38 research outputs found
Strong Normalization for HA + EM1 by Non-Deterministic Choice
We study the strong normalization of a new Curry-Howard correspondence for HA
+ EM1, constructive Heyting Arithmetic with the excluded middle on
Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda
calculus plus an operator ||_a which represents, from the viewpoint of
programming, an exception operator with a delimited scope, and from the
viewpoint of logic, a restricted version of the excluded middle. We give a
strong normalization proof for the system based on a technique of
"non-deterministic immersion".Comment: In Proceedings COS 2013, arXiv:1309.092
A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
Basic proof-search tactics in logic and type theory can be seen as the
root-first applications of rules in an appropriate sequent calculus, preferably
without the redundancies generated by permutation of rules. This paper
addresses the issues of defining such sequent calculi for Pure Type Systems
(PTS, which were originally presented in natural deduction style) and then
organizing their rules for effective proof-search. We introduce the idea of
Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the
syntax of a permutation-free sequent calculus for propositional logic due to
Herbelin, which is strongly related to natural deduction and already well
adapted to proof-search. The operational semantics is adapted from Herbelin's
and is defined by a system of local rewrite rules as in cut-elimination, using
explicit substitutions. We prove confluence for this system. Restricting our
attention to PTSC, a type system for the ground terms of this system, we obtain
the Subject Reduction property and show that each PTSC is logically equivalent
to its corresponding PTS, and the former is strongly normalising iff the latter
is. We show how to make the logical rules of PTSC into a syntax-directed system
PS for proof-search, by incorporating the conversion rules as in
syntax-directed presentations of the PTS rules for type-checking. Finally, we
consider how to use the explicitly scoped meta-variables of PTSCalpha to
represent partial proof-terms, and use them to analyse interactive proof
construction. This sets up a framework PE in which we are able to study
proof-search strategies, type inhabitant enumeration and (higher-order)
unification
Monadic translation of classical sequent calculus
International audienceWe study monadic translations of the call-by-name (cbn) and call-by-value (cbv) fragments of the classical sequent calculus due to Curien and Herbelin, and give modular and syntactic proofs of strong normalisation. The target of the translations is a new meta-language for classical logic, named monadic λμ. This language is a monadic reworking of Parigot's λμ-calculus, where the monadic binding is confined to commands, thus integrating the monad with the classical features. Also, its μ-reduction rule is replaced by a rule expressing the interaction between monadic binding and μ-abstraction.Our monadic translations produce very tight simulations of the respective fragments of within monadic λμ, with reduction steps of being translated in a 1–1 fashion, except for β steps, which require two steps. The monad of monadic λμ can be instantiated to the continuations monad so as to ensure strict simulation of monadic λμ within simply typed λ-calculus with β- and η-reduction. Through strict simulation, the strong normalisation of simply typed λ-calculus is inherited by monadic λμ, and then by cbn and cbv , thus reproving strong normalisation in an elementary syntactical way for these fragments of , and establishing it for our new calculus. These results extend to second-order logic, with polymorphic λ-calculus as the target, giving new strong normalisation results for classical second-order logic in sequent calculus style.CPS translations of cbn and cbv with the strict simulation property are obtained by composing our monadic translations with the continuations-monad instantiation. In an appendix to the paper, we investigate several refinements of the continuations-monad instantiation in order to obtain in a modular way improvements of the CPS translations enjoying extra properties like simulation by cbv β-reduction or reduction of administrative redexes at compile time
Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi
The intuitionistic fragment of the call-by-name version of Curien and
Herbelin's \lambda\_mu\_{\~mu}-calculus is isolated and proved strongly
normalising by means of an embedding into the simply-typed lambda-calculus. Our
embedding is a continuation-and-garbage-passing style translation, the
inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's
\lambda\_mu-calculus. The embedding strictly simulates reductions while usual
continuation-passing-style transformations erase permutative reduction steps.
