2,302 research outputs found
A Case Study on Logical Relations using Contextual Types
Proofs by logical relations play a key role to establish rich properties such
as normalization or contextual equivalence. They are also challenging to
mechanize. In this paper, we describe the completeness proof of algorithmic
equality for simply typed lambda-terms by Crary where we reason about logically
equivalent terms in the proof environment Beluga. There are three key aspects
we rely upon: 1) we encode lambda-terms together with their operational
semantics and algorithmic equality using higher-order abstract syntax 2) we
directly encode the corresponding logical equivalence of well-typed
lambda-terms using recursive types and higher-order functions 3) we exploit
Beluga's support for contexts and the equational theory of simultaneous
substitutions. This leads to a direct and compact mechanization, demonstrating
Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
Nominal Logic Programming
Nominal logic is an extension of first-order logic which provides a simple
foundation for formalizing and reasoning about abstract syntax modulo
consistent renaming of bound names (that is, alpha-equivalence). This article
investigates logic programming based on nominal logic. We describe some typical
nominal logic programs, and develop the model-theoretic, proof-theoretic, and
operational semantics of such programs. Besides being of interest for ensuring
the correct behavior of implementations, these results provide a rigorous
foundation for techniques for analysis and reasoning about nominal logic
programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as
of July 23, 200
Multi-level Contextual Type Theory
Contextual type theory distinguishes between bound variables and
meta-variables to write potentially incomplete terms in the presence of
binders. It has found good use as a framework for concise explanations of
higher-order unification, characterize holes in proofs, and in developing a
foundation for programming with higher-order abstract syntax, as embodied by
the programming and reasoning environment Beluga. However, to reason about
these applications, we need to introduce meta^2-variables to characterize the
dependency on meta-variables and bound variables. In other words, we must go
beyond a two-level system granting only bound variables and meta-variables.
In this paper we generalize contextual type theory to n levels for arbitrary
n, so as to obtain a formal system offering bound variables, meta-variables and
so on all the way to meta^n-variables. We obtain a uniform account by
collapsing all these different kinds of variables into a single notion of
variabe indexed by some level k. We give a decidable bi-directional type system
which characterizes beta-eta-normal forms together with a generalized
substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
Lincx: A Linear Logical Framework with First-class Contexts
Linear logic provides an elegant framework for modelling stateful, imperative and con- current systems by viewing a context of assumptions as a set of resources. However, mech- anizing the meta-theory of such systems remains a challenge, as we need to manage and reason about mixed contexts of linear and intuitionistic assumptions. We present Lincx, a contextual linear logical framework with first-class mixed contexts. Lincx allows us to model (linear) abstract syntax trees as syntactic structures that may depend on intuitionistic and linear assumptions. It can also serve as a foundation for reasoning about such structures. Lincx extends the linear logical framework LLF with first-class (linear) contexts and an equational theory of context joins that can otherwise be very tedious and intricate to develop. This work may be also viewed as a generalization of contextual LF that supports both intuitionistic and linear variables, functions, and assumptions. We describe a decidable type-theoretic foundation for Lincx that only characterizes canonical forms and show that our equational theory of context joins is associative and commu- tative. Finally, we outline how Lincx may serve as a practical foundation for mechanizing the meta-theory of stateful systems.La logique lineĢaire represente une structure eĢleĢgante pour modeler des systeĢmes im- peĢratifs, concurrents et avec des systeĢmes a eĢtats, en repreĢsentant un contexte d'hypotheĢses comme une collection de ressources. Cependant, la meĢcanisation de la meĢtatheĢorie de ces systeĢmes demeure un deĢfi, puisque nous devons geĢrer et raisonner aĢ propos de contextes d'hypotheĢses mixtes lineĢaires et intuitionistiques. Nous preĢsentons Lincx, une structure logique lineĢaire et contextuelle avec des contextes mixtes de premieĢre classe. Lincx nous permet d'eĢtablir des modeĢles (lineĢaires) d'arbres de syntaxe abstraits en tant que structures syntactiques qui peuvent dependre d'hypotheĢses intuitionistiques et lineĢaires. Lincx peut eĢgalement servir de fondation pour raisonner aĢ propos de telles structures. Lincx eĢtend la structure logique lineĢaire LLF avec des contextes (lineĢaires) de premier ordre et une theĢorie d'equations d'assemblage de contextes qui peut autrement eĢtre treĢs fastidieux et complexe aĢ deĢvelopper. Cet oeuvre peut eĢgalement eĢtre percĢ§u comme une geĢneĢralisation du LF contextuel qui supporte les fonctions, les hypotheĢses et les variables intuitionistiques et lineĢaires. Nous deĢcrivons une fondation de la theĢorie des types deĢcidable pour Lincx qui ne deĢcrit que les formes canoniques et montrons que notre theorie d'eĢquations d'assemblage de contextes est associative et commutative. Finalement, nous donnons un apercĢ§u de comment Lincx peut servir de fondation pratique pour la meĢcanisation de la meĢtatheĢorie de systeĢmes aĢ eĢtats
Cut Elimination for a Logic with Induction and Co-induction
Proof search has been used to specify a wide range of computation systems. In
order to build a framework for reasoning about such specifications, we make use
of a sequent calculus involving induction and co-induction. These proof
principles are based on a proof theoretic (rather than set-theoretic) notion of
definition. Definitions are akin to logic programs, where the left and right
rules for defined atoms allow one to view theories as "closed" or defining
fixed points. The use of definitions and free equality makes it possible to
reason intentionally about syntax. We add in a consistent way rules for pre and
post fixed points, thus allowing the user to reason inductively and
co-inductively about properties of computational system making full use of
higher-order abstract syntax. Consistency is guaranteed via cut-elimination,
where we give the first, to our knowledge, cut-elimination procedure in the
presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi
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