit of work surrounding the complexity of algorithmic problems?

Answer: foundations were laid out

<hl> Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. <hl> Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous <hl> foundations were laid out <hl> by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.

question: What tactic did researchers employ to offset the former deficit of work surrounding the complexity of algorithmic problems?, context: Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.


Question: Who was the most influential researcher among those grappling with the deficit of work surrounding the complexity posed by algorithmic problems?

Answer: Alan Turing

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. <hl> Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer. <hl>

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by <hl> Alan Turing <hl> in 1936, which turned out to be a very robust and flexible simplification of a computer.

question: Who was the most influential researcher among those grappling with the deficit of work surrounding the complexity posed by algorithmic problems?, context: Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.


Question: What theoretical device is attributed to Alan Turing?

Answer: Turing machines

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. <hl> Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer. <hl>

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of <hl> Turing machines <hl> by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.

question: What theoretical device is attributed to Alan Turing?, context: Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.


Question: In what year was the Alan Turing's definitional model of a computing device received?

Answer: 1936

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. <hl> Most influential among these was the definition of Turing machines by Alan Turing in 1936 , which turned out to be a very robust and flexible simplification of a computer. <hl>

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in <hl> 1936 <hl>, which turned out to be a very robust and flexible simplification of a computer.

Most influential among these was the definition of Turing machines by Alan Turing in <hl> 1936 <hl> , which turned out to be a very robust and flexible simplification of a computer.

question: In what year was the Alan Turing's definitional model of a computing device received?, context: Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.


Question: In the most basic sense what did a Turing machine emulate?

Answer: a computer

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. <hl> Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer . <hl>

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of <hl> a computer <hl>.

Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of <hl> a computer <hl> .

question: In the most basic sense what did a Turing machine emulate?, context: Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.


Question: What paper is commonly considered the bellwether ushering in systematic studies computational complexity?

Answer: On the Computational Complexity of Algorithms

<hl> As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper " On the Computational Complexity of Algorithms " by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. <hl> Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "<hl> On the Computational Complexity of Algorithms <hl>" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper " <hl> On the Computational Complexity of Algorithms <hl> " by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems.

question: What paper is commonly considered the bellwether ushering in systematic studies computational complexity?, context: As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.


Question: What individuals were responsible for authoring "On the Computational Complexity of Algorithms"?

Answer: Juris Hartmanis and Richard Stearns

<hl> As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. <hl> Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by <hl> Juris Hartmanis and Richard Stearns <hl> (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

question: What individuals were responsible for authoring "On the Computational Complexity of Algorithms"?, context: As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.


Question: In what year was Hatmanis and Stearn's seminal work in computational complexity received?

Answer: 1965

<hl> As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns ( 1965 ), which laid out the definitions of time and space complexity and proved the hierarchy theorems. <hl> Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (<hl> 1965 <hl>), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns ( <hl> 1965 <hl> ), which laid out the definitions of time and space complexity and proved the hierarchy theorems.

question: In what year was Hatmanis and Stearn's seminal work in computational complexity received?, context: As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.


Question: What complex measurements were defined by "On the Computational Complexity of Algorithms"?

Answer: time and space

<hl> As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. <hl> Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of <hl> time and space <hl> complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

question: What complex measurements were defined by "On the Computational Complexity of Algorithms"?, context: As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.


Question: In what year did Edmond's characterize a "good" algorithm?

Answer: 1965

<hl> As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns ( 1965 ), which laid out the definitions of time and space complexity and proved the hierarchy theorems. <hl> Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (<hl> 1965 <hl>), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.

As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns ( <hl> 1965 <hl> ), which laid out the definitions of time and space complexity and proved the hierarchy theorems.

question: In what year did Edmond's characterize a "good" algorithm?, context: As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.


Question: Who provided a definition of linear bounded automata in 1960?

Answer: John Myhill

<hl> Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill 's <hl> definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

Earlier papers studying problems solvable by Turing machines with specific bounded resources include <hl> John Myhill <hl>'s definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

Earlier papers studying problems solvable by Turing machines with specific bounded resources include <hl> John Myhill <hl> 's

question: Who provided a definition of linear bounded automata in 1960?, context: Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:


Question: In what year did Raymond Sullivan publish a study of rudimentary sets?

