Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Answer Set Programming Modulo Theories and Reasoning about Continuous Changes Joohyung Lee and Yunsong Meng School of Computing, Informatics and Decision Systems Engineering Arizona State University, Tempe, USA fjoolee, [email protected] Abstract grounding rule (1) yields a large set of ground instances when Speed ranges over a large integer domain. Moreover, a real Answer Set Programming Modulo Theories is a number domain is not supported because grounding cannot new framework of tight integration of answer set be applied. programming (ASP) and satisfiability modulo the- In order to alleviate this “grounding problem,” there have ories (SMT). Similar to the relationship between been several recent efforts to integrate ASP with constraint first-order logic and SMT, it is based on a recent solving and satisfiability modulo theories (SMT), where func- proposal of the functional stable model semantics tional fluents can be expressed by variables in Constraint Sat- by fixing interpretations of background theories. isfaction Problems or uninterpreted constants in SMT. Bal- Analogously to a known relationship between ASP duccini [2009] and Gebser et al. [2009] combine ASP and and SAT, “tight” ASPMT programs can be trans- constraint solving in a way that is similar to the “lazy ap- lated into SMT instances. We demonstrate the use- proach” of SMT solvers. Niemela¨ [2008] shows that ASP fulness of ASPMT by enhancing action language programs can be translated into a particular SMT fragment, C+ to handle continuous changes as well as discrete namely difference logic. Janhunen et al. [2011] presents changes. We reformulate the semantics of C+ in a tight integration of ASP and SMT. All these approaches terms of ASPMT, and show that SMT solvers can aim at addressing the issue (i) above by avoiding grounding be used to compute the language. We also show and exploiting efficient constraint solving methods applied to how the language can represent cumulative effects functions. However, they do not address the issue (ii) be- on continuous resources. cause, like the standard ASP, nonmonotonicity of those ex- tensions has to do with predicates only. For instance, a natu- 1 Introduction ral counterpart of (1) in the language of CLINGCON [Gebser ] The success of answer set programming (ASP) [Lifschitz, et al., 2009 , 2008; Brewka et al., 2011] is in part thanks to efficiency of Speed =$ x Speed =$ x; not Speed 6=$ x ; ASP solvers that utilize (i) intelligent grounding—the process 1 0 1 that replaces variables with ground terms—and (ii) efficient does not correctly represent inertia. search methods that originated from propositional satisfiabil- In this paper, we present a framework of combining answer ity solvers (SAT solvers). However, this method is not scal- set programming with satisfiability modulo theories, which able when we need to represent functional fluents that range we call Answer Set Programming Modulo Theories (ASPMT). over large numeric domains. Since answer sets (a.k.a. stable Just as SMT is a generalization of SAT and, at the same time, models) are limited to Herbrand models, functional fluents a special case of first-order logic in which certain predicate are represented by predicates, but not by functions, as in the and function constants in background theories (such as dif- following example that represents the inertia assumption on ference logic and the theory of reals) have fixed interpreta- Speed: tions, ASPMT is a generalization of the standard ASP and, at the same time, a special case of the functional stable model Speed (x) Speed (x); not :Speed (x) : (1) 1 0 1 semantics [Bartholomew and Lee, 2012] that assumes back- Here the subscripts 0 and 1 are time stamps, and x is a ground theories. Like the known relationship between SAT variable ranging over the value domain of fluent Speed.A and ASP that tight ASP programs can be translated into SAT more natural representation using functions which replaces instances, tight ASPMT programs can be translated into SMT Speedi(x) with Speedi = x does not work under the ASP se- instances, which allows SMT solvers for computing ASPMT mantics because (i) answer sets are Herbrand models (e.g., programs. Speed1 = Speed0 is always false under any Herbrand inter- These results allow us to enhance action language C+ pretation), and (ii) nonmonotonicity of the stable model se- [Giunchiglia et al., 2004] to handle reasoning about con- mantics has to do with minimizing the extents of predicates tinuous changes. Language C+ is an expressive action de- but has nothing to do with functions. On the other hand, scription language but its semantics was defined in terms of 990 propositional causal