## Reformulating the Situation Calculus and the Event Calculus in the General Theory of Stable Models and in Answer Set Programming

Citations: | 7 - 5 self |

### BibTeX

@MISC{Lee_reformulatingthe,

author = {Joohyung Lee and Ravi Palla},

title = {Reformulating the Situation Calculus and the Event Calculus in the General Theory of Stable Models and in Answer Set Programming},

year = {}

}

### OpenURL

### Abstract

Circumscription and logic programs under the stable model semantics are two wellknown nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis of a family of action languages, such as language A and several of its descendants. Based on the discovery that circumscription and the stable model semantics coincide on a class of canonical formulas, we reformulate the situation calculus and the event calculus in the general theory of stable models. We also present a translation that turns the reformulations further into answer set programs, so that efficient answer set solvers can be applied to compute the situation calculus and the event calculus. 1.

### Citations

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(Show Context)
Citation Context ...answer set solvers can be applied to compute the situation calculus and the event calculus. 1. Introduction Circumscription (McCarthy, 1980, 1986) and logic programs under the stable model semantics (=-=Gelfond & Lifschitz, 1988-=-) are two well-known nonmonotonic formalisms. As one of the oldest nonmonotonic formalisms, circumscription has found many applications in commonsense reasoning and model-based diagnoses (e.g., McCart... |

1490 | Some philosophical problems from the standpoint of arti cial intelligence
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Citation Context ...tions in knowledge representation research. In particular, in the area of temporal reasoning, the former has served as a basis of classical logic based action calculi, such as the situation calculus (=-=McCarthy & Hayes, 1969-=-; Reiter, 2001), the event calculus (Shanahan, 1995) and temporal action logics (Doherty, c○2012 AI Access Foundation. All rights reserved.Lee & Palla Gustafsson, Karlsson, & Kvarnström, 1998), where... |

1423 |
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Citation Context ...erms of translation into classical (second-order) logic, stable models proposed by Gelfond and Lifschitz (1988) are defined in terms of grounding and fixpoints in the style of Reiter’s default logic (=-=Reiter, 1980-=-). These differences in part account for the fact that the two formalisms have formed rather disparate traditions in knowledge representation research. In particular, in the area of temporal reasoning... |

815 |
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Citation Context ...rns the reformulations further into answer set programs, so that efficient answer set solvers can be applied to compute the situation calculus and the event calculus. 1. Introduction Circumscription (=-=McCarthy, 1980-=-, 1986) and logic programs under the stable model semantics (Gelfond & Lifschitz, 1988) are two well-known nonmonotonic formalisms. As one of the oldest nonmonotonic formalisms, circumscription has fo... |

618 |
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Citation Context ... classical logic and logic program traditions in temporal reasoning. Interestingly, the development of the event calculus has spanned over both traditions. The original version of the event calculus (=-=Kowalski & Sergot, 1986-=-) was formulated in logic programs, but not under the stable model semantics (that was the time before the invention of the stable model semantics). More extensive developments have been later carried... |

550 | The frame problem in the situation calculus: a simple solution (sometimes) and a completeness result for goal regression - Reiter - 1991 |

485 | Applications of circumscription to formalizing common sense knowledge
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Citation Context ..., 1988) are two well-known nonmonotonic formalisms. As one of the oldest nonmonotonic formalisms, circumscription has found many applications in commonsense reasoning and model-based diagnoses (e.g., =-=McCarthy, 1986-=-; Shanahan, 1995; Besnard & Cordier, 1994). The stable model semantics is the mathematical basis of Answer Set Programming (ASP) (Marek & Truszczyński, 1999; Niemelä, 1999; Lifschitz, 2008), which is ... |

431 | Planning as satisfiability - Kautz, Selman - 1992 |

414 |
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Citation Context ...sentation research. In particular, in the area of temporal reasoning, the former has served as a basis of classical logic based action calculi, such as the situation calculus (McCarthy & Hayes, 1969; =-=Reiter, 2001-=-), the event calculus (Shanahan, 1995) and temporal action logics (Doherty, c○2012 AI Access Foundation. All rights reserved.Lee & Palla Gustafsson, Karlsson, & Kvarnström, 1998), whereas the latter ... |