Answer: 1961

<hl> Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets ( 1961 ), as well as Hisao Yamada's paper on real-time computations (1962). <hl> Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (<hl> 1961 <hl>), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets ( <hl> 1961 <hl> ), as well as Hisao Yamada's paper on real-time computations (1962).

question: In what year did Raymond Sullivan publish a study of rudimentary sets?, context: Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:


Question: In 1962, who was responsible for the authorship of a paper published on real time-computations?

Answer: Hisao Yamada

<hl> Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada 's paper on real-time computations (1962). <hl> Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as <hl> Hisao Yamada <hl>'s paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as <hl> Hisao Yamada <hl> 's paper on real-time computations (1962).

question: In 1962, who was responsible for the authorship of a paper published on real time-computations?, context: Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:


Question: What is the concrete choice typically assumed by most complexity-theoretic theorems?

Answer: input encoding

<hl> Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding , one tries to keep the discussion abstract enough to be independent of the choice of encoding. <hl> This can be achieved by ensuring that different representations can be transformed into each other efficiently.

Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of <hl> input encoding <hl>, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.

Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of <hl> input encoding <hl> , one tries to keep the discussion abstract enough to be independent of the choice of encoding.

question: What is the concrete choice typically assumed by most complexity-theoretic theorems?, context: Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.


Question: In the effort of maintaining a level of abstraction, what choice is typically left independent?

Answer: encoding

<hl> Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding , one tries to keep the discussion abstract enough to be independent of the choice of encoding. <hl> This can be achieved by ensuring that different representations can be transformed into each other efficiently.

Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input <hl> encoding <hl>, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.

Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input <hl> encoding <hl> , one tries to keep the discussion abstract enough to be independent of the choice of encoding.

question: In the effort of maintaining a level of abstraction, what choice is typically left independent?, context: Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.


Question: Who is responsible for axiomatic complexity theory?

Answer: Manuel Blum

<hl> In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. <hl> The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

In 1967, <hl> Manuel Blum <hl> developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

question: Who is responsible for axiomatic complexity theory?, context: In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.


Question: What theorem was implicated by Manuel Blum's axioms?

Answer: speed-up theorem

<hl> In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem . <hl> The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, <hl> speed-up theorem <hl>. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, <hl> speed-up theorem <hl> .

question: What theorem was implicated by Manuel Blum's axioms?, context: In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.


Question: What is the paper written by Richard Karp in 1972 that ushered in a new era of understanding between intractability and NP-complete problems?

Answer: "Reducibility Among Combinatorial Problems"

In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. <hl> In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems" , in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete. <hl>

In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, <hl> "Reducibility Among Combinatorial Problems" <hl>, in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

In 1972, Richard Karp took this idea a leap forward with his landmark paper, <hl> "Reducibility Among Combinatorial Problems" <hl> , in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

question: What is the paper written by Richard Karp in 1972 that ushered in a new era of understanding between intractability and NP-complete problems?, context: In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.


Question: How many combinatory and graph theoretical problems, formerly believed to be plagued by intractability, did Karp's paper address?

Answer: 21

In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. <hl> In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete. <hl>

In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that <hl> 21 <hl> diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

question: How many combinatory and graph theoretical problems, formerly believed to be plagued by intractability, did Karp's paper address?, context: In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.


Question: What is a course of study called?

Answer: the curriculum.

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. <hl> Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum. <hl>

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called <hl> the curriculum. <hl>

question: What is a course of study called?, context: The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.


Question: What is another name to describe the science of teaching?

Answer: pedagogy

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. <hl> These professional qualifications may include the study of pedagogy , the science of teaching. <hl> Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of <hl> pedagogy <hl>, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.

These professional qualifications may include the study of <hl> pedagogy <hl> , the science of teaching.

question: What is another name to describe the science of teaching?, context: The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.


Question: Where do most teachers get their credentials from?

Answer: university or college.

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. <hl> In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. <hl> Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a <hl> university or college. <hl> These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.

question: Where do most teachers get their credentials from?, context: The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.


Question: What can a teacher use to help students learn?

Answer: lesson plan

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. <hl> Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum. <hl>

The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a <hl> lesson plan <hl> to facilitate student learning, providing a course of study which is called the curriculum.

question: What can a teacher use to help students learn?, context: The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.