283 research outputs found
Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models
The -logic (which is called E-logic in this paper) of
Kuyper and Terwijn is a variant of first order logic with the same syntax, in
which the models are equipped with probability measures and in which the
quantifier is interpreted as "there exists a set of measure
such that for each , ...." Previously, Kuyper and
Terwijn proved that the general satisfiability and validity problems for this
logic are, i) for rational , respectively
-complete and -hard, and ii) for ,
respectively decidable and -complete. The adjective "general" here
means "uniformly over all languages."
We extend these results in the scenario of finite models. In particular, we
show that the problems of satisfiability by and validity over finite models in
E-logic are, i) for rational , respectively
- and -complete, and ii) for , respectively
decidable and -complete. Although partial results toward the countable
case are also achieved, the computability of E-logic over countable
models still remains largely unsolved. In addition, most of the results, of
this paper and of Kuyper and Terwijn, do not apply to individual languages with
a finite number of unary predicates. Reducing this requirement continues to be
a major point of research.
On the positive side, we derive the decidability of the corresponding
problems for monadic relational languages --- equality- and function-free
languages with finitely many unary and zero other predicates. This result holds
for all three of the unrestricted, the countable, and the finite model cases.
Applications in computational learning theory, weighted graphs, and neural
networks are discussed in the context of these decidability and undecidability
results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger
Kuype
Flowchart Programs, Regular Expressions, and Decidability of Polynomial Growth-Rate
We present a new method for inferring complexity properties for a class of
programs in the form of flowcharts annotated with loop information.
Specifically, our method can (soundly and completely) decide if computed values
are polynomially bounded as a function of the input; and similarly for the
running time. Such complexity properties are undecidable for a Turing-complete
programming language, and a common work-around in program analysis is to settle
for sound but incomplete solutions. In contrast, we consider a class of
programs that is Turing-incomplete, but strong enough to include several
challenges for this kind of analysis. For a related language that has
well-structured syntax, similar to Meyer and Ritchie's LOOP programs, the
problem has been previously proved to be decidable. The analysis relied on the
compositionality of programs, hence the challenge in obtaining similar results
for flowchart programs with arbitrary control-flow graphs. Our answer to the
challenge is twofold: first, we propose a class of loop-annotated flowcharts,
which is more general than the class of flowcharts that directly represent
structured programs; secondly, we present a technique to reuse the ideas from
the work on tructured programs and apply them to such flowcharts. The technique
is inspired by the classic translation of non-deterministic automata to regular
expressions, but we obviate the exponential cost of constructing such an
expression, obtaining a polynomial-time analysis. These ideas may well be
applicable to other analysis problems.Comment: In Proceedings VPT 2016, arXiv:1607.0183
The submonoid and rational subset membership problems for graph groups
We show that the membership problem in a finitely generated submonoid of a
graph group (also called a right-angled Artin group or a free partially
commutative group) is decidable if and only if the independence graph
(commutation graph) is a transitive forest. As a consequence we obtain the
first example of a finitely presented group with a decidable generalized word
problem that does not have a decidable membership problem for finitely
generated submonoids. We also show that the rational subset membership problem
is decidable for a graph group if and only if the independence graph is a
transitive forest, answering a question of Kambites, Silva, and the second
author. Finally we prove that for certain amalgamated free products and
HNN-extensions the rational subset and submonoid membership problems are
recursively equivalent. In particular, this applies to finitely generated
groups with two or more ends that are either torsion-free or residually finite
Splicing Systems from Past to Future: Old and New Challenges
A splicing system is a formal model of a recombinant behaviour of sets of
double stranded DNA molecules when acted on by restriction enzymes and ligase.
In this survey we will concentrate on a specific behaviour of a type of
splicing systems, introduced by P\u{a}un and subsequently developed by many
researchers in both linear and circular case of splicing definition. In
particular, we will present recent results on this topic and how they stimulate
new challenging investigations.Comment: Appeared in: Discrete Mathematics and Computer Science. Papers in
Memoriam Alexandru Mateescu (1952-2005). The Publishing House of the Romanian
Academy, 2014. arXiv admin note: text overlap with arXiv:1112.4897 by other
author
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