Situation calculus
The situation calculus is a logic formalism designed for representing and reasoning about dynamical domains. It was first introduced by John McCarthy in 1963. The main version of the situational calculus that is presented in this article is based on that introduced by Ray Reiter in 1991. It is followed by sections about McCarthy's 1986 version and a logic programming formulation. Overview. The situation calculus represents changing scenarios as a set of first-order logic formulae. The basic elements of the calculus are: A domain is formalized by a number of formulae, namely: A simple robot world  will be modeled as a running example. In this world there is a single robot and several inanimate objects. The world is laid out according to a grid so that locations can be specified in terms of formula_1 coordinate points. It is possible for the robot to move around the world, and to pick up and drop items. Some items may be too heavy for the robot to pick up, or fragile so that they break when they are dropped. The robot also has the ability to repair any broken items that it is holding. Elements. The main elements of  the situation calculus are the actions, fluents and the situations. A number of objects are also typically involved in the description of the world. The situation calculus is based on a sorted domain with three sorts: actions, situations, and objects, where the objects include everything that is not an action or a situation. Variables of each sort can be used. While actions, situations, and objects are elements of the domain, the fluents are modeled as either predicates or functions. Actions. The actions form a sort of the domain. Variables of sort action can be used. Actions can be quantified. In  the example robot world, possible action terms would be formula_2 to model the robot moving to a new location formula_1, and formula_4 to model the robot picking up an object . A special predicate is used to indicate when an action is executable. Situations. In the situation calculus, a dynamic world is modeled as progressing through a series of situations as a result of various actions being performed within the world. A situation represents a history of action occurrences. In the Reiter version of the situation calculus described here, a situation does not represent a state, contrarily to the literal meaning  of the term and contrarily to the original definition by McCarthy and Hayes. This point has been summarized by Reiter as follows: The situation before any actions have been performed is typically denoted and called the initial situation. The new situation resulting from the performance of an action is denoted using the function symbol (Some other references also use ). This function symbol has a situation and an action as arguments, and a situation as a result, the latter being the situation that results from performing the given action in the given situation. The fact that situations are sequences of actions  and not states is enforced by an axiom stating that formula_5 is equal to formula_6 if and only if formula_7 and formula_8. This condition makes no sense if situations were states, as two different actions executed in two different states can result in the same state. In the example robot world, if the robot's first action is to move to location formula_9, the first action is formula_10 and the resulting situation is formula_11. If its next action is to pick up the ball, the resulting situation is formula_12. Situations terms like formula_11 and formula_12 denote the sequences of executed  actions, and not the description of the state that result from execution. Fluents. Statements whose truth value may change are modeled by "relational fluents", predicates which take a situation as their final argument. Also possible are "functional fluents", functions which take a situation as their final argument and return a situation-dependent value. Fluents may be thought of as "properties of the world"'. In the example, the fluent formula_15 can be used to indicate that the robot is carrying a particular object in a particular situation. If the robot initially carries nothing, formula_16 is false while formula_17 is true. The  location of the robot can be modeled using a functional fluent formula_18 which returns the location formula_1 of the robot in a particular situation. Formulae. The description of a dynamic world is encoded in second order logics using three kinds of formulae: formulae about actions (preconditions and effects), formulae about the state of the world, and foundational axioms. Action preconditions. Some actions may not be executable in a given situation. For example, it is impossible to put down an object unless one is in fact carrying it. The restrictions on the performance of actions are modeled by literals of the  form formula_20, where is an action, a situation, and is a special binary predicate denoting executability of actions. In the example, the condition that dropping an object is only possible when one is carrying it is modeled by: As a more complex example, the following models that the robot can carry only one object at a time, and that some objects are too heavy for the robot to lift (indicated by the predicate ): Action effects. Given that an action is possible in a situation, one must specify the effects of that action on the fluents. This is done by the  effect axioms. For example, the fact that picking up an object causes the robot to be carrying it can be modeled as: It is also possible to specify conditional effects, which are effects that depend on the current state. The following models that some objects are fragile (indicated by the predicate ) and dropping them causes them to be broken (indicated by the fluent ): While this formula correctly describes the effect of the actions, it is not sufficient to correctly describe the action in logic, because of the frame problem. The frame problem. While the above formulae seem suitable for reasoning  about the effects of actions, they have a critical weakness - they cannot be used to derive the "non-effects" of actions. For example, it is not possible to deduce that after picking up an object, the robot's location remains unchanged. This requires a so-called frame axiom, a formula like: The need to specify frame axioms has long been recognised as a problem in axiomatizing dynamic worlds, and is known as the frame problem. As there are generally a very large number of such axioms, it is very easy for the designer to leave out a necessary frame  axiom, or to forget to modify all appropriate axioms when a change to the world description is made. The successor state axioms. The successor state axioms "solve" the frame problem in the situation calculus. According to this