Program Equivalence by Circular Reasoning

We propose a logic and a deductive system for stating and automatically proving the equivalence of programs written in deterministic languages having a rewriting-based operational semantics. The chosen equivalence is parametric in a so-called observation relation, and it says that two programs satisfying the observation relation will inevitably be, in the future, in the observation relation again. This notion of equivalence generalises several well-known equivalences, and is shown to be appropriate for deterministic programs. The deductive system is circular in nature and is proved sound and weakly complete; together, these results say that, when it terminates, our system correctly solves the given program-equivalence problem. We show that our approach is suitable for proving equivalence for terminating and non-terminating programs as well as for concrete and symbolic programs. The latter are programs in which some statements or expressions are symbolic variables. By proving the equivalence between symbolic programs, one proves the equivalence of (infinitely) many concrete programs obtained by replacing the variables by concrete statements or expressions. The approach is illustrated by proving program equivalence in two languages from different programming paradigms. The examples in the paper, as well as other examples, can be checked using an online tool.Nous proposons une logique et un système déductif pour exprimer et prouver automatiquement l'équivalence de programmes dans des langages déterministes munis de sé- mantiques opérationnelles définies par réécriture. Le système déductif proposé est de nature circulaire; nous démontrons qu'il est correct et faiblement complet. Ces deux résultats sig- nifient que, lorsqu'il termine, notre système résout correctement le problème d'équivalence de programmes tels que nous l'avons posé. Nous montrons que ce système fonctionne autant pour des programmes qui terminent que pour des programmes qui ne terminent pas. Les programmes dits symboliques, dans lesquels certaines expressions ou instructions restent non-interprétés, peu- vent également être traités par notre approche. La démonstration d'une équivalence entre deux programmes symboliques revient à démontrer l'équivalence entre une infinité potentielle de pro- grammes concrets, qui sont des instances des programmes symboliques obtenues en remplaçant les variables symboliques par des instructions ou des expressions concrètes. L'approche est illustrée par la preuve d'équivalence de programmes dans deux langages appartenant à des paradigmes de programmation différents. Les exemples contenus dans l'article, ainsi que d'autres exemples, peuvent être essayés dans un outil en ligne

(Co-)Inductive semantics for Constraint Handling Rules

In this paper, we address the problem of defining a fixpoint semantics for Constraint Handling Rules (CHR) that captures the behavior of both simplification and propagation rules in a sound and complete way with respect to their declarative semantics. Firstly, we show that the logical reading of states with respect to a set of simplification rules can be characterized by a least fixpoint over the transition system generated by the abstract operational semantics of CHR. Similarly, we demonstrate that the logical reading of states with respect to a set of propagation rules can be characterized by a greatest fixpoint. Then, in order to take advantage of both types of rules without losing fixpoint characterization, we present an operational semantics with persistent. We finally establish that this semantics can be characterized by two nested fixpoints, and we show the resulting language is an elegant framework to program using coinductive reasoning.Comment: 17 page

Relating Weight Constraint and Aggregate Programs: Semantics and Representation

Weight constraint and aggregate programs are among the most widely used logic programs with constraints. In this paper, we relate the semantics of these two classes of programs, namely the stable model semantics for weight constraint programs and the answer set semantics based on conditional satisfaction for aggregate programs. Both classes of programs are instances of logic programs with constraints, and in particular, the answer set semantics for aggregate programs can be applied to weight constraint programs. We show that the two semantics are closely related. First, we show that for a broad class of weight constraint programs, called strongly satisfiable programs, the two semantics coincide. When they disagree, a stable model admitted by the stable model semantics may be circularly justified. We show that the gap between the two semantics can be closed by transforming a weight constraint program to a strongly satisfiable one, so that no circular models may be generated under the current implementation of the stable model semantics. We further demonstrate the close relationship between the two semantics by formulating a transformation from weight constraint programs to logic programs with nested expressions which preserves the answer set semantics. Our study on the semantics leads to an investigation of a methodological issue, namely the possibility of compact representation of aggregate programs by weight constraint programs. We show that almost all standard aggregates can be encoded by weight constraints compactly. This makes it possible to compute the answer sets of aggregate programs using the ASP solvers for weight constraint programs. This approach is compared experimentally with the ones where aggregates are handled more explicitly, which show that the weight constraint encoding of aggregates enables a competitive approach to answer set computation for aggregate programs.Comment: To appear in Theory and Practice of Logic Programming (TPLP), 2011. 30 page

Rewriting recursive aggregates in answer set programming: back to monotonicity

Aggregation functions are widely used in answer set programming for representing and reasoning on knowledge involving sets of objects collectively. Current implementations simplify the structure of programs in order to optimize the overall performance. In particular, aggregates are rewritten into simpler forms known as monotone aggregates. Since the evaluation of normal programs with monotone aggregates is in general on a lower complexity level than the evaluation of normal programs with arbitrary aggregates, any faithful translation function must introduce disjunction in rule heads in some cases. However, no function of this kind is known. The paper closes this gap by introducing a polynomial, faithful, and modular translation for rewriting common aggregation functions into the simpler form accepted by current solvers. A prototype system allows for experimenting with arbitrary recursive aggregates, which are also supported in the recent version 4.5 of the grounder gringo, using the methods presented in this paper

A Framework to Synergize Partial Order Reduction with State Interpolation

We address the problem of reasoning about interleavings in safety verification of concurrent programs. In the literature, there are two prominent techniques for pruning the search space. First, there are well-investigated trace-based methods, collectively known as "Partial Order Reduction (POR)", which operate by weakening the concept of a trace by abstracting the total order of its transitions into a partial order. Second, there is state-based interpolation where a collection of formulas can be generalized by taking into account the property to be verified. Our main contribution is a framework that synergistically combines POR with state interpolation so that the sum is more than its parts