143 research outputs found
Decidability for Non-Standard Conversions in Typed Lambda-Calculi
This thesis studies the decidability of conversions in typed lambda-calculi, along with the algorithms allowing for this decidability. Our study takes in consideration conversions going beyond the traditional beta, eta, or permutative conversions (also called commutative conversions). To decide these conversions, two classes of algorithms compete, the algorithms based on rewriting, here the goal is to decompose and orient the conversion so as to obtain a convergent system, these algorithms then boil down to rewrite the terms until they reach an irreducible forms; and the "reduction free" algorithms where the conversion is decided recursively by a detour via a meta-language. Throughout this thesis, we strive to explain the latter thanks to the former
Superposition for Lambda-Free Higher-Order Logic
We introduce refutationally complete superposition calculi for intentional and extensional clausal -free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the -free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone towards complete, highly efficient automatic theorem provers for full higher-order logic
Elementary data structures in ALGOL-like languages
AbstractJ.C. Reynolds has pointed out that ALGOL 60 has a set of properties not shared by most of the languages usually regarded as being its successors. We propose to use this ALGOL-like framework to design a language that could adequately support both applicative and imperative programming while also retaining the advantages of each of the “pure” frameworks. This paper discusses elementary data-structuring facilities (products, arrays, sums) for such a language, taking advantage of recent developments, such as this author's “quantification” notation, and the notion of “conjunctive type” proposed by Coppo and Dezani, and adapted to explicitly-typed languages by Reynolds
Polynomial Size Analysis of First-Order Shapely Functions
We present a size-aware type system for first-order shapely function
definitions. Here, a function definition is called shapely when the size of the
result is determined exactly by a polynomial in the sizes of the arguments.
Examples of shapely function definitions may be implementations of matrix
multiplication and the Cartesian product of two lists. The type system is
proved to be sound w.r.t. the operational semantics of the language. The type
checking problem is shown to be undecidable in general. We define a natural
syntactic restriction such that the type checking becomes decidable, even
though size polynomials are not necessarily linear or monotonic. Furthermore,
we have shown that the type-inference problem is at least semi-decidable (under
this restriction). We have implemented a procedure that combines run-time
testing and type-checking to automatically obtain size dependencies. It
terminates on total typable function definitions.Comment: 35 pages, 1 figur
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