1,286 research outputs found
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
Multiplicative-Additive Proof Equivalence is Logspace-complete, via Binary Decision Trees
Given a logic presented in a sequent calculus, a natural question is that of
equivalence of proofs: to determine whether two given proofs are equated by any
denotational semantics, ie any categorical interpretation of the logic
compatible with its cut-elimination procedure. This notion can usually be
captured syntactically by a set of rule permutations.
Very generally, proofnets can be defined as combinatorial objects which
provide canonical representatives of equivalence classes of proofs. In
particular, the existence of proof nets for a logic provides a solution to the
equivalence problem of this logic. In certain fragments of linear logic, it is
possible to give a notion of proofnet with good computational properties,
making it a suitable representation of proofs for studying the cut-elimination
procedure, among other things.
It has recently been proved that there cannot be such a notion of proofnets
for the multiplicative (with units) fragment of linear logic, due to the
equivalence problem for this logic being Pspace-complete.
We investigate the multiplicative-additive (without unit) fragment of linear
logic and show it is closely related to binary decision trees: we build a
representation of proofs based on binary decision trees, reducing proof
equivalence to decision tree equivalence, and give a converse encoding of
binary decision trees as proofs. We get as our main result that the complexity
of the proof equivalence problem of the studied fragment is Logspace-complete.Comment: arXiv admin note: text overlap with arXiv:1502.0199
Dialectica Categories for the Lambek Calculus
We revisit the old work of de Paiva on the models of the Lambek Calculus in
dialectica models making sure that the syntactic details that were sketchy on
the first version got completed and verified. We extend the Lambek Calculus
with a \kappa modality, inspired by Yetter's work, which makes the calculus
commutative. Then we add the of-course modality !, as Girard did, to
re-introduce weakening and contraction for all formulas and get back the full
power of intuitionistic and classical logic. We also present the categorical
semantics, proved sound and complete. Finally we show the traditional
properties of type systems, like subject reduction, the Church-Rosser theorem
and normalization for the calculi of extended modalities, which we did not have
before
Applications of Intuitionistic Logic in Answer Set Programming
We present some applications of intermediate logics in the field of Answer
Set Programming (ASP). A brief, but comprehensive introduction to the answer
set semantics, intuitionistic and other intermediate logics is given. Some
equivalence notions and their applications are discussed. Some results on
intermediate logics are shown, and applied later to prove properties of answer
sets. A characterization of answer sets for logic programs with nested
expressions is provided in terms of intuitionistic provability, generalizing a
recent result given by Pearce.
It is known that the answer set semantics for logic programs with nested
expressions may select non-minimal models. Minimal models can be very important
in some applications, therefore we studied them; in particular we obtain a
characterization, in terms of intuitionistic logic, of answer sets which are
also minimal models. We show that the logic G3 characterizes the notion of
strong equivalence between programs under the semantic induced by these models.
Finally we discuss possible applications and consequences of our results. They
clearly state interesting links between ASP and intermediate logics, which
might bring research in these two areas together.Comment: 30 pages, Under consideration for publication in Theory and Practice
of Logic Programmin
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