Ilkka Niemelä

Extending and implementing the stable model semantics

A novel logic program like language, weight constraint rules, is developed for answer set programming purposes. It generalizes normal logic programs by allowing weight constraints in place of literals to represent, e.g., cardinality and resource constraints and by providing optimization capabilities. A declarative semantics is developed which extends the stable model semantics of normal programs. The computational complexity of the language is shown to be similar to that of normal programs under the stable model semantics. A simple embedding of general weight constraint rules to a small subclass of the language called basic constraint rules is devised. An implementation of the language, the SMODELS system, is developed based on this embedding. It uses a two level architecture consisting of a front-end and a kernel language implementation. The front-end allows restricted use of variables and functions and compiles general weight constraint rules to basic constraint rules. A major part of the work is the development of an efficient search procedure for computing stable models for this kernel language. The procedure is compared with and empirically tested against satisfiability checkers and an implementation of the stable model semantics. It offers a competitive implementation of the stable model semantics for normal programs and attractive performance for problems where the new types of rules provide a compact representation.

Logic programs with stable model semantics as a constraint programming paradigm

Logic programming with the stable model semantics is put forward as a novel constraint programming paradigm. This paradigm is interesting because it bring advantages of logic programming based knowledge representation techniques to constraint programming and because implementation methods for the stable model semantics for ground (variable‐free) programs have advanced significantly in recent years. For a program with variables these methods need a grounding procedure for generating a variable‐free program. As a practical approach to handling the grounding problem a subclass of logic programs, domain restricted programs, is proposed. This subclass enables efficient grounding procedures and serves as a basis for integrating built‐in predicates and functions often needed in applications. It is shown that the novel paradigm embeds classical logical satisfiability and standard (finite domain) constraint satisfaction problems but seems to provide a more expressive framework from a knowledge representation point of view. The first steps towards a programming methodology for the new paradigm are taken by presenting solutions to standard constraint satisfaction problems, combinatorial graph problems and planning problems. An efficient implementation of the paradigm based on domain restricted programs has been developed. This is an extension of a previous implementation of the stable model semantics, the Smodels system, and is publicly available. It contains, e.g., built‐in integer arithmetic integrated to stable model computation. The implementation is described briefly and some test results illustrating the current level of performance are reported.

Smodels — an implementation of the stable model and well-founded semantics for normal logic programs

The Smodels system is a C++ implementation of the well-founded and stable model semantics for range-restricted function-free normal programs. The system includes two modules: (i) smodels which implements the two semantics for ground programs and (ii) parse which computes a grounded version of a range-restricted function-free normal program. The latter module does not produce the whole set of ground instances of the program but a subset that is sufficient in the sense that no stable models are lost. The implementation of the stable model semantics for ground programs is based on bottom-up backtracking search where a powerful pruning method is employed. The pruning method exploits an approximation technique for stable models which is closely related to the well-founded semantics. One of the advantages of this novel technique is that it can be implemented to work in linear space. This makes it possible to apply the stable model semantics also in areas where resulting programs are highly non-stratified and can possess a large number of stable models. The implementation has been tested extensively and compared with a state of the art implementation of the stable model semantics, the SLG system. In tests involving ground programs it clearly outperforms SLG.

Unfolding partiality and disjunctions in stable model semantics

This article studies an implementation methodology for partial and disjunctive stable models where partiality and disjunctions are unfolded from a logic program so that an implementation of stable models for normal (disjunction-free) programs can be used as the core inference engine. The unfolding is done in two separate steps. First, it is shown that partial stable models can be captured by total stable models using a simple linear and modular program transformation. Hence, reasoning tasks concerning partial stable models can be solved using an implementation of total stable models. Disjunctive partial stable models have been lacking implementations which now become available as the translation handles also the disjunctive case. Second, it is shown how total stable models of disjunctive programs can be determined by computing stable models for normal programs. Thus an implementation of stable models of normal programs can be used as a core engine for implementing disjunctive programs. The feasibility of the approach is demonstrated by constructing a system for computing stable models of disjunctive programs using the SMODELS system as the core engine. The performance of the resulting system is compared to that of DLV, which is a state-of-the-art system for disjunctive programs.

