1,641 research outputs found
A linear proof language for second-order intuitionistic linear logic
We present a polymorphic linear lambda-calculus as a proof language for
second-order intuitionistic linear logic. The calculus includes addition and
scalar multiplication, enabling the proof of a linearity result at the
syntactic level.Comment: 18 pages + appendix. arXiv admin note: text overlap with
arXiv:2201.1122
Call-by-value non-determinism in a linear logic type discipline
We consider the call-by-value lambda-calculus extended with a may-convergent
non-deterministic choice and a must-convergent parallel composition. Inspired
by recent works on the relational semantics of linear logic and non-idempotent
intersection types, we endow this calculus with a type system based on the
so-called Girard's second translation of intuitionistic logic into linear
logic. We prove that a term is typable if and only if it is converging, and
that its typing tree carries enough information to give a bound on the length
of its lazy call-by-value reduction. Moreover, when the typing tree is minimal,
such a bound becomes the exact length of the reduction
A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4
We propose a modal linear logic to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality, and show that it admits a sound translation from IS4. Through the Curry-Howard correspondence we further obtain a Geometry of Interaction Machine semantics of the modal lambda-calculus by Pfenning and Davies for staged computation
Weak Typed Boehm Theorem on IMLL
In the Boehm theorem workshop on Crete island, Zoran Petric called Statman's
``Typical Ambiguity theorem'' typed Boehm theorem. Moreover, he gave a new
proof of the theorem based on set-theoretical models of the simply typed lambda
calculus. In this paper, we study the linear version of the typed Boehm theorem
on a fragment of Intuitionistic Linear Logic. We show that in the
multiplicative fragment of intuitionistic linear logic without the
multiplicative unit 1 (for short IMLL) weak typed Boehm theorem holds. The
system IMLL exactly corresponds to the linear lambda calculus without
exponentials, additives and logical constants. The system IMLL also exactly
corresponds to the free symmetric monoidal closed category without the unit
object. As far as we know, our separation result is the first one with regard
to these systems in a purely syntactical manner.Comment: a few minor correction
A new graphical calculus of proofs
We offer a simple graphical representation for proofs of intuitionistic
logic, which is inspired by proof nets and interaction nets (two formalisms
originating in linear logic). This graphical calculus of proofs inherits good
features from each, but is not constrained by them. By the Curry-Howard
isomorphism, the representation applies equally to the lambda calculus,
offering an alternative diagrammatic representation of functional computations.Comment: In Proceedings TERMGRAPH 2011, arXiv:1102.226
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
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
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