130 research outputs found
Frameworks for logically classifying polynomial-time optimisation problems.
We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems
Randomisation and Derandomisation in Descriptive Complexity Theory
We study probabilistic complexity classes and questions of derandomisation
from a logical point of view. For each logic L we introduce a new logic BPL,
bounded error probabilistic L, which is defined from L in a similar way as the
complexity class BPP, bounded error probabilistic polynomial time, is defined
from PTIME. Our main focus lies on questions of derandomisation, and we prove
that there is a query which is definable in BPFO, the probabilistic version of
first-order logic, but not in Cinf, finite variable infinitary logic with
counting. This implies that many of the standard logics of finite model theory,
like transitive closure logic and fixed-point logic, both with and without
counting, cannot be derandomised. Similarly, we present a query on ordered
structures which is definable in BPFO but not in monadic second-order logic,
and a query on additive structures which is definable in BPFO but not in FO.
The latter of these queries shows that certain uniform variants of AC0
(bounded-depth polynomial sized circuits) cannot be derandomised. These results
are in contrast to the general belief that most standard complexity classes can
be derandomised. Finally, we note that BPIFP+C, the probabilistic version of
fixed-point logic with counting, captures the complexity class BPP, even on
unordered structures
Counter Machines and Distributed Automata: A Story about Exchanging Space and Time
We prove the equivalence of two classes of counter machines and one class of
distributed automata. Our counter machines operate on finite words, which they
read from left to right while incrementing or decrementing a fixed number of
counters. The two classes differ in the extra features they offer: one allows
to copy counter values, whereas the other allows to compute copyless sums of
counters. Our distributed automata, on the other hand, operate on directed path
graphs that represent words. All nodes of a path synchronously execute the same
finite-state machine, whose state diagram must be acyclic except for
self-loops, and each node receives as input the state of its direct
predecessor. These devices form a subclass of linear-time one-way cellular
automata.Comment: 15 pages (+ 13 pages of appendices), 5 figures; To appear in the
proceedings of AUTOMATA 2018
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
Computing with cells: membrane systems - some complexity issues.
Membrane computing is a branch of natural computing which abstracts computing models from the structure and the functioning of the living cell. The main ingredients of membrane systems, called P systems, are (i) the membrane structure, which consists of a hierarchical arrangements of membranes which delimit compartments where (ii) multisets of symbols, called objects, evolve according to (iii) sets of rules which are localised and associated with compartments. By using the rules in a nondeterministic/deterministic maximally parallel manner, transitions between the system configurations can be obtained. A sequence of transitions is a computation of how the system is evolving. Various ways of controlling the transfer of objects from one membrane to another and applying the rules, as well as possibilities to dissolve, divide or create membranes have been studied. Membrane systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems once future biotechnology gives way to a practical bio-realization. In this paper we survey some interesting and fundamental complexity issues such as universality vs. nonuniversality, determinism vs. nondeterminism, membrane and alphabet size hierarchies, characterizations of context-sensitive languages and other language classes and various notions of parallelism
Translation from Classical Two-Way Automata to Pebble Two-Way Automata
We study the relation between the standard two-way automata and more powerful
devices, namely, two-way finite automata with an additional "pebble" movable
along the input tape. Similarly as in the case of the classical two-way
machines, it is not known whether there exists a polynomial trade-off, in the
number of states, between the nondeterministic and deterministic pebble two-way
automata. However, we show that these two machine models are not independent:
if there exists a polynomial trade-off for the classical two-way automata, then
there must also exist a polynomial trade-off for the pebble two-way automata.
Thus, we have an upward collapse (or a downward separation) from the classical
two-way automata to more powerful pebble automata, still staying within the
class of regular languages. The same upward collapse holds for complementation
of nondeterministic two-way machines.
These results are obtained by showing that each pebble machine can be, by
using suitable inputs, simulated by a classical two-way automaton with a linear
number of states (and vice versa), despite the existing exponential blow-up
between the classical and pebble two-way machines
Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata
After an apparent hiatus of roughly 30 years, we revisit a seemingly
neglected subject in the theory of (one-dimensional) cellular automata:
sublinear-time computation. The model considered is that of ACAs, which are
language acceptors whose acceptance condition depends on the states of all
cells in the automaton. We prove a time hierarchy theorem for sublinear-time
ACA classes, analyze their intersection with the regular languages, and,
finally, establish strict inclusions in the parallel computation classes
and (uniform) . As an addendum, we introduce and
investigate the concept of a decider ACA (DACA) as a candidate for a decider
counterpart to (acceptor) ACAs. We show the class of languages decidable in
constant time by DACAs equals the locally testable languages, and we also
determine as the (tight) time complexity threshold for DACAs
up to which no advantage compared to constant time is possible.Comment: 16 pages, 2 figures, to appear at DLT 202
Computer aided synthesis: a game theoretic approach
In this invited contribution, we propose a comprehensive introduction to game
theory applied in computer aided synthesis. In this context, we give some
classical results on two-player zero-sum games and then on multi-player non
zero-sum games. The simple case of one-player games is strongly related to
automata theory on infinite words. All along the article, we focus on general
approaches to solve the studied problems, and we provide several illustrative
examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language
Theory" (DLT 2017
Recent Advances in Σ-definability over Continuous Data Types
The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas
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