327 | Logic programs with stable model semantics as a constraint programming paradigm
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Citation Context ...odel-based diagnoses (e.g., McCarthy, 1986; Shanahan, 1995; Besnard & Cordier, 1994). The stable model semantics is the mathematical basis of Answer Set Programming (ASP) (Marek & Truszczyński, 1999; =-=Niemelä, 1999-=-; Lifschitz, 2008), which is being widely applied thanks to the availability of several efficient implementations, known as answer set solvers. While the two nonmonotonic formalisms have been applied ... |

248 | Stable models and an alternative logic programming paradigm, in: The Logic Programming Paradigm: a 25-Year Perspective
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Citation Context ... commonsense reasoning and model-based diagnoses (e.g., McCarthy, 1986; Shanahan, 1995; Besnard & Cordier, 1994). The stable model semantics is the mathematical basis of Answer Set Programming (ASP) (=-=Marek & Truszczyński, 1999-=-; Niemelä, 1999; Lifschitz, 2008), which is being widely applied thanks to the availability of several efficient implementations, known as answer set solvers. While the two nonmonotonic formalisms hav... |

210 | Nonmonotonic causal theories
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Citation Context ...here ΠA(x, s) is the disjunction of πA(x, s) for all axioms (29) in Dprecond. 11. The axioms Dinertia and Dexogenous0 are also closely related to the translation of C+ into nonmonotonic causal logic (=-=Giunchiglia, Lee, Lifschitz, McCain, & Turner, 2004-=-). 584Reformulating the Situation Calculus and the Event Calculus Theorem 4 Let T be a theory (26) of signature σ Holds , and I a coherent interpretation of σ Holds that is complete on Holds. If I sa... |

195 | Strongly equivalent logic programs
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Citation Context ...while preserving satisfiability, such transformations do not necessarily preserve stable models. This is due to the fact that the notion of equivalence is “stronger” under the stable model semantics (=-=Lifschitz, Pearce, & Valverde, 2001-=-). Definition 4 (Ferraris et al., 2011) A formula F is strongly equivalent to formula G if, for any formula H containing F as a subformula (and possibly containing object, function and predicate const... |

184 | Action languages
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- 1998
(Show Context)
Citation Context ... c○2012 AI Access Foundation. All rights reserved.Lee & Palla Gustafsson, Karlsson, & Kvarnström, 1998), whereas the latter has served as a basis of a family of action languages, such as language A (=-=Gelfond & Lifschitz, 1998-=-) and several of its descendants which can be translated into logic programs under the stable model semantics. However, a recent generalization of the stable model semantics shed new light on the rela... |

180 | Solving the Frame Problem: A Mathematical Investigation of the Common Sense Law of Inertia - Shanahan - 1997 |

178 | Embracing causality in specifying the indirect effects of actions - Lin - 1995 |

143 | Consistency of Clark’s completion and existence of stable models
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- 1994
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Citation Context ...mportant in understanding the relationship between them. It is known that, for tight logic programs and tight first-order formulas, the stable model semantics coincides with the completion semantics (=-=Fages, 1994-=-; Erdem & Lifschitz, 2003; Ferraris et al., 2011). This fact helps us understand the relationship between the two semantics, and led to the design of the answer set solver cmodels-1 23 that computes a... |

129 | The event calculus explained
- Shanahan
- 1999
(Show Context)
Citation Context ...the invention of the stable model semantics). More extensive developments have been later carried out on the classical logic foundation via circumscription (e.g., Shanahan, 1995, 1997, 1999; Miller & =-=Shanahan, 1999-=-; Mueller, 2004), but the relation to 572Reformulating the Situation Calculus and the Event Calculus the logic program formulation remained implicit. Based on the reduction of circumscription to comp... |

112 | Nested expressions in logic programs
- Lifschitz, Tang, et al.
- 1999
(Show Context)
Citation Context ...ect constants (and no other function constants), if we replace ∃x(p(x) ∧ q(x)) with multiple disjunctions and then turn the resulting program with nested expressions into a usual disjunctive program (=-=Lifschitz, Tang, & Turner, 1999-=-), 2n rules are generated. For instance, if n = 3, the 12. Pearce and Valverde (2005) show that a sentence in QN c 5, the monotonic basis of Quantified Equilibrium Logic, can be turned into prenex for... |

