470 research outputs found
Two-Way Automata Making Choices Only at the Endmarkers
The question of the state-size cost for simulation of two-way
nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was
raised in 1978 and, despite many attempts, it is still open. Subsequently, the
problem was attacked by restricting the power of 2DFAs (e.g., using a
restricted input head movement) to the degree for which it was already possible
to derive some exponential gaps between the weaker model and the standard
2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the
degree for which it is still possible to obtain a subexponential conversion
from the stronger model to the standard 2DFAs. In particular, it turns out that
subexponential conversion is possible for two-way automata that make
nondeterministic choices only when the input head scans one of the input tape
endmarkers. However, there is no restriction on the input head movement. This
implies that an exponential gap between 2NFAs and 2DFAs can be obtained only
for unrestricted 2NFAs using capabilities beyond the proposed new model. As an
additional bonus, conversion into a machine for the complement of the original
language is polynomial in this model. The same holds for making such machines
self-verifying, halting, or unambiguous. Finally, any superpolynomial lower
bound for the simulation of such machines by standard 2DFAs would imply LNL.
In the same way, the alternating version of these machines is related to L =?
NL =? P, the classical computational complexity problems.Comment: 23 page
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
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
The descriptive complexity approach to LOGCFL
Building upon the known generalized-quantifier-based first-order
characterization of LOGCFL, we lay the groundwork for a deeper investigation.
Specifically, we examine subclasses of LOGCFL arising from varying the arity
and nesting of groupoidal quantifiers. Our work extends the elaborate theory
relating monoidal quantifiers to NC1 and its subclasses. In the absence of the
BIT predicate, we resolve the main issues: we show in particular that no single
outermost unary groupoidal quantifier with FO can capture all the context-free
languages, and we obtain the surprising result that a variant of Greibach's
``hardest context-free language'' is LOGCFL-complete under quantifier-free
BIT-free projections. We then prove that FO with unary groupoidal quantifiers
is strictly more expressive with the BIT predicate than without. Considering a
particular groupoidal quantifier, we prove that first-order logic with majority
of pairs is strictly more expressive than first-order with majority of
individuals. As a technical tool of independent interest, we define the notion
of an aperiodic nondeterministic finite automaton and prove that FO
translations are precisely the mappings computed by single-valued aperiodic
nondeterministic finite transducers.Comment: 10 pages, 1 figur
Converting Nondeterministic Two-Way Automata into Small Deterministic Linear-Time Machines
In 1978 Sakoda and Sipser raised the question of the cost, in terms of size
of representations, of the transformation of two-way and one-way
nondeterministic automata into equivalent two-way deterministic automata.
Despite all the attempts, the question has been answered only for particular
cases (e.g., restrictions of the class of simulated automata or of the class of
simulating automata). However the problem remains open in the general case, the
best-known upper bound being exponential. We present a new approach in which
unrestricted nondeterministic finite automata are simulated by deterministic
models extending two-way deterministic finite automata, paying a polynomial
increase of size only. Indeed, we study the costs of the conversions of
nondeterministic finite automata into some variants of one-tape deterministic
Turing machines working in linear time, namely Hennie machines, weight-reducing
Turing machines, and weight-reducing Hennie machines. All these variants are
known to share the same computational power: they characterize the class of
regular languages
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
Classical and quantum Merlin-Arthur automata
We introduce Merlin-Arthur (MA) automata as Merlin provides a single
certificate and it is scanned by Arthur before reading the input. We define
Merlin-Arthur deterministic, probabilistic, and quantum finite state automata
(resp., MA-DFAs, MA-PFAs, MA-QFAs) and postselecting MA-PFAs and MA-QFAs
(resp., MA-PostPFA and MA-PostQFA). We obtain several results using different
certificate lengths.
We show that MA-DFAs use constant length certificates, and they are
equivalent to multi-entry DFAs. Thus, they recognize all and only regular
languages but can be exponential and polynomial state efficient over binary and
unary languages, respectively. With sublinear length certificates, MA-PFAs can
recognize several nonstochastic unary languages with cutpoint 1/2. With linear
length certificates, MA-PostPFAs recognize the same nonstochastic unary
languages with bounded error. With arbitrarily long certificates, bounded-error
MA-PostPFAs verify every unary decidable language. With sublinear length
certificates, bounded-error MA-PostQFAs verify several nonstochastic unary
languages. With linear length certificates, they can verify every unary
language and some NP-complete binary languages. With exponential length
certificates, they can verify every binary language.Comment: 14 page
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