## Model Expansion as a Framework for Modelling and Solving Search Problems

Citations: | 3 - 3 self |

### BibTeX

@MISC{Mitchell_modelexpansion,

author = {David Mitchell and Eugenia Ternovska and Faraz Hach and Raheleh Mohebali},

title = {Model Expansion as a Framework for Modelling and Solving Search Problems},

year = {}

}

### OpenURL

### Abstract

We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework. 1.

### Citations

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Citation Context ...trary inductive definitions by handling only a “well-behaved” fragment, in which each definition can either be used to compute the defined predicate during grounding, or replaced with its completion [=-=Clark 1978-=-], which is a classical formula. The features of our implementation include multiple sorts, ordered structures, bounded quantifiers, etc. 1.3 Paper Outline In Section 2 we give formal preliminaries an... |

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Citation Context ...ernational Planning Competition [Gerevini et al. 2006]. Reduction to SAT has also been shown to be an effective method for model checking (perhaps better called bug-finding) in hardware verification [=-=Biere et al. 1999-=-]. SAT-based “bounded model checking” (BMC) is now standard in the electronic design industry, and of growing importance in software verification. Implementations of other logic-based tools routinely ... |

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Citation Context ...ily directed at NPsearch problems. There, modelling is in an extended first-order logic, and solving is based on reduction to the satisfiability problem for propositional logic (SAT). Cook’s theorem [=-=Cook 1971-=-] that every problem in the complexity class NP can be reduced in polynomial time to SAT suggested a scheme for solving problems in NP: (1) For each problem Π, implement a polytime reduction to SAT; (... |

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Citation Context ...ed NP-SPEC, which used an extension of Datalog as a modelling language. For results on expressiveness and complexity of ASP we refer the reader to [Marek and Truszczynski 1999; Marek and Remmel 2003; =-=Leone et al. 2006-=-]. We have recently learned about the diploma project by Alban Rrustemi, under the supervision of Anuj Dawar, which explores a solving scheme very similar to ours, ∃SO as the representation language, ... |

273 |
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Citation Context ...or other complexity classes, most notably for P [Immerman 1982; Vardi 1982; Livchak 1983]. This study of the relationship between logical definability and complexity is called descriptive complexity [=-=Immerman 1999-=-]. The results show that logics can be seen as modelling languages for problems in corresponding complexity classes — another idea that has widely been considered of theoretical, but not practical, in... |

270 |
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267 |
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Citation Context ... choice for our framework [Mitchell and Ternovska 2005b; 2005c]. The following properties for MX in the parameterized setting are immediate from the definition of MX and results in [Grädel 1992] and [=-=Stockmeyer 1977-=-]. ◦ On ordered structures, FO universal Horn MX captures P, ◦ On ordered structures, FO universal Krom MX captures NL, 2 For theoretical considerations, this assumption can always be removed by the s... |

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Citation Context ...y[x = x ∧ y = y ∧ E(x,y)]; and the formula ∃x∃y[R(x) ∧ E(x,y)] can be rewritten as ∃x∃y[R(x)∧y = y∧E(x,y)], where R is an instance predicate, and E is an expansion predicate. Extending the result of [=-=Kolaitis and Vardi 1998-=-], Flum et al. [Flum et al. 2002] showed that FO k , the set of FO sentences with at most k distinct variables, has the same expressive power as FO formulas of treewidth bounded by k. The fragment RGF... |

109 |
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Citation Context ... structures; FO does not capture NP-search. The statement follows from the definition of NP, the two lemmas above, and the fact that FO(IFP) captures P [Immerman 1982; Vardi 1986], while FO does not [=-=Ajtai 1983-=-; Furst et al. 1984]. Corollary 3.7. Given an NP-search problem represented by a σ ∪ ε checking relation R, there is a parameterized model expansion axiomatization φ ∈ FO(LFP) such that the expansion ... |

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Citation Context ...AC0: capturing AC0 requires extending the logic, for example to FO(<,BIT) or FO(+,×) (see [Immerman 1999]). FO MX captures NP. In the combined setting (combined complexity), FO MC is PSPACE-complete [=-=Stockmeyer 1974-=-], and FO MX is NEXPTIME-complete. FO satisfiability in the finite is undecidable [Trakhtenbrot 1950]. Model expansion avoids undecidability of FO by specifying the finite universe as part of the inst... |

