93 research outputs found
Light types for polynomial time computation in lambda-calculus
We propose a new type system for lambda-calculus ensuring that well-typed
programs can be executed in polynomial time: Dual light affine logic (DLAL).
DLAL has a simple type language with a linear and an intuitionistic type
arrow, and one modality. It corresponds to a fragment of Light affine logic
(LAL). We show that contrarily to LAL, DLAL ensures good properties on
lambda-terms: subject reduction is satisfied and a well-typed term admits a
polynomial bound on the reduction by any strategy. We establish that as LAL,
DLAL allows to represent all polytime functions. Finally we give a type
inference procedure for propositional DLAL.Comment: 20 pages (including 10 pages of appendix). (revised version; in
particular section 5 has been modified). A short version is to appear in the
proceedings of the conference LICS 2004 (IEEE Computer Society Press
A lambda calculus for quantum computation with classical control
The objective of this paper is to develop a functional programming language
for quantum computers. We develop a lambda calculus for the classical control
model, following the first author's work on quantum flow-charts. We define a
call-by-value operational semantics, and we give a type system using affine
intuitionistic linear logic. The main results of this paper are the safety
properties of the language and the development of a type inference algorithm.Comment: 15 pages, submitted to TLCA'05. Note: this is basically the work done
during the first author master, his thesis can be found on his webpage.
Modifications: almost everything reformulated; recursion removed since the
way it was stated didn't satisfy lemma 11; type inference algorithm added;
example of an implementation of quantum teleportation adde
A feasible algorithm for typing in Elementary Affine Logic
We give a new type inference algorithm for typing lambda-terms in Elementary
Affine Logic (EAL), which is motivated by applications to complexity and
optimal reduction. Following previous references on this topic, the variant of
EAL type system we consider (denoted EAL*) is a variant without sharing and
without polymorphism. Our algorithm improves over the ones already known in
that it offers a better complexity bound: if a simple type derivation for the
term t is given our algorithm performs EAL* type inference in polynomial time.Comment: 20 page
On an Intuitionistic Logic for Pragmatics
We reconsider the pragmatic interpretation of intuitionistic logic [21]
regarded as a logic of assertions and their justications and its relations with classical
logic. We recall an extension of this approach to a logic dealing with assertions
and obligations, related by a notion of causal implication [14, 45]. We focus on
the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on
polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the
S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic
that correctly represents the duality between intuitionistic and co-intuitionistic logic,
correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism
as a distributed calculus of coroutines is then used to give an operational
interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear
calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation
of linear co-intuitionism is given as in [9]. Also we remark that by extending the
language of intuitionistic logic we can express the notion of expectation, an assertion
that in all situations the truth of p is possible and that in a logic of expectations
the law of double negation holds. Similarly, extending co-intuitionistic logic, we can
express the notion of conjecture that p, dened as a hypothesis that in some situation
the truth of p is epistemically necessary
The ILLTP Library for Intuitionistic Linear Logic
Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems
The First-Order Hypothetical Logic of Proofs
The Propositional Logic of Proofs (LP) is a modal logic in which the modality â–ˇA is revisited as [​[t]​]​A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[​[t]​]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [​[t]​]​A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de ComputaciĂłn; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y TecnologĂa; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; Argentin
From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics
We extend to natural deduction the approach of Linear Nested Sequents and
2-sequents. Formulas are decorated with a spatial coordinate, which allows a
formulation of formal systems in the original spirit of natural
deduction---only one introduction and one elimination rule per connective, no
additional (structural) rule, no explicit reference to the accessibility
relation of the intended Kripke models. We give systems for the normal modal
logics from K to S4. For the intuitionistic versions of the systems, we define
proof reduction, and prove proof normalisation, thus obtaining a syntactical
proof of consistency. For logics K and K4 we use existence predicates
(following Scott) for formulating sound deduction rules
- …