69 |
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- 1930
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Citation Context ...& Shoham, 1992). These definitions are also equivalent to the definition of Quantified Equilibrium Logic given by Pearce and Valverde (2005), which is defined in terms of the logic of Here-and-There (=-=Heyting, 1930-=-). The new definition of a stable model motivates us to investigate the relationship between stable model reasoning and minimal model reasoning. In particular, we focus on their relationship in the ar... |

69 |
Commonsense Reasoning
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- 2006
(Show Context)
Citation Context ...ate constraints, event occurrence constraints, and the set of domainindependent axioms in the event calculus, such as EC (for the continuous event calculus) and DEC (for the discrete event calculus) (=-=Mueller, 2006-=-, Chapter 2). It also 579Lee & Palla includes the following definitions of the predicates used in the causal constraints in ∆: Started(f, t) def ↔ (HoldsAt(f, t) ∨ ∃e(Happens(e, t) ∧ Initiates(e, f, ... |

68 | A Circumscriptive Calculus of Events
- Shanahan
- 1995
(Show Context)
Citation Context ...well-known nonmonotonic formalisms. As one of the oldest nonmonotonic formalisms, circumscription has found many applications in commonsense reasoning and model-based diagnoses (e.g., McCarthy, 1986; =-=Shanahan, 1995-=-; Besnard & Cordier, 1994). The stable model semantics is the mathematical basis of Answer Set Programming (ASP) (Marek & Truszczyński, 1999; Niemelä, 1999; Lifschitz, 2008), which is being widely app... |

66 | A New Perspective on Stable Models
- Ferraris, Lee, et al.
- 2007
(Show Context)
Citation Context ..., r)) ∧ u(a) ∧ v(b) ∧∀x((u(x) ∧ (¬v(x) ∧ ¬q(x)) → w(x)) ∧ (p(x) ∧ ¬q(x) → r(x)))), which is equivalent to the first-order sentence ∀x(p(x) ↔ x = a) ∧ ∀x(q(x) ↔ x = b) ∧ ∀x(r(x) ↔ (p(x) ∧ ¬q(x))) (2) (=-=Ferraris et al., 2007-=-, Example 3). The stable models of F are any first-order models of (2). The only answer set of F is the Herbrand model {p(a), q(b), r(a)}. According to Ferraris et al. (2011), this definition of an an... |

50 | Stable models and circumscription
- Ferraris, Lee, et al.
- 2011
(Show Context)
Citation Context ...s do not necessarily preserve stable models. This is due to the fact that the notion of equivalence is “stronger” under the stable model semantics (Lifschitz, Pearce, & Valverde, 2001). Definition 4 (=-=Ferraris et al., 2011-=-) A formula F is strongly equivalent to formula G if, for any formula H containing F as a subformula (and possibly containing object, function and predicate constants that do not occur in F , G), and ... |

49 | GrinGo: A New Grounder for Answer Set Programming
- Gebser, Schaub, et al.
(Show Context)
Citation Context ..., do(am−1, do(. . . , do(a1, s0))))) for all possible sequences of actions a1, . . . , am for m = 0, . . . , maxdepth. Though this program does not satisfy syntactic conditions, such as λ-restricted (=-=Gebser, Schaub, & Thiele, 2007-=-), ω-restricted (Syrjänen, 2004), or finite domain programs (Calimeri, Cozza, Ianni, & Leone, 2008), that answer set solvers usually impose in order to ensure finite grounding, the rules can still be ... |

48 | Representing transition systems by logic programs
- Lifschitz, Turner
- 1999
(Show Context)
Citation Context ...is the conjunction of for all fluent names R. Holds(R(x), S0)∨ ∼Holds(R(x), S0) Note that axioms in Dinertia are typically used in answer set programming to represent the common sense law of inertia (=-=Lifschitz & Turner, 1999-=-). Similarly, Dexogenous0 is used to represent that the initial value of a fluent is arbitrary. 11 We will show how this ASP-style BAT is related to Reiter’s BAT. First, since we use strong negation, ... |