Extending the Smodels system with cardinality and weight constraints

The Smodels system is one of the state-of-the-art implementations of stable model computation for normal logic programs. In order to enable more realistic applications, the basic modeling language of normal programs has been extended with new constructs including cardinality and weight constraints and corresponding implementation techniques have been developed. This paper summarizes the extensions that have been included in the system, demonstrates their use, provides basic application methodology, illustrates the current level of performance of the system, and compares it to state-of-the-art satisfiability checkers.

Planning as satisfiability: parallel plans and algorithms for plan search

We address two aspects of constructing plans efficiently by means of satisfiability testing: efficient encoding of the problem of existence of plans of a given number t of time points in the propositional logic and strategies for finding plans, given these formulae for different values of t. For the first problem we consider three semantics for plans with parallel operator application in order to make the search for plans more efficient. The standard semantics requires that parallel operators are independent and can therefore be executed in any order. We consider a more relaxed definition of parallel plans which was first proposed by Dimopoulos et al., as well as a normal form for parallel plans that requires every operator to be executed as early as possible. We formalize the semantics of parallel plans emerging in this setting and present translations of these semantics into the propositional logic. The sizes of the translations are asymptotically optimal. Each of the semantics is constructed in such a way that there is a plan following the semantics exactly when there is a sequential plan, and moreover, the existence of a parallel plan implies the existence of a sequential plan with as many operators as in the parallel one. For the second problem we consider strategies based on testing the satisfiability of several formulae representing plans of n time steps for several values of n concurrently by several processes. We show that big efficiency gains can be obtained in comparison to the standard strategy of sequentially testing the satisfiability of formulae for an increasing number of time steps.

Developing a Declarative Rule Language for Applications in Product Configuration

A rule-based language is proposed for product configuration applications. It is equipped with a declarative semantics providing formal definitions for main concepts in product configuration, including configuration models, requirements and valid configurations. The semantics uses Horn clause derivability to guarantee that each element in a configuration has a justification. This leads to favorable computational properties. For example, the validity of a configuration can be decided in linear time and other computational tasks remain in NP. It is shown that CSP and dynamic CSP can be embedded in the proposed language which seems to be more suitable for representing configuration knowledge. The rule language is closely related to normal logic programs with the stable model semantics. This connection is exploited in the first implementation which is based on a translator from rules to normal programs and on an existing high performance implementation of the stable model semantics, the Smodels system.

Stable Model Semantics of Weight Constraint Rules

A generalization of logic program rules is proposed where rules are built from weight constraints with type information for each predicate instead of simple literals. These kinds of constraints are useful for concisely representing different kinds of choices as well as cardinality, cost and resource constraints in combinatorial problems such as product configuration. A declarative semantics for the rules is presented which generalizes the stable model semantics of normal logic programs. It is shown that for ground rules the complexity of the relevant decision problems stays in NP. The first implementation of the language handles a decidable subset where function symbols are not allowed. It is based on a new procedure for computing stable models for ground rules extending normal programs with choice and weight constructs and a compilation technique where a weight rule with variables is transformed to a set of such simpler ground rules.

A Tableau Calculus for Minimal Model Reasoning

The paper studies the automation of minimal model inference, i.e., determining whether a formula is true in every minimal model of the premises. A novel tableau calculus for prepositional minimal model reasoning is presented in two steps. First an analytic clausal tableau calculus employing a restricted cut rule is introduced. Then the calculus is extended to handle minimal model inference by employing a groundedness property of minimal models. A decision procedure based on the basic calculus is devised and then it is extended to minimal model inference. The basic decision procedure and its extension enjoy some interesting properties. When deciding logical consequence, the basic procedure explores the search space of counter-models with a preference to minimal models and each counter-model is not generated more than once. The procedures can be implemented to run in polynomial space, and they provide polynomial time decision procedures for Horn clauses. The extended decision procedure can also be used to finding all minimal models of a set of clauses.

The Smodels System

The Smodels system is an Answer Set Programming (ASP) implementation based on the stable model semantics of normal logic programs. The basic idea of ASP is to encode the constraints of a problem as a logic program such that the answer sets (stable models) of the program correspond to the solutions of the problem. Then we can solve the problem by letting a logic program engine to find the answer sets of the program.