Extensions to the Estimation Calculus

Walther’s estimation calculus was designed to prove the termination of functional programs, and can also be used to solve the similar problem of proving the well-foundedness of induction rules. However, there are certain features of the goal formulae which are more common to the problem of induction rule well-foundedness than the problem of termination, and which the calculus cannot handle. We present a sound extension of the calculus that is capable of dealing with these features. The extension develops Walther’s concept of an argument bounded function in two ways: firstly, so that the function may be bounded below by its argument, and secondly, so that a bound may exist between two arguments of a predicate. Our calculus enables automatic proofs of the well-foundedness of a large class of induction rules not captured by the original calculus

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

Relational Rippling: a General Approach

We propose a new version of rippling, called relational rippling. Rippling is a heuristic for guiding proof search, especially in the step cases of inductive proofs. Relational rippling is designed for representations in which value passing is by shared existential variables, as opposed to function nesting. Thus relational rippling can be used to guide reasoning about logic programs or circuits represented as relations. We give an informal motivation and introduction to relational rippling. More details, including formal definitions and termination proofs can be found in the longer version of this paper, [Bundy and Lombart, 1995]

Automatic Generation of Proof Tactics for Finite-Valued Logics

A number of flexible tactic-based logical frameworks are nowadays available that can implement a wide range of mathematical theories using a common higher-order metalanguage. Used as proof assistants, one of the advantages of such powerful systems resides in their responsiveness to extensibility of their reasoning capabilities, being designed over rule-based programming languages that allow the user to build her own `programs to construct proofs' - the so-called proof tactics. The present contribution discusses the implementation of an algorithm that generates sound and complete tableau systems for a very inclusive class of sufficiently expressive finite-valued propositional logics, and then illustrates some of the challenges and difficulties related to the algorithmic formation of automated theorem proving tactics for such logics. The procedure on whose implementation we will report is based on a generalized notion of analyticity of proof systems that is intended to guarantee termination of the corresponding automated tactics on what concerns theoremhood in our targeted logics

Size-Change Termination as a Contract

Termination is an important but undecidable program property, which has led to a large body of work on static methods for conservatively predicting or enforcing termination. One such method is the size-change termination approach of Lee, Jones, and Ben-Amram, which operates in two phases: (1) abstract programs into "size-change graphs," and (2) check these graphs for the size-change property: the existence of paths that lead to infinite decreasing sequences. We transpose these two phases with an operational semantics that accounts for the run-time enforcement of the size-change property, postponing (or entirely avoiding) program abstraction. This choice has two key consequences: (1) size-change termination can be checked at run-time and (2) termination can be rephrased as a safety property analyzed using existing methods for systematic abstraction. We formulate run-time size-change checks as contracts in the style of Findler and Felleisen. The result compliments existing contracts that enforce partial correctness specifications to obtain contracts for total correctness. Our approach combines the robustness of the size-change principle for termination with the precise information available at run-time. It has tunable overhead and can check for nontermination without the conservativeness necessary in static checking. To obtain a sound and computable termination analysis, we apply existing abstract interpretation techniques directly to the operational semantics, avoiding the need for custom abstractions for termination. The resulting analyzer is competitive with with existing, purpose-built analyzers