45 | TAL) temporal action logics: Language specification and tutorial
- Doherty, Gustafsson, et al.
- 1998
(Show Context)
Citation Context ...r has served as a basis of classical logic based action calculi, such as the situation calculus (McCarthy & Hayes, 1969; Reiter, 2001), the event calculus (Shanahan, 1995) and temporal action logics (=-=Doherty, Gustafsson, Karlsson, & Kvarnström, 1998-=-), whereas the latter has served as a basis of a 1family of action languages, such as language A (Gelfond & Lifschitz, 1998) and several of its descendants which can be translated into logic programs... |

45 | The event calculus in classical logic—Alternative axiomatisations
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- 1999
(Show Context)
Citation Context ...e before the invention of the stable model semantics). More extensive developments have been later carried out on the classical logic foundation via circumscription (e.g., Shanahan, 1995, 1997, 1999; =-=Miller & Shanahan, 1999-=-; Mueller, 2004), but the relation to 572Reformulating the Situation Calculus and the Event Calculus the logic program formulation remained implicit. Based on the reduction of circumscription to comp... |

41 | What is answer set programming
- Lifschitz
- 2008
(Show Context)
Citation Context ...noses (e.g., McCarthy, 1986; Shanahan, 1995; Besnard & Cordier, 1994). The stable model semantics is the mathematical basis of Answer Set Programming (ASP) (Marek & Truszczyński, 1999; Niemelä, 1999; =-=Lifschitz, 2008-=-), which is being widely applied thanks to the availability of several efficient implementations, known as answer set solvers. While the two nonmonotonic formalisms have been applied to overlapping cl... |

35 |
V.: Tight logic programs. Theory and Practice of Logic Programming 3
- Erdem, Lifschitz
- 2003
(Show Context)
Citation Context ...nderstanding the relationship between them. It is known that, for tight logic programs and tight first-order formulas, the stable model semantics coincides with the completion semantics (Fages, 1994; =-=Erdem & Lifschitz, 2003-=-; Ferraris et al., 2011). This fact helps us understand the relationship between the two semantics, and led to the design of the answer set solver cmodels-1 23 that computes answer sets using completi... |

28 | A logic of non-monotone inductive definitions
- Denecker, Ternovska
(Show Context)
Citation Context ...re is based on completion and as such cannot handle recursive axioms unlike our approach. Denecker and Ternovska (2007) present an inductive variant of the situation calculus represented in ID-logic (=-=Denecker & Ternovska, 2008-=-)—classical logic extended with inductive definitions. ID-logic and the first-order stable model semantics appear to be closely related, but the precise relationship between them has yet to be shown. ... |

27 | Representing the zoo world and the traffic world in the language of the causal calculator - Akman, Erdogan, et al. |

27 | Inductive situation calculus - Denecker, Ternovska - 2007 |

27 | Event calculus reasoning through satisfiability
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Citation Context ... the stable model semantics). More extensive developments have been later carried out on the classical logic foundation via circumscription (e.g., Shanahan, 1995, 1997, 1999; Miller & Shanahan, 1999; =-=Mueller, 2004-=-), but the relation to 572Reformulating the Situation Calculus and the Event Calculus the logic program formulation remained implicit. Based on the reduction of circumscription to completion, SAT-bas... |

26 |
A logic of knowledge and justified assumptions
- Lin, Shoham
- 1992
(Show Context)
Citation Context ...first-order level without limiting attention to Herbrand models. Essentially the same characterization was independently given by Lin and Zhou (2011), via logic of knowledge and justified assumption (=-=Lin & Shoham, 1992-=-). These definitions are also equivalent to the definition of Quantified Equilibrium Logic given by Pearce and Valverde (2005), which is defined in terms of the logic of Here-and-There (Heyting, 1930)... |

25 | R.: A reductive semantics for counting and choice in answer set programming - Lee, Lifschitz, et al. - 2008 |

22 |
Propositional theories are strongly equivalent to logic programs. Theory and Practice of Logic Programming
- Cabalar, Ferraris
(Show Context)
Citation Context ...odels. For instance, consider p and ¬¬p. When p is intensional, the former has stable models and the latter does not. It is known that every propositional formula can be rewritten as a logic program (=-=Cabalar & Ferraris, 2007-=-; Cabalar, Pearce, & Valverde, 2005; Lee & Palla, 2007), and such translations can be extended to quantifier-free formulas in a straightforward way (Section 5.1). However, the method does not work in ... |