81 | Query evaluation via tree-decompositions
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Citation Context ...ula ∃x∃y[R(x) ∧ E(x,y)] can be rewritten as ∃x∃y[R(x)∧y = y∧E(x,y)], where R is an instance predicate, and E is an expansion predicate. Extending the result of [Kolaitis and Vardi 1998], Flum et al. [=-=Flum et al. 2002-=-] showed that FO k , the set of FO sentences with at most k distinct variables, has the same expressive power as FO formulas of treewidth bounded by k. The fragment RGFk contains FO k , and thus conta... |

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Citation Context ...Our approach is natural in light of descriptive complexity results, but is not typical. Historically problem solving in logic was often cast as validity checking, as in the influential work of Green [=-=Green 1969-=-]. There, the user axiomatizes the claim that a solution exists, and a resolution theorem prover produces, as a side-effect of proving the claim, a term describing the solution. This idea was the basi... |

58 | Extending classical logic with inductive definitions - Denecker - 2000 |

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Citation Context ...[Cadoli and Schaerf 2005], but this handled only the fragment of the specification language without aggregates, recursion, or negative occurrences of defined predicates in rule bodies. 8.5 ESRA ESRA [=-=Flener et al. 2003-=-] is The ESRA modelling language [Flener et al. 2003] is syntactically FO with extensions, provided with a denotational semantics. Extensions include domain declarations (sorts), arithmetic over finit... |

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Citation Context ...n 1999]). FO MX captures NP. In the combined setting (combined complexity), FO MC is PSPACE-complete [Stockmeyer 1974], and FO MX is NEXPTIME-complete. FO satisfiability in the finite is undecidable [=-=Trakhtenbrot 1950-=-]. Model expansion avoids undecidability of FO by specifying the finite universe as part of the instance. We discuss the complexity of MC, MX and satisfiability for some fragments and extensions of FO... |

44 |
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Citation Context ...pend on φ). Supposing otherwise, we would have for every class K of structures in NP, an O(n k ) time reduction to SAT, so membership in K can be decided in NTIME(n k ). But by the hierarchy theorem [=-=Cook 1973-=-], for every k, there are infinitely many problems in NP which are not in NTIME(n k ). One can obtain better complexity results for particular classes of formulas, as illustrated later in this section... |

42 |
Capturing complexity classes by fragments of second-order logic
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(Show Context)
Citation Context ...as the language of choice for our framework [Mitchell and Ternovska 2005b; 2005c]. The following properties for MX in the parameterized setting are immediate from the definition of MX and results in [=-=Grädel 1992-=-] and [Stockmeyer 1977]. ◦ On ordered structures, FO universal Horn MX captures P, ◦ On ordered structures, FO universal Krom MX captures NL, 2 For theoretical considerations, this assumption can alwa... |

37 | np-spec: An executable specification language for solving all problems in NP
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Citation Context ...languages, in which one describes the problem independently of particular instances. A solver takes as input a pair consisting of a problem specification and a distinct instance description. NP-SPEC [=-=Cadoli et al. 2000-=-] was among the first proposals for a logic-based system which makes this distinction. In Answer Set Programming (ASP), some authors have argued for the importance of this separation [Marek and Truszc... |

36 | Compiling problem specifications into SAT
- Cadolia, Schaerf
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(Show Context)
Citation Context ...rted finite structure. The initial implementation was provided by reduction to Prolog, with apparently poor performance. Later an implementation, called Spec2SAT, was provided using reduction to SAT [=-=Cadoli and Schaerf 2005-=-], but this handled only the fragment of the specification language without aggregates, recursion, or negative occurrences of defined predicates in rule bodies. 8.5 ESRA ESRA [Flener et al. 2003] is T... |

35 |
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- 1972
(Show Context)
Citation Context ...ce ∃R1,...,∃Rm φ, by associating a universe of size n with n. Thus, if σ = ∅, model expansion coincides with the spectrum problem. The case of Theorem 2.2 with σ = ∅ is equivalent to the result from [=-=Jones and Selman 1974-=-] that a set X ⊂ N is a FO spectrum iff it is in NEXPTIME. In [Fagin 1974] the class of models of an ∃SO formula were called a generalized spectrum. There is a simple relationship between MX and datab... |

31 | The Design of ESSENCE: A Constraint Language for Specifying Combinatorial Problems
- Frisch, Grum, et al.
- 2007
(Show Context)
Citation Context ...999]. Formally, though, both parts are formulas (logic programs), and the distinction is made only as a convention, and it is not fully followed. In CSP, there a number of proposals, such as Essence [=-=Frisch et al. 2007-=-], although there is no widely-used standard to date. 1.1.4 Expressive Logics as Languages for Modelling Search. The logics proposed for modelling languages have been primarily non-classical, presumab... |

29 | A logic of nonmonotone inductive definitions and its modularity properties - Denecker, Ternovska - 2004 |

29 | A framework for representing and solving np search problems - Mitchell, Ternovska - 2005 |

28 | A logic of non-monotone inductive definitions - Denecker, Ternovska |

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

20 | Predicate-calculus based logics for modeling and solving search problems
- East, Truszczy`nski
(Show Context)
Citation Context ...h the classical part is FO. We proposed this as the logic of choice for modelling NP-search problems as model expansion in [Mitchell and Ternovska 2005b; 2005c]. East and Truszczynski’s system ASPPS [=-=East and Truszczynski 2004-=-] may be viewed as a precursor of our work in the sense that it is the first implemented system for solving NP-search problems whose modelling language may be seen as a restriction of FO(ID) to a lang... |

20 | Finite-model theory --- a personal perspective
- Fagin
- 1990
(Show Context)
Citation Context ...ions, without appealing to fixpoint logics. Fully understanding this would answer an open question of Fagin, regarding the number and arities of second order variables required for ∃SO to capture NP [=-=Fagin 1993-=-]. The question is closely related to the problems of recognizing those ∃SO formulas which define NP-complete or polytime classes of structures (see [Medina and Immerman 1994]). This is an interesting... |

17 |
The OPL Optimization Programming Language
- Hentenryck, Lustig, et al.
- 1999
(Show Context)
Citation Context ...sumes, the choice of a denotational rather than model-theoretic semantics. ESRA is implemented in the form of compilation to OPL, a related modelling language with an effective solver implementation [=-=Hentenryck 1999-=-]. The ESRA authors aim to provide a simple, high-level language “capture common modelling idioms”, while expressing dis-interest in any formal notion of completeness, such as capturing a class of pro... |

16 |
Stable logic programming - an alternative logic programming paradigm
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Citation Context ...doli et al. 2000] was among the first proposals for a logic-based system which makes this distinction. In Answer Set Programming (ASP), some authors have argued for the importance of this separation [=-=Marek and Truszczynski 1999-=-]. Formally, though, both parts are formulas (logic programs), and the distinction is made only as a convention, and it is not fully followed. In CSP, there a number of proposals, such as Essence [Fri... |

10 | Fixed point logics
- Dawar, Gurevich
(Show Context)
Citation Context ...lts from finite model theory, we refer the reader to [Immerman 1999; Libkin 2004; Ebbinghaus and Flum 1995]. A detailed discussion of fixpoint logics such as FO(LFP) and FO(IFP) can be found in e.g. [=-=Dawar and Gurevich 2002-=-] and in [Ebbinghaus and Flum 1995]. 2.2 Main Definition: Model Expansion (MX) For any logic L, (e.g., FO, FO(ID), FO(LFP), SO, etc.,) the L model expansion problem, L-MX, is: Instance: A pair 〈φ, A〉,... |

9 |
Relational queries computable in polytime
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(Show Context)
Citation Context ...formula φ; (2) Given instance π of Π, check if π |= φ. A possible implementation of (2) is by reduction to SAT. Results analogous to Fagin’s followed for other complexity classes, most notably for P [=-=Immerman 1982-=-; Vardi 1982; Livchak 1983]. This study of the relationship between logical definability and complexity is called descriptive complexity [Immerman 1999]. The results show that logics can be seen as mo... |

9 | On the expressibility of stable logic programming
- Marek, Remmel
- 2003
(Show Context)
Citation Context ...adoli et al. 2000] called NP-SPEC, which used an extension of Datalog as a modelling language. For results on expressiveness and complexity of ASP we refer the reader to [Marek and Truszczynski 1999; =-=Marek and Remmel 2003-=-; Leone et al. 2006]. We have recently learned about the diploma project by Alban Rrustemi, under the supervision of Anuj Dawar, which explores a solving scheme very similar to ours, ∃SO as the repres... |

8 | The IDP framework for declarative problem solving
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(Show Context)
Citation Context ...ska 2005]). Alternately, one could reduce to an extension of propositional logic with inductive definitions, but building a propositional solver for this language is complex (See [Mariën et al. 2005; =-=Mariën et al. 2006-=-]). In our current implementation, called MXG, we perform a reduction to SAT, and side-step the complexity of dealing with arbitrary inductive definitions by handling only a “well-behaved” fragment, i... |

