Proving termination through conditional termination

We present a constraint-based method for proving conditional termination of integer programs. Building on this, we construct a framework to prove (unconditional) program termination using a powerful mechanism to combine conditional termination proofs. Our key insight is that a conditional termination proof shows termination for a subset of program execution states which do not need to be considered in the remaining analysis. This facilitates more effective termination as well as non-termination analyses, and allows handling loops with different execution phases naturally. Moreover, our method can deal with sequences of loops compositionally. In an empirical evaluation, we show that our implementation VeryMax outperforms state-of-the-art tools on a range of standard benchmarks.Peer ReviewedPostprint (author's final draft

Goal-Driven Query Answering for Existential Rules with Equality

Inspired by the magic sets for Datalog, we present a novel goal-driven approach for answering queries over terminating existential rules with equality (aka TGDs and EGDs). Our technique improves the performance of query answering by pruning the consequences that are not relevant for the query. This is challenging in our setting because equalities can potentially affect all predicates in a dataset. We address this problem by combining the existing singularization technique with two new ingredients: an algorithm for identifying the rules relevant to a query and a new magic sets algorithm. We show empirically that our technique can significantly improve the performance of query answering, and that it can mean the difference between answering a query in a few seconds or not being able to process the query at all

A Symbolic Transformation Language and its Application to a Multiscale Method

The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple package for rule-based programming, rewriting strategies and their combination with standard Maple code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.Comment: 36 page

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Inference of termination conditions for numerical loops in Prolog

We present a new approach to termination analysis of numerical computations in logic programs. Traditional approaches fail to analyse them due to non well-foundedness of the integers. We present a technique that allows overcoming these difficulties. Our approach is based on transforming a program in a way that allows integrating and extending techniques originally developed for analysis of numerical computations in the framework of query-mapping pairs with the well-known framework of acceptability. Such an integration not only contributes to the understanding of termination behaviour of numerical computations, but also allows us to perform a correct analysis of such computations automatically, by extending previous work on a constraint-based approach to termination. Finally, we discuss possible extensions of the technique, including incorporating general term orderings.Comment: To appear in Theory and Practice of Logic Programming. To appear in Theory and Practice of Logic Programmin

Automated verification of termination certificates

In order to increase user confidence, many automated theorem provers provide certificates that can be independently verified. In this paper, we report on our progress in developing a standalone tool for checking the correctness of certificates for the termination of term rewrite systems, and formally proving its correctness in the proof assistant Coq. To this end, we use the extraction mechanism of Coq and the library on rewriting theory and termination called CoLoR

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Mechanized semantics

The goal of this lecture is to show how modern theorem provers---in this case, the Coq proof assistant---can be used to mechanize the specification of programming languages and their semantics, and to reason over individual programs and over generic program transformations, as typically found in compilers. The topics covered include: operational semantics (small-step, big-step, definitional interpreters); a simple form of denotational semantics; axiomatic semantics and Hoare logic; generation of verification conditions, with application to program proof; compilation to virtual machine code and its proof of correctness; an example of an optimizing program transformation (dead code elimination) and its proof of correctness

Strict General Setting for Building Decision Procedures into Theorem Provers

The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of well-known or new schemes within this framework a very easy task even for a non-expert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas