Vitaly Lagoon

Testing for termination with monotonicity constraints

Termination analysis is often performed over the abstract domains of monotonicity constraints or of size change graphs. First, the transition relation for a given program is approximated by a set of descriptions. Then, this set is closed under a composition operation. Finally, termination is determined if all of the idempotent loop descriptions in this closure have (possibly different) ranking functions. This approach is complete for size change graphs: An idempotent loop description has a ranking function if and only if it has one which indicates that some specific argument decreases in size. In this paper we generalize the size change criteria for size change graphs which are not idempotent. We also illustrate that proving termination with monotonicity constraints is more powerful than with size change graphs and demonstrate that the size change criteria is incomplete for monotonicity constraints. Finally, we provide a complete termination test for monotonicity constraints.

Solving set constraint satisfaction problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems.

Type dependencies for logic programs using ACI-unification

Abstract This paper presents a new notion of typing for logic programs which generalizes the notion of directional types. The generation of type dependencies for a logic program is fully automatic with respect to a given domain of types. The analysis method is based on a novel combination of program abstraction and ACI-unification which is shown to be correct and optimal. Type dependencies are obtained by abstracting programs, replacing concrete terms by their types, and evaluating the meaning of the abstract programs using a standard semantics for logic programs enhanced by ACI-unification. This approach is generic and can be used with any standard semantics. The method is both theoretically clean and easy to implement using general purpose tools. The proposed domain of types is condensing which means that analyses can be carried out in both top-down or bottom-up frameworks with no loss of precision for goal-independent analyses. The proposed method has been fully implemented within a bottom-up approach and the experimental results are promising.

Logic programming with satisfiability

This paper presents a Prolog interface to the MiniSat satisfiability solver. Logic programming with satisfiability combines the strengths of the two paradigms: logic programming for encoding search problems into satisfiability on the one hand and efficient SAT solving on the other. This synergy between these two exposes a programming paradigm that we propose here as a logic programming pearl. To illustrate logic programming with SAT solving, we give an example Prolog program that solves instances of Partial MAXSAT.

Combining Norms to Prove Termination

Automatic termination analysers typically measure the size of terms applying norms which are mappings from terms to the natural numbers. This paper illustrates howt o enable the use of size functions defined as tuples of these simpler norm functions. This approach enables us to simplify the problem of deriving automatically a candidate norm with which to prove termination. Instead of deriving a single, complex norm function, it is sufficient to determine a collection of simpler norms, some combination of which, leads to a proof of termination. We propose that a collection of simple norms, one for each of the recursive data-types in the program, is often a suitable choice. We first demonstrate the power of combining norm functions and then the adequacy of combining norms based on regular types.

An algebraic approach to sharing analysis of logic programs

Abstract This paper describes an algebraic approach to the sharing analysis of logic programs based on an abstract domain of set logic programs . Set logic programs are logic programs in which the terms are sets of variables and unification is based on an associative, commutative, and idempotent equality theory. All of the basic operations required for sharing analyses, as well as their formal justification, are based on simple algebraic properties of set substitutions and set-based atoms. An ordering on set-based syntactic objects, similar to “less general” on concrete syntactic objects, is shown to reflect the notion of “less sharing” information. The (abstract) unification of a pair of set-based terms corresponds to finding their most general ACI1 unifier with respect to this ordering. The unification of a set of equations between set-based terms is defined exactly as in the concrete case, by solving the equations one by one and repeatedly applying their solutions to the remaining equations. We demonstrate that all of the operations in a sharing analysis have natural definitions which are both correct and optimal.

Termination Analysis with Types Is More Accurate

In this paper we show how we can use size and groundness analyses lifted to regular and (polymorphic) Hindley/Milner typed programs to determine more accurate termination of (type correct) programs. Type information for programs may be either inferred automatically or declared by the programmer. The analysis of the typed logic programs is able to completely reuse a framework for termination analysis of untyped logic programs by using abstract compilation of the type abstraction. We show that our typed termination analysis is uniformly more accurate than untyped termination analysis for regularly typed programs, and demonstrate how it is able to prove termination of programs which the untyped analysis can not.

Boolean equi-propagation for optimized SAT encoding

We present an approach to propagation based SAT encoding, Boolean equi-propagation, where constraints are modelled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied as a form of partial evaluation to simplify constraints prior to their encoding as CNF formulae. We demonstrate for a variety of benchmarks that our approach leads to a considerable reduction in the size of CNF encodings and subsequent speed-ups in SAT solving times.

A Framework for Analysis of Typed Logic Programs

The paper presents a novel approach to the analysis of typed logic programs. We assume regular type descriptions of logic program variables provided by regular tree grammars. Types are used to identify components of terms which can be equated during the programs execution. This information is reflected in form of set constraints in the generic abstract domain called Type(X). The domain allows for abstract compilation of typed logic programs into set logic programs. We demonstrate how the analyzers of groundness and sharing of typed logic programs are obtained by applying the existing (untyped) techniques based on Pos and Sharing to set logic programs. The corresponding analyses in Type(Pos) and Type(Sharing) are more precise than those in Pos and Sharing.

Fast set bounds propagation using a BDD-SAT hybrid

Binary Decision Diagram (BDD) based set bounds propagation is a powerful approach to solving set-constraint satisfaction problems. However, prior BDD based techniques incur the significant overhead of constructing and manipulating graphs during search. We present a set-constraint solver which combines BDD-based set-bounds propagators with the learning abilities of a modern SAT solver. Together with a number of improvements beyond the basic algorithm, this solver is highly competitive with existing propagation based set constraint solvers.