161 research outputs found
Regular realizability problems and context-free languages
We investigate regular realizability (RR) problems, which are the problems of
verifying whether intersection of a regular language -- the input of the
problem -- and fixed language called filter is non-empty. In this paper we
focus on the case of context-free filters. Algorithmic complexity of the RR
problem is a very coarse measure of context-free languages complexity. This
characteristic is compatible with rational dominance. We present examples of
P-complete RR problems as well as examples of RR problems in the class NL. Also
we discuss RR problems with context-free filters that might have intermediate
complexity. Possible candidates are the languages with polynomially bounded
rational indices.Comment: conference DCFS 201
Tight polynomial worst-case bounds for loop programs
In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid
Realizability and uniqueness in graphs
AbstractConsider a finite graph G(V,E). Let us associate to G a finite list P(G) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P, when is P=P(G) for some graph G?, (U) Uniqueness. Suppose P(G)=P(H) for graphs G and H. When does this imply G ≅ H? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs
Quantitative Models and Implicit Complexity
We give new proofs of soundness (all representable functions on base types
lies in certain complexity classes) for Elementary Affine Logic, LFPL (a
language for polytime computation close to realistic functional programming
introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs
are based on a common semantical framework which is merely instantiated in four
different ways. The framework consists of an innovative modification of
realizability which allows us to use resource-bounded computations as realisers
as opposed to including all Turing computable functions as is usually the case
in realizability constructions. For example, all realisers in the model for
LFPL are polynomially bounded computations whence soundness holds by
construction of the model. The work then lies in being able to interpret all
the required constructs in the model. While being the first entirely semantical
proof of polytime soundness for light logi cs, our proof also provides a
notable simplification of the original already semantical proof of polytime
soundness for LFPL. A new result made possible by the semantic framework is the
addition of polymorphism and a modality to LFPL thus allowing for an internal
definition of inductive datatypes.Comment: 29 page
Tight Polynomial Worst-Case Bounds for Loop Programs
In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple
programming language - representing non-deterministic imperative programs with
bounded loops, and arithmetics limited to addition and multiplication - it is
possible to decide precisely whether a program has certain growth-rate
properties, in particular whether a computed value, or the program's running
time, has a polynomial growth rate.
A natural and intriguing problem was to move from answering the decision
problem to giving a quantitative result, namely, a tight polynomial upper
bound. This paper shows how to obtain asymptotically-tight, multivariate,
disjunctive polynomial bounds for this class of programs. This is a complete
solution: whenever a polynomial bound exists it will be found.
A pleasant surprise is that the algorithm is quite simple; but it relies on
some subtle reasoning. An important ingredient in the proof is the forest
factorization theorem, a strong structural result on homomorphisms into a
finite monoid
Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions
We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an
arithmetical theory based on basic logic which is weaker than intuitionistic
logic. We show that the class of the provably recursive functions of BA is a
proper sub-class of primitive recursive functions. Three extensions of BA,
called BA+U, BA_c and EBA are investigated with relation to their provably
recursive functions. It is shown that the provably recursive functions of these
three extensions of BA are exactly primitive recursive functions. Moreover,
among other things, it is shown that the well-known MRDP theorem doesn't hold
in BA, BA+U, BA_c, but holds in EBA
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