12 research outputs found
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
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
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 rooted in proof
theory and more specifically linear logic and geometry of interaction. We show
how to build a model of computation in the unification algebra and then, by
means of a syntactic representation of finite permutations in the algebra, we
prove that whether an observation (the algebraic counterpart of a program)
accepts a word can be decided within logarithmic space. Finally, we show that
the construction naturally corresponds to pointer machines, a convenient way of
understanding logarithmic space computation.Comment: arXiv admin note: text overlap with arXiv:1402.432
Unification and Logarithmic Space: Journal Version
Soumis au numéro spécial de LMCS pour RTA/TLCA 2014 ( http://www.lmcs-online.org/ojs/specialIssues.php?id=67 )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 rooted in proof theory and more specifically linear logic and geometry of interaction. We show how to build a model of computation in the unification algebra and then, by means of a syntactic representation of finite permutations in the algebra, we prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. Finally, we show that the construction naturally corresponds to pointer machines, an convenient way of understanding logarithmic space computation
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
On the Resolution Semiring
In this thesis, we study a semiring structure with a product based on theresolution rule of logic programming. This mathematical object was introducedinitially in the setting of the geometry of interaction program in order to modelthe cut-elimination procedure of linear logic. It provides us with an algebraicand abstract setting, while being presented in a syntactic and concrete way, inwhich a theoretical study of computation can be carried on.We will review first the interactive interpretation of proof theory withinthis semiring via the categorical axiomatization of the geometry of interactionapproach. This interpretation establishes a way to translate functional programsinto a very simple form of logic programs.Secondly, complexity theory problematics will be considered: while thenilpotency problem in the semiring we study is undecidable in general, it willappear that certain restrictions allow for characterizations of (deterministicand non-deterministic) logarithmic space and (deterministic) polynomial timecomputation
Logarithmic space and permutations
In a recent work, Girard [1] proposed a new and innovative approach to computational complexity based on the proofs-as-programs correspondence. In a previous paper [2], the authors showed how Girard proposal succeeds in obtaining a new characterization of co-NL languages as a set of operators acting on a Hilbert Space. In this paper, we extend this work by showing that it is also possible to define a set of operators characterizing the class L of logarithmic space languages