For our intuitionistic sequent calculus, we even only need "units of garbage"
to be passed. We apply the same method to other calculi, namely successive
extensions of the simply-typed λ-calculus leading to our intuitionistic
system, and already for the simplest extension we consider (λ-calculus
with generalised application), this yields the first proof of strong
normalisation through a reduction-preserving embedding. The results obtained
extend to second and higher-order calculi
Characterization of strong normalizability for a sequent lambda calculus with co-control
We study strong normalization in a lambda calculus of proof-terms
with co-control for the intuitionistic sequent calculus. In this sequent
lambda calculus, the management of formulas on the left hand
side of typing judgements is “dual" to the management of formulas
on the right hand side of the typing judgements in Parigot’s lambdamu
calculus - that is why our system has first-class “co-control".
The characterization of strong normalization is by means of intersection
types, and is obtained by analyzing the relationship with
another sequent lambda calculus, without co-control, for which a
characterization of strong normalizability has been obtained before.
The comparison of the two formulations of the sequent calculus,
with or without co-control, is of independent interest. Finally, since
it is known how to obtain bidirectional natural deduction systems
isomorphic to these sequent calculi, characterizations are obtained
of the strongly normalizing proof-terms of such natural deduction
systems.The authors would like to thank the anonymous
referees for their valuable comments and helpful suggestions.
This work was partly supported by FCT—Fundação para a Ciência
e a Tecnologia, within the project UID-MAT-00013/2013; by
COST Action CA15123 - The European research network on types
for programming and verification (EUTypes) via STSM; and by the
Ministry of Education, Science and Technological Development,
Serbia, under the projects ON174026 and III44006.info:eu-repo/semantics/publishedVersio
Computational interpretation of classical logic with explicit structural rules
We present a calculus providing a Curry-Howard correspondence to classical logic represented in the sequent calculus with explicit structural rules, namely weakening and contraction. These structural rules introduce explicit erasure and duplication of terms, respectively. We present a type system for which we prove the type-preservation under reduction. A mutual relation with classical calculus featuring implicit structural rules has been studied in detail. From this analysis we derive strong normalisation property
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Programming and proving with classical types
The propositions-as-types correspondence is ordinarily presen-
ted as linking the metatheory of typed λ-calculi and the proof theory
of intuitionistic logic. Griffin observed that this correspondence could
be extended to classical logic through the use of control operators. This
observation set off a flurry of further research, leading to the development
of Parigot’s λμ-calculus. In this work, we use the λμ-calculus as the
foundation for a system of proof terms for classical first-order logic. In
particular, we define an extended call-by-value λμ-calculus with a type
system in correspondence with full classical logic. We extend the language
with polymorphic types, add a host of data types in ‘direct style’, and
prove several metatheoretical properties. All of our proofs and definitions
are mechanised in Isabelle/HOL, and we automatically obtain an inter-
preter for a system of proof terms cum programming language—called
μML—using Isabelle’s code generation mechanism. Atop our proof terms,
we build a prototype LCF-style interactive theorem prover—called μTP—
for classical first-order logic, capable of synthesising μML programs from
completed tactic-driven proofs. We present example closed μML programs
with classical tautologies for types, including some inexpressible as closed
programs in the original λμ-calculus, and some example tactic-driven
μTP proofs of classical tautologies
A Theory of Explicit Substitutions with Safe and Full Composition
Many different systems with explicit substitutions have been proposed to
implement a large class of higher-order languages. Motivations and challenges
that guided the development of such calculi in functional frameworks are
surveyed in the first part of this paper. Then, very simple technology in named
variable-style notation is used to establish a theory of explicit substitutions
for the lambda-calculus which enjoys a whole set of useful properties such as
full composition, simulation of one-step beta-reduction, preservation of
beta-strong normalisation, strong normalisation of typed terms and confluence
on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference
"International Colloquium on Automata, Languages and Programming 2008" edited
by Giuseppe Castagna and Igor Walukiewic