184 research outputs found
A Recursion-Theoretic Characterization of the Probabilistic Class PP
Probabilistic complexity classes, despite capturing the notion of feasibility, have escaped any treatment by the tools of so-called implicit-complexity. Their inherently semantic nature is of course a barrier to the characterization of classes like BPP or ZPP, but not all classes are semantic. In this paper, we introduce a recursion-theoretic characterization of the probabilistic class PP, using recursion schemata with pointers
A Recursion-Theoretic Characterization of the Probabilistic Class PP
Probabilistic complexity classes, despite capturing the notion of feasibility, have escaped any treatment by the tools of so-called implicit-complexity. Their inherently semantic nature is of course a barrier to the characterization of classes like BPP or ZPP, but not all classes are semantic. In this paper, we introduce a recursion-theoretic characterization of the probabilistic class PP, using recursion schemata with pointers
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
Logic Programming and Logarithmic Space
We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation
The Safe Lambda Calculus
Safety is a syntactic condition of higher-order grammars that constrains
occurrences of variables in the production rules according to their
type-theoretic order. In this paper, we introduce the safe lambda calculus,
which is obtained by transposing (and generalizing) the safety condition to the
setting of the simply-typed lambda calculus. In contrast to the original
definition of safety, our calculus does not constrain types (to be
homogeneous). We show that in the safe lambda calculus, there is no need to
rename bound variables when performing substitution, as variable capture is
guaranteed not to happen. We also propose an adequate notion of beta-reduction
that preserves safety. In the same vein as Schwichtenberg's 1976
characterization of the simply-typed lambda calculus, we show that the numeric
functions representable in the safe lambda calculus are exactly the
multivariate polynomials; thus conditional is not definable. We also give a
characterization of representable word functions. We then study the complexity
of deciding beta-eta equality of two safe simply-typed terms and show that this
problem is PSPACE-hard. Finally we give a game-semantic analysis of safety: We
show that safe terms are denoted by `P-incrementally justified strategies'.
Consequently pointers in the game semantics of safe lambda-terms are only
necessary from order 4 onwards
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our preservation theorem states that,
over an aleph-zero categorical structure, a relation is positive Horn definable
if and only if it is preserved by all periomorphisms of the structure. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
Does Treewidth Help in Modal Satisfiability?
Many tractable algorithms for solving the Constraint Satisfaction Problem
(CSP) have been developed using the notion of the treewidth of some graph
derived from the input CSP instance. In particular, the incidence graph of the
CSP instance is one such graph. We introduce the notion of an incidence graph
for modal logic formulae in a certain normal form. We investigate the
parameterized complexity of modal satisfiability with the modal depth of the
formula and the treewidth of the incidence graph as parameters. For various
combinations of Euclidean, reflexive, symmetric and transitive models, we show
either that modal satisfiability is FPT, or that it is W[1]-hard. In
particular, modal satisfiability in general models is FPT, while it is
W[1]-hard in transitive models. As might be expected, modal satisfiability in
transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1:
improved section 5 to avoid exponential blow-up in formula siz
A generic imperative language for polynomial time
The ramification method in Implicit Computational Complexity has been
associated with functional programming, but adapting it to generic imperative
programming is highly desirable, given the wider algorithmic applicability of
imperative programming. We introduce a new approach to ramification which,
among other benefits, adapts readily to fully general imperative programming.
The novelty is in ramifying finite second-order objects, namely finite
structures, rather than ramifying elements of free algebras. In so doing we
bridge between Implicit Complexity's type theoretic characterizations of
feasibility, and the data-flow approach of Static Analysis.Comment: 18 pages, submitted to a conferenc
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