scholarly journals Classical Automata on Promise Problems

2015 ◽  
Vol Vol. 17 no.2 (Automata, Logic and Semantics) ◽  
Author(s):  
Viliam Geffert ◽  
Abuzer Yakaryilmaz
Keyword(s):  
Classical Model ◽  
Automata Theory ◽  
Language Recognition ◽  
Quantum Automata ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Bounded Error ◽  
Deterministic Automata ◽  
Promise Problems ◽  
Alternating Automata

International audience Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.

scholarly journals Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

2020 ◽  
Vol 30 (1) ◽  
pp. 175-192
Author(s):  
NathanaËl Fijalkow
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Numbers ◽  
Regular Languages ◽  
Automata Theory ◽  
Seminal Paper ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Probabilistic Languages ◽  
Alternating Automata

Abstract This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

Descriptional Complexity of the Forever Operator

2019 ◽  
Vol 30 (01) ◽  
pp. 115-134 ◽  
Author(s):  
Michal Hospodár ◽  
Galina Jirásková ◽  
Peter Mlynárčik
Keyword(s):  
Regular Language ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Descriptional Complexity ◽  
Trade Offs ◽  
Number Formula ◽  
Deterministic Automata ◽  
Initial States ◽  
Nondeterministic State

We examine the descriptional complexity of the forever operator, which assigns the language [Formula: see text] to a regular language [Formula: see text], and we investigate the trade-offs between various models of finite automata. We consider complete and partial deterministic finite automata, nondeterministic finite automata with single or multiple initial states, alternating, and Boolean finite automata. We assume that the argument and the result of this operation are accepted by automata belonging to one of these six models. We investigate all possible trade-offs and provide a tight upper bound for 32 of 36 of them. The most interesting result is the trade-off from nondeterministic to deterministic automata given by the Dedekind number [Formula: see text]. We also prove that the nondeterministic state complexity of [Formula: see text] is [Formula: see text] which solves an open problem stated by Birget [The state complexity of [Formula: see text] and its connection with temporal logic, Inform. Process. Lett. 58 (1996) 185–188].

scholarly journals Superiority of exact quantum automata for promise problems

2012 ◽  
Vol 112 (7) ◽  
pp. 289-291 ◽  
Author(s):  
Andris Ambainis ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Quantum Automata ◽  
Exact Quantum ◽  
Promise Problems

scholarly journals Why These Automata Types?

10.29007/c3bj ◽  
2018 ◽  
Author(s):  
Udi Boker
Keyword(s):  
Polynomial Time ◽  
Single Type ◽  
Natural Question ◽  
Boolean Operations ◽  
Standard Type ◽  
Complexity Bounds ◽  
Infinite Words ◽  
Deterministic Automata ◽  
Alternating Automata ◽  
Exponential Size

There are various types of automata on infinite words, differing in their acceptance conditions. The most classic ones are weak, Bu ̈chi, co-Bu ̈chi, parity, Rabin, Streett, and Muller. This is opposed to the case of automata on finite words, in which there is only one standard type. The natural question is why—Why not a single type? Why these particular types? Shall we further look into additional types?For answering these questions, we clarify the succinctness of the different automata types and the size blowup involved in performing boolean operations on them. To this end, we show that unifying or intersecting deterministic automata of the classic ω-regular- complete types, namely parity, Rabin, Streett, and Muller, involves an exponential size blowup.We argue that there are good reasons for the classic types, mainly in the case of nondeterministic and alternating automata. They admit good size and complexity bounds with respect to succinctness, boolean operations, and decision procedures, and they are closely connected to various logics.Yet, we also argue that there is place for additional types, especially in the case of deterministic automata. In particular, generalized-Rabin, which was recently introduced, as well as a disjunction of Streett conditions, which we call hyper-Rabin, where the latter further generalizes the former, are interesting to consider. They may be exponentially more succinct than the classic types, they allow for union and intersection with only a quadratic size blowup, and their nonemptiness can be checked in polynomial time.

scholarly journals Unary probabilistic and quantum automata on promise problems

2017 ◽  
Vol 17 (2) ◽  
Author(s):  
Aida Gainutdinova ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Quantum Automata ◽  
Promise Problems

scholarly journals Two Extensions of Cover Automata

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 338
Author(s):  
Cezar Câmpeanu
Keyword(s):  
Computational Complexity ◽  
Finite Automata ◽  
The State ◽  
Finite Cover ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Compact Representations ◽  
Minimization Algorithms ◽  
Deterministic Automata ◽  
Representation Transformation

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA.

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