6 research outputs found
Linear-algebraic lambda-calculus
With a view towards models of quantum computation and/or the interpretation
of linear logic, we define a functional language where all functions are linear
operators by construction. A small step operational semantic (and hence an
interpreter/simulator) is provided for this language in the form of a term
rewrite system. The linear-algebraic lambda-calculus hereby constructed is
linear in a different (yet related) sense to that, say, of the linear
lambda-calculus. These various notions of linearity are discussed in the
context of quantum programming languages. KEYWORDS: quantum lambda-calculus,
linear lambda-calculus, -calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language
interpreter/simulator file (see "other formats"). See the more recent
arXiv:quant-ph/061219
On the Most Suitable Axiomatization of Signed Integers
Part 4: Regular PapersInternational audienceThe standard mathematical definition of signed integers, based on set theory, is not well-adapted to the needs of computer science. For this reason, many formal specification languages and theorem provers have designed alternative definitions of signed integers based on term algebras , by extending the Peano-style construction of unsigned naturals using "zero" and "succ" to the case of signed integers. We compare the various approaches used in CADP, CASL, Coq, Isabelle/HOL, KIV, Maude, mCRL2, PSF, SMT-LIB, TLA+, etc. according to objective criteria and suggest an "optimal" definition of signed integers
Lineal: A linear-algebraic lambda-calculus
International audienceWe provide a computational de nition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the lambda-calculus with the possibility to make arbitrary linear combinations of terms : alpha t + beta u. We describe how to \execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the eld of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus
Lineal: A linear-algebraic Lambda-calculus
We provide a computational definition of the notions of vector space and
bilinear functions. We use this result to introduce a minimal language
combining higher-order computation and linear algebra. This language extends
the Lambda-calculus with the possibility to make arbitrary linear combinations
of terms alpha.t + beta.u. We describe how to "execute" this language in terms
of a few rewrite rules, and justify them through the two fundamental
requirements that the language be a language of linear operators, and that it
be higher-order. We mention the perspectives of this work in the field of
quantum computation, whose circuits we show can be easily encoded in the
calculus. Finally, we prove the confluence of the entire calculus.Comment: The complementary note "On the critical pairs of a rewrite system for
vector spaces" is provided in the source files. Short version :
"Linear-algebraic Lambda-calculus : higher-order and confluence", Proceedings
of RTA 08, Hagenberg, July 2008. LNCS 5117, 17, (2008). Long version : LMC
Lineal: A linear-algebraic lambda-calculus
International audienceWe provide a computational de nition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the lambda-calculus with the possibility to make arbitrary linear combinations of terms : alpha t + beta u. We describe how to \execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the eld of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus
An efficient representation of arithmetic for term rewriting
We give a locally confluent set of rewrite rules for integer (positive and negative) arithmetic using the familiar system of place notation. We are unable to prove its termination at present, but we strongly conjecture that rewriting with this system terminates and give our reasons. We show that every term has a normal form and so the rewrite system is normalising. We justify our choice of representation in terms of both space efficiency and speed of rewriting. Finally we give several examples of the use of our system