Formalising Confluence in PVS

Confluence is a critical property of computational systems which is related with determinism and non ambiguity and thus with other relevant computational attributes of functional specifications and rewriting system as termination and completion. Several criteria have been explored that guarantee confluence and their formalisations provide further interesting information. This work discusses topics and presents personal positions and views related with the formalisation of confluence properties in the Prototype Verification System PVS developed at our research group.Comment: In Proceedings DCM 2015, arXiv:1603.0053

Type Soundness for Path Polymorphism

Path polymorphism is the ability to define functions that can operate uniformly over arbitrary recursively specified data structures. Its essence is captured by patterns of the form $x\,y$ which decompose a compound data structure into its parts. Typing these kinds of patterns is challenging since the type of a compound should determine the type of its components. We propose a static type system (i.e. no run-time analysis) for a pattern calculus that captures this feature. Our solution combines type application, constants as types, union types and recursive types. We address the fundamental properties of Subject Reduction and Progress that guarantee a well-behaved dynamics. Both these results rely crucially on a notion of pattern compatibility and also on a coinductive characterisation of subtyping

Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices

The lambda-calculus with de Bruijn indices assembles each alpha-class of lambda-terms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable lambda-terms through the property that a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in beta-contractions, several calculi of explicit substitution were developed mostly with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborate type systems such as intersection types. In previous work, we introduced a de Bruijn version of the lambda-calculus with an intersection type system and proved that it preserves subject reduction, a basic property of type systems. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for beta-normal forms is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss the failure of the subject reduction property and some possible solutions for it

A deductive calculus for conditional equational systems with built-in predicates as premises

Conditional equationally defined classes of many-sorted algebras, whose premises are conjunctions of (positive) equations and builtin predicates (constraints) in a basic first-order theory, are introduced. These classes are important in the field of algebraic specification because the combination of equational and built-in premises give rise to a type of clauses which is more expressive than purely conditional equations. A sound and complete deductive system is presented and algebraic aspects of these classes are investigated. In particular, the existence of free algebras is examined

Formalizing the Confluence of Orthogonal Rewriting Systems

Orthogonality is a discipline of programming that in a syntactic manner guarantees determinism of functional specifications. Essentially, orthogonality avoids, on the one side, the inherent ambiguity of non determinism, prohibiting the existence of different rules that specify the same function and that may apply simultaneously (non-ambiguity), and, on the other side, it eliminates the possibility of occurrence of repetitions of variables in the left-hand side of these rules (left linearity). In the theory of term rewriting systems (TRSs) determinism is captured by the well-known property of confluence, that basically states that whenever different computations or simplifications from a term are possible, the computed answers should coincide. Although the proofs are technically elaborated, confluence is well-known to be a consequence of orthogonality. Thus, orthogonality is an important mathematical discipline intrinsic to the specification of recursive functions that is naturally applied in functional programming and specification. Starting from a formalization of the theory of TRSs in the proof assistant PVS, this work describes how confluence of orthogonal TRSs has been formalized, based on axiomatizations of properties of rules, positions and substitutions involved in parallel steps of reduction, in this proof assistant. Proofs for some similar but restricted properties such as the property of confluence of non-ambiguous and (left and right) linear TRSs have been fully formalized.Comment: In Proceedings LSFA 2012, arXiv:1303.713

Nominal C-Unification

Nominal unification is an extension of first-order unification that takes into account the \alpha-equivalence relation generated by binding operators, following the nominal approach. We propose a sound and complete procedure for nominal unification with commutative operators, or nominal C-unification for short, which has been formalised in Coq. The procedure transforms nominal C-unification problems into simpler (finite families) of fixpoint problems, whose solutions can be generated by algebraic techniques on combinatorics of permutations.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854

On Nominal Syntax and Permutation Fixed Points

We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new $\alpha$-equivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. Although it may seem less natural than the standard notion of nominal unifier based on freshness constraints, the notion of unifier based on fixed-point constraints behaves better when equational theories are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts. We provide a definition of $\alpha$-equivalence modulo equational theories that take into account A, C and AC theories. Based on this notion of equivalence, we show that C-unification is finitary and we provide a sound and complete C-unification algorithm, as a first step towards the development of nominal unification modulo AC and other equational theories with permutative properties