921 research outputs found
A Correspondence between Maximal Abelian Sub-Algebras and Linear Logic Fragments
We show a correspondence between a classification of maximal abelian
sub-algebras (MASAs) proposed by Jacques Dixmier and fragments of linear logic.
We expose for this purpose a modified construction of Girard's hyperfinite
geometry of interaction which interprets proofs as operators in a von Neumann
algebra. The expressivity of the logic soundly interpreted in this model is
dependent on properties of a MASA which is a parameter of the interpretation.
We also unveil the essential role played by MASAs in previous geometry of
interaction constructions
Unification and Logarithmic Space
We present an algebraic characterization of the complexity classes Logspace
and NLogspace, using an algebra with a composition law based on unification.
This new bridge between unification and complexity classes is inspired from
proof theory and more specifically linear logic and Geometry of Interaction.
We show how unification can be used to build a model of computation by means
of specific subalgebras associated to finite permutations groups. We then prove
that whether an observation (the algebraic counterpart of a program) accepts a
word can be decided within logarithmic space. We also show that the
construction can naturally represent pointer machines, an intuitive way of
understanding logarithmic space computing
-Conformal Algebras
-conformal algebra is an axiomatic description of the operator
product expansion of chiral fields with simple poles at finitely many points.
We classify these algebras and their representations in terms of Lie algebras
and their representations with an action of the group . To every
-conformal algebra and a character of we associate a Lie
algebra generated by fields with the OPE with simple poles. Examples include
twisted affine Kac-Moody algebras, the sin algebra (which is a
``-conformal'' analodue of the general linear algebra) and its
analogues, the algebra of pseudodifferential operators on the circle, etc.Comment: 23 pages, AMSLatex Repotr-no: ITEP-TH-28/9
Memoization for Unary Logic Programming: Characterizing PTIME
We give a characterization of deterministic polynomial time computation based
on an algebraic structure called the resolution semiring, whose elements can be
understood as logic programs or sets of rewriting rules over first-order terms.
More precisely, we study the restriction of this framework to terms (and logic
programs, rewriting rules) using only unary symbols. We prove it is complete
for polynomial time computation, using an encoding of pushdown automata. We
then introduce an algebraic counterpart of the memoization technique in order
to show its PTIME soundness. We finally relate our approach and complexity
results to complexity of logic programming. As an application of our
techniques, we show a PTIME-completeness result for a class of logic
programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201
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