2,367 research outputs found
Minimizing finite automata is computationally hard
It is known that deterministic finite automata (DFAs) can be algorithmically minimized, i.e., a DFA M can be converted to an equivalent DFA M' which has a minimal number of states. The minimization can be done efficiently [6]. On the other hand, it is known that unambiguous finite automata (UFAs) and nondeterministic finite automata (NFAs) can be algorithmically minimized too, but their minimization problems turn out to be NP-complete and PSPACE-complete [8]. In this paper, the time complexity of the minimization problem for two restricted types of finite automata is investigated. These automata are nearly deterministic, since they only allow a small amount of non determinism to be used. On the one hand, NFAs with a fixed finite branching are studied, i.e., the number of nondeterministic moves within every accepting computation is bounded by a fixed finite number. On the other hand, finite automata are investigated which are essentially deterministic except that there is a fixed number of different initial states which can be chosen nondeterministically. The main result is that the minimization problems for these models are computationally hard, namely NP-complete. Hence, even the slightest extension of the deterministic model towards a nondeterministic one, e.g., allowing at most one nondeterministic move in every accepting computation or allowing two initial states instead of one, results in computationally intractable minimization problems
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
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
LTLf and LDLf Synthesis under Partial Observability
In this paper, we study synthesis under partial observability for logical specifications over finite traces expressed in LTLf/LDLf. This form of synthesis can be seen as a generalization of planning under partial observability in nondeterministic domains, which is known to be 2EXPTIME-complete. We start by showing that the usual "belief-state construction" used in planning under partial observability works also for general LTLf/LDLf synthesis, though with a jump in computational complexity from 2EXPTIME to 3EXPTIME. Then we show that the belief-state construction can be avoided in favor of a direct automata construction which exploits projection to hide unobservable propositions. This allow us to prove that the problem remains 2EXPTIME-complete. The new synthesis technique proposed is effective and readily implementable
Can Nondeterminism Help Complementation?
Complementation and determinization are two fundamental notions in automata
theory. The close relationship between the two has been well observed in the
literature. In the case of nondeterministic finite automata on finite words
(NFA), complementation and determinization have the same state complexity,
namely Theta(2^n) where n is the state size. The same similarity between
determinization and complementation was found for Buchi automata, where both
operations were shown to have 2^\Theta(n lg n) state complexity. An intriguing
question is whether there exists a type of omega-automata whose determinization
is considerably harder than its complementation. In this paper, we show that
for all common types of omega-automata, the determinization problem has the
same state complexity as the corresponding complementation problem at the
granularity of 2^\Theta(.).Comment: In Proceedings GandALF 2012, arXiv:1210.202
MeGARA: Menu-based Game Abstraction and Abstraction Refinement of Markov Automata
Markov automata combine continuous time, probabilistic transitions, and
nondeterminism in a single model. They represent an important and powerful way
to model a wide range of complex real-life systems. However, such models tend
to be large and difficult to handle, making abstraction and abstraction
refinement necessary. In this paper we present an abstraction and abstraction
refinement technique for Markov automata, based on the game-based and
menu-based abstraction of probabilistic automata. First experiments show that a
significant reduction in size is possible using abstraction.Comment: In Proceedings QAPL 2014, arXiv:1406.156
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
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
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