theories, which limits the language to ex- This definition of a stable model is a proper generalization press discrete changes only. By reformulating C+ in terms of the one from [Ferraris et al., 2011]: in the absence of in- of ASPMT, we naturally extend the language to overcome tensional function constants, it reduces to the one in [Ferraris the limitation, and use SMT solvers to compute the language. et al., 2011]. Our experiments show that this approach outperforms the ex- isting implementations of C+ by several orders of magnitude 2.2 Answer Set Programming Modulo Theories for some benchmark problems. Formally, an SMT instance is a formula in many-sorted first- Section 2 reviews the functional stable model semantics order logic, where some designated function and predicate by Bartholomew and Lee [2012], and defines ASPMT as its constants are constrained by some fixed background interpre- special case. Section 3 reformulates language C+ in terms tation. SMT is the problem of determining whether such a of ASPMT, and Section 4 shows how the reformulation can formula has a model that expands the background interpreta- be used to represent continuous changes. The language is tion [Barrett et al., 2009]. further extended in Section 5 to represent cumulative effects The syntax of ASPMT is the same as that of SMT. Let σbg on continuous resources. Related work and the conclusion be the (many-sorted) signature of the background theory bg. are presented in Section 6. An interpretation of σbg is called a background interpreta- tion if it satisfies the background theory. For instance, in the 2 Functional SM and ASPMT theory of reals, we assume that σbg contains the set R of sym- 2.1 Review: Functional Stable Model Semantics bols for all real numbers, the set of arithmetic functions over real numbers, and the set f<; >; ≤; ≥} of binary predicates We review the semantics from [Bartholomew and Lee, 2012]. over real numbers. Background interpretations interpret these Formulas are built the same as in first-order logic. A signa- symbols in the standard way. ture consists of function constants and predicate constants. Let σ be a signature that is disjoint from σbg. We say that Function constants of arity 0 are called object constants, and an interpretation I of σ satisfies F w.r.t. the background the- predicate constants of arity 0 are called propositional con- ory bg, denoted by I j=bg F , if there is a background inter- stants. bg Similar to circumscription, for predicate symbols (con- pretation J of σ that has the same universe as I, and I [ J stants or variables) u and c, expression u ≤ c is defined as satisfies F . For any ASPMT sentence F with background theory σbg, interpretation I is a stable model of F relative shorthand for 8x(u(x) ! c(x)). Expression u = c is defined bg as 8x(u(x) $ c(x)) if u and c are predicate symbols, and to c (w.r.t. background theory σ ) if I j=bg SM[F ; c]. 8x(u(x) = c(x)) if they are function symbols. For lists of We will often write G F , in a rule form as in logic pro- grams, to denote the universal closure of F ! G. A finite set symbols u = (u1; : : : ; un) and c = (c1; : : : ; cn), expression of formulas is identified with the conjunction of the formulas u ≤ c is defined as (u1 ≤ c1)^· · ·^(un ≤ cn), and similarly, in the set. expression u = c is defined as (u1 = c1) ^ · · · ^ (un = cn). Let c be a list of distinct predicate and function constants, Example 1 The following set F of formulas describes the in- and let c be a list of distinct predicate and function variables ertia assumption on the speed of a car and the effect of accel- b pred func corresponding to c. By c (c , respectively) we mean eration assuming the theory of reals as the background the- the list of all predicate constants (function constants, respec- ory. pred tively) in c, and by bc the list of the corresponding predi- cate variables in bc. Speed1 =x ::(Speed1 =x) ^ Speed0 =x For any formula F , expression SM[F ; c] is defined as Speed1 =x (x = Speed0 +3× Duration) (2) ∗ ^ Accelerate= TRUE : F ^ :9bc(bc < c ^ F (bc)); where c < c is shorthand for (cpred ≤ cpred ) ^ :(c = c), Here x is a variable of sort R≥0; Speed0, Speed1 ∗b b b and Duration are object constants with value sort and F (bc) is defined recursively as follows. ∗ 0 0 R≥0 and Accelerate is an object constant with • When F is an atomic formula, F is F ^ F where F value sort Boolean. For interpretation I of signa- is obtained from F by replacing all intensional (function ture fSpeed0; Speed1; Duration; Accelerateg such that and predicate) constants c in it with the corresponding I I I Accelerate = FALSE, Speed = 1, Speed = 1, (function and predicate) variables from c; 0 1 b DurationI = 1:5, we check that I j= SM[F ; Speed ].