Closed nominal rewriting and efficiently computable nominal algebra equality

We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of the lambda-calculus and first-order logic.Comment: In Proceedings LFMTP 2010, arXiv:1009.218

Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201

Consistency and Completeness of Rewriting in the Calculus of Constructions

Adding rewriting to a proof assistant based on the Curry-Howard isomorphism, such as Coq, may greatly improve usability of the tool. Unfortunately adding an arbitrary set of rewrite rules may render the underlying formal system undecidable and inconsistent. While ways to ensure termination and confluence, and hence decidability of type-checking, have already been studied to some extent, logical consistency has got little attention so far. In this paper we show that consistency is a consequence of canonicity, which in turn follows from the assumption that all functions defined by rewrite rules are complete. We provide a sound and terminating, but necessarily incomplete algorithm to verify this property. The algorithm accepts all definitions that follow dependent pattern matching schemes presented by Coquand and studied by McBride in his PhD thesis. It also accepts many definitions by rewriting, containing rules which depart from standard pattern matching.Comment: 20 page

Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

The Sigma-Semantics: A Comprehensive Semantics for Functional Programs

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the sigma-semantics, especially call-by-value and call-by-name semantics, are highlighted. Furthermore, a property can be validated for all instances of the sigma-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the sigma-semantics. We present and apply means for very simple proofs of equivalence with the denotational sigma-semantics for a large class of reduction-based sigma-semantics. Our basis are simple first-order constructor-based functional programs with patterns