7 | Incremental algorithms for local search from existential second-order logic
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(Show Context)
Citation Context ...ropriate algorithms to translate to those forms, for efficient grounding and solving. We also plan to explore alternate schemes for solver construction, such as the local search approach proposed in [=-=Agren et al. 2005-=-], and application of general finite-model finding tools, such as used in the theorem proving community. 9.4 Future Work: Modelling Languages For many industrial practitioners, classical logic may not... |

7 | guards: game theoretic and logical characterizations of hypertree width - Gottlob, Leone, et al. |

7 |
The relational model for process control
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- 1983
(Show Context)
Citation Context ...ce π of Π, check if π |= φ. A possible implementation of (2) is by reduction to SAT. Results analogous to Fagin’s followed for other complexity classes, most notably for P [Immerman 1982; Vardi 1982; =-=Livchak 1983-=-]. This study of the relationship between logical definability and complexity is called descriptive complexity [Immerman 1999]. The results show that logics can be seen as modelling languages for prob... |

7 | Satisfiability checking for PC(ID
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- 2005
(Show Context)
Citation Context ...ee [Pelov and Ternovska 2005]). Alternately, one could reduce to an extension of propositional logic with inductive definitions, but building a propositional solver for this language is complex (See [=-=Mariën et al. 2005-=-; Mariën et al. 2006]). In our current implementation, called MXG, we perform a reduction to SAT, and side-step the complexity of dealing with arbitrary inductive definitions by handling only a “well-... |

7 | Grounding for model expansion in k-guarded formulas with inductive definitions
- Patterson, Liu, et al.
- 2007
(Show Context)
Citation Context ... of axiomatizing the problem in an appropriate logic. The framework was proposed in [Mitchell and Ternovska 2005b; 2005c; 2005a] and certain aspects were further developed in [Kolokolova et al. 2006; =-=Patterson et al. 2006-=-; 2007]. Here, we report on the progress to date. We explain the framework and investigate a number of properties. We also report on an implementation which is primarily directed at NPsearch problems.... |

5 |
A syntactic characterization of NP-completeness
- Medina, Immerman
- 1994
(Show Context)
Citation Context ...riables required for ∃SO to capture NP [Fagin 1993]. The question is closely related to the problems of recognizing those ∃SO formulas which define NP-complete or polytime classes of structures (see [=-=Medina and Immerman 1994-=-]). This is an interesting topic for further research. 4. SOLVING BY GROUNDING One approach to solving problems cast as MX is through propositionalization, or grounding, in which the question of exist... |

5 | Reducing inductive definitions to propositional satisfiability
- Pelov, Ternovska
- 2005
(Show Context)
Citation Context ...olvers An earlier solver produced at SFU used a slightly modified version of LPARSE, the standard ASP grounder, together with a translation from propositional logic with inductive definitions to SAT [=-=Pelov and Ternovska 2005-=-]. The formula φ was in the input syntax of LPARSE, but with FO(ID) semantics. The output of LPARSE was transformed to SAT using a level-mapping reduction for inductive definitions. While performance ... |

4 | Complexity of expanding a finite structure and related tasks
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- 2006
(Show Context)
Citation Context ...lling consists primarily of axiomatizing the problem in an appropriate logic. The framework was proposed in [Mitchell and Ternovska 2005b; 2005c; 2005a] and certain aspects were further developed in [=-=Kolokolova et al. 2006-=-; Patterson et al. 2006; 2007]. Here, we report on the progress to date. We explain the framework and investigate a number of properties. We also report on an implementation which is primarily directe... |

4 | Declarative constraint modelling and specification-level reasoning
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- 2005
(Show Context)
Citation Context ...000sModelling and Solving Search Problems with Logic · 35 not aware of any serious attempt to develop the idea further and to address issues which must be handled to produce practical tools. Mancini [=-=Mancini 2004-=-] has explored various ways in which ∃SO axiomatizations can be manipulated to support effective search. Mancini views ∃SO as a natural abstraction of concrete modelling languages proposed in the cons... |

2 |
5th international planning competition (ipc-5), web page: http://zeus.ing.unibs.it/ipc-5
- Gerevini, A, et al.
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Citation Context ...l-purpose planning programs of the day [Kautz and Selman 1992]. SAT-based techniques are currently the best, or among the best, in several categories of the annual International Planning Competition [=-=Gerevini et al. 2006-=-]. Reduction to SAT has also been shown to be an effective method for model checking (perhaps better called bug-finding) in hardware verification [Biere et al. 1999]. SAT-based “bounded model checking... |