21 |
Computable Functions in ASP: Theory and Implementation
- Calimeri, Cozza, et al.
- 2008
(Show Context)
Citation Context ...or m = 0, . . . , maxdepth. Though this program does not satisfy syntactic conditions, such as λ-restricted (Gebser, Schaub, & Thiele, 2007), ω-restricted (Syrjänen, 2004), or finite domain programs (=-=Calimeri, Cozza, Ianni, & Leone, 2008-=-), that answer set solvers usually impose in order to ensure finite grounding, the rules can still be finitely grounded 596Reformulating the Situation Calculus and the Event Calculus % File: suitcase... |

21 | Symmetric splitting in the general theory of stable models
- Ferraris, Lee, et al.
- 2009
(Show Context)
Citation Context ... ∀yG(y, x) is not strongly equivalent to ¬∃y¬G(y, x), and in general such a classically equivalent transformation may not necessarily preserve stable models. However, the Theorem on Double Negations (=-=Ferraris et al., 2009-=-, also reviewed in Appendix C) tells us that such a transformation is ensured to preserve p-stable models if the replaced occurrence is p-negated in the given formula. Now we are ready to present our ... |

21 | Loop formulas for circumscription - Lee, Lin |

19 | Circumscriptive event calculus as answer set programming - Kim, Lee, et al. - 2009 |

17 | Compiling causal theories to successor state axioms and STRIPS-like systems - Lin |

17 | Event calculus planning through satisfiability - Shanahan, Witkowski |

16 | Reconciling the event calculus with the situation calculus
- Kowalski, Sadri
- 1997
(Show Context)
Citation Context ...the event calculus are widely studied action formalisms, and there are several papers that compare and relate them (e.g., Belleghem, Denecker, & Schreye, 1995; Provetti, 1996; Belleghem et al., 1997; =-=Kowalski & Sadri, 1997-=-). Prolog provides a natural implementation for basic action theories since definitional axioms can be represented by Prolog rules according to the Clark’s theorem (Reiter, 2001, Chapter 5). The Lloyd... |

15 | Hypothetical reasoning about actions: From situation calculus to event calculus
- Provetti
- 1996
(Show Context)
Citation Context ... structures. The situation calculus and the event calculus are widely studied action formalisms, and there are several papers that compare and relate them (e.g., Belleghem, Denecker, & Schreye, 1995; =-=Provetti, 1996-=-; Belleghem et al., 1997; Kowalski & Sadri, 1997). Prolog provides a natural implementation for basic action theories since definitional axioms can be represented by Prolog rules according to the Clar... |

13 | Capturing parallel circumscription with disjunctive logic programs - Janhunen, Oikarinen - 2004 |

13 | From answer set logic programming to circumscription via logic of GK - Lin, Zhou |

12 | A.: Reducing propositional theories in equilibrium logic to logic programs
- Cabalar, Pearce, et al.
- 2005
(Show Context)
Citation Context ...der p and ¬¬p. When p is intensional, the former has stable models and the latter does not. It is known that every propositional formula can be rewritten as a logic program (Cabalar & Ferraris, 2007; =-=Cabalar, Pearce, & Valverde, 2005-=-; Lee & Palla, 2007), and such translations can be extended to quantifier-free formulas in a straightforward way (Section 5.1). However, the method does not work in the presence of arbitrary quantifie... |

12 | A First Order Nonmonotonic Extension of Constructive Logic - Pearce, Valverde - 2005 |

12 |
Cardinality constraint programs
- Syrjänen
- 2004
(Show Context)
Citation Context ...ssible sequences of actions a1, . . . , am for m = 0, . . . , maxdepth. Though this program does not satisfy syntactic conditions, such as λ-restricted (Gebser, Schaub, & Thiele, 2007), ω-restricted (=-=Syrjänen, 2004-=-), or finite domain programs (Calimeri, Cozza, Ianni, & Leone, 2008), that answer set solvers usually impose in order to ensure finite grounding, the rules can still be finitely grounded 596Reformula... |