solution, the designer must enumerate as effect axioms all the ways in which the value of a particular fluent can be changed. The effect axioms affecting the value of fluent formula_26 can be written in generalised form as a positive and a negative effect axiom: The formula formula_29 describes the conditions under which action in situation makes the fluent become true in the  successor situation formula_5. Likewise, formula_31 describes the conditions under which performing action in situation makes fluent false in the successor situation. If this pair of axioms describe all the ways in which fluent can change value, they can be rewritten as a single axiom: In words, this formula states: "given that it is possible to perform action in situation , the fluent would be true in the resulting situation formula_5 if and only if performing in would make it true, or it is true in situation and performing in would not make it false." By way of example, the value of  the fluent introduced above is given by the following successor state axiom: States. The properties of the initial or any other situation can be specified by simply stating them as formulae. For example, a fact about the initial state is formalized by making assertions about formula_35 (which is not a state, but a "situation"). The following statements model that initially, the robot carries nothing, is at location formula_36, and there are no broken objects: Foundational axioms. The foundational axioms of the situation calculus formalize the idea that situations are histories by having formula_40. They also include other properties such as  the second order induction on situations. Regression. Regression is a mechanism for proving consequences in the situation calculus. It is based on expressing a formula containing the situation formula_5 in terms of a formula containing the action and the situation , but not the situation formula_5. By iterating this procedure, one can end up with an equivalent formula containing only the initial situation . Proving consequences is supposedly simpler from this formula than from the original one. GOLOG. GOLOG is a logic programming language based on the situation calculus. The original version of the situation calculus. The main difference between the original  situation calculus by McCarthy and Hayes and the one in use today is the interpretation of situations. In the modern version of the situational calculus, a situation is a sequence of actions. Originally, situations were defined as "the complete state of the universe at an instant of time". It was clear from the beginning that such situations could not be completely described; the idea was simply to give some statements about situations, and derive consequences from them. This is also different from the approach that is taken by the fluent calculus, where a state can be a collection of known  facts, that is, a possibly "incomplete" description of the universe. In the original version of the situation calculus, fluents are not reified. In other words, conditions that can change are represented by predicates and not by functions. Actually, McCarthy and Hayes defined a fluent as a function that depends on the situation, but they then proceeded always using predicates to represent fluents. For example, the fact that it is raining at place in the situation is represented by the literal formula_43. In the 1986 version of the situation calculus by McCarthy, functional fluents are used. For example, the position of  an object in the situation is represented by the value of formula_44, where is a function. Statements about such functions can be given using equality: formula_45 means that the location of the object is the same in the two situations and formula_46. The execution of actions is represented by the function : the execution of the action in the situation is the situation formula_47. The effects of actions are expressed by formulae relating fluents in situation and fluents in situations formula_47. For example, that the action of opening the door results in the door being open if not locked is represented  by: The predicates and represent the conditions of a door being locked and open, respectively. Since these condition may vary, they are represented by predicates with a situation argument. The formula says that if the door is not locked in a situation, then the door is open after executing the action of opening, this action being represented by the constant . These formulae are not sufficient to derive everything that is considered plausible. Indeed, fluents at different situations are only related if they are preconditions and effects of actions; if a fluent is not affected by an action, there is no  way to deduce it did not change. For example, the formula above does not imply that formula_50 follows from formula_51, which is what one would expect (the door is not made locked by opening it). In order for inertia to hold, formulae called "frame axioms" are needed. These formulae specify all non-effects of actions: In the original formulation of the situation calculus, the initial situation, later denoted by , is not explicitly identified. The initial situation is not needed if situations are taken to be descriptions of the world. For example, to represent the scenario in which the door was  closed but not locked and the action of opening it is performed is formalized by taking a constant to mean the initial situation and making statements about it (e.g., formula_51). That the door is open after the change is reflected by formula formula_54 being entailed. The initial situation is instead necessary if, like in the modern situation calculus, a situation is taken to be a history of actions, as the initial situation represents the empty sequence of actions. The version of the situation calculus introduced by McCarthy in 1986 differs to the original one for the use of functional fluents  (e.g., formula_44 is a term representing the position of in the situation ) and for an attempt to use circumscription to replace the frame axioms. The situation calculus as a logic program. It is also possible (e.g. Kowalski 1979, Apt and Bezem 1990, Shanahan 1997) to write the situation calculus as a logic program: Here is a meta-predicate and the variable ranges over fluents. The predicates , and correspond to the predicates , formula_58, and formula_59 respectively. The left arrow ← is half of the equivalence ↔. The other half is implicit in the completion of the program, in which negation is  interpreted as negation as failure. Induction axioms are also implicit, and are needed only to prove program properties. Backward reasoning as in SLD resolution, which is the usual mechanism used to execute logic programs, implements regression implicitly.