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 Copyful Streaming String Transducers

2021 ◽  
Vol 178 (1-2) ◽  
pp. 59-76
Author(s):  
Emmanuel Filiot ◽  
Pierre-Alain Reynier
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Expressive Power ◽  
The Other ◽  
Finite State Automata ◽  
Finite State ◽  
Variable Content ◽  
Deterministic Automata ◽  
Intermediate Output ◽  
Linear Manner

Copyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Černý in 2010 as a one-way deterministic automata model to define transductions of finite strings. Copyless SST extend deterministic finite state automata with a set of variables in which to store intermediate output strings, and those variables can be combined and updated all along the run, in a linear manner, i.e., no variable content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string transductions, and are as expressive as deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decidable equivalence problem (in PSpace). On the other hand, HDT0L systems have been introduced for a while, the most prominent result being the decidability of the equivalence problem. In this paper, we propose a semantics of HDT0L systems in terms of transductions, and use it to study the class of deterministic copyful SST. Our contributions are as follows: (i)HDT0L systems and total deterministic copyful SST have the same expressive power, (ii)the equivalence problem for deterministic copyful SST and the equivalence problem for HDT0L systems are inter-reducible, in quadratic time. As a consequence, equivalence of deterministic SST is decidable, (iii)the functionality of non-deterministic copyful SST is decidable, (iv)determining whether a non-deterministic copyful SST can be transformed into an equivalent non-deterministic copyless SST is decidable in polynomial time.

scholarly journals Geometry and complexity of O'Hara's algorithm

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Matjaž Konvalinka ◽  
Igor Pak
Keyword(s):  
Polynomial Time ◽  
Finite Support ◽  
Type Result ◽  
Complexity Bounds ◽  
International Audience

International audience In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we see that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we present a number of new complexity bounds, proving that O'Hara's bijection is efficient in most cases and mildly exponential in general. Finally, we see that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.

scholarly journals Minimization of Limit-Average Automata

2021 ◽  
Author(s):  
Jakub Michaliszyn ◽  
Jan Otop
Keyword(s):  
Polynomial Time ◽  
Equivalence Relation ◽  
Minimization Problem ◽  
Value Function ◽  
Equivalence Classes ◽  
Minimization Algorithm ◽  
Infinite Words ◽  
Average Value ◽  
Weighted Automata

LimAvg-automata are weighted automata over infinite words that aggregate weights along runs with the limit-average value function. In this paper, we study the minimization problem for (deterministic) LimAvg-automata. Our main contribution is an equivalence relation on words characterizing LimAvg-automata, i.e., the equivalence classes of this relation correspond to states of an equivalent LimAvg-automaton. In contrast to relations characterizing DFA, our relation depends not only on the function defined by the target automaton, but also on its structure. We show two applications of this relation. First, we present a minimization algorithm for LimAvg-automata, which returns a minimal LimAvg-automaton among those equivalent and structurally similar to the input one. Second, we present an extension of Angluin's L^*-algorithm with syntactic queries, which learns in polynomial time a LimAvg-automaton equivalent to the target one.

FAMILIES OF NONCOUNTING LANGUAGES AND THEIR LEARNABILITY FROM POSITIVE DATA

1996 ◽  
Vol 07 (04) ◽  
pp. 309-327 ◽  
Author(s):  
SATOSHI KOBAYASHI ◽  
TAKASHI YOKOMORI
Keyword(s):  
Polynomial Time ◽  
Close Relation ◽  
Boolean Operations ◽  
Positive Data ◽  
Language Classes ◽  
Identification In The Limit ◽  
The Relationship

This paper introduces some subclasses of noncounting languages and presents some results on the learnability of the classes from positive data. We first establish several relationships among the language classes introduced and the class of reversible languages. Especially, we introduce the notion of local parsability, and define a class (k, l)-CLTS, which is a subclass of the class of concatenations of strictly locally testable languages. We show its close relation to the class of reversible languages. We then study on the relationship between the closure of the Boolean operations and the learnability in the limit from positive data only. Further, we explore the learnability question of some subclasses of noncounting languages in the model of identification in the limit from positive data. In particular, we show that, for each k, l≥1, (k, l)-CLTS is identifiable in the limit from positive data using reversible automata with the conjectures updated in polynomial time. Some possible applications of the result are also briefly discussed.

scholarly journals On hereditary Helly classes of graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Marina Groshaus ◽  
Jayme Luiz Szwarcfiter
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Natural Question ◽  
Fixed Size ◽  
Forbidden Subgraphs ◽  
Induced Subgraph ◽  
Helly Property ◽  
International Audience

Graphs and Algorithms International audience In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.

scholarly journals Computational Collapse of Quantum State with Application to Oblivious Transfer

2003 ◽  
Vol 10 (37) ◽  
Author(s):  
Claude Crépeau ◽  
Paul Dumais ◽  
Dominic Mayers ◽  
Louis Salvail
Keyword(s):  
Quantum State ◽  
Polynomial Time ◽  
Quantum Measurement ◽  
Oblivious Transfer ◽  
Black Box ◽  
The State ◽  
Natural Question ◽  
Full Information ◽  
Classical Counterpart ◽  
Commitment Scheme

Quantum 2-party cryptography differs from its classical counterpart in at least one important way: Given black-box access to a perfect commitment scheme there exists a secure 1-2 <em>quantum</em> oblivious transfer. This reduction proposed by Crépeau and Kilian was proved secure against any receiver by Yao, in the case where perfect commitments are used. However, quantum commitments would normally be based on computational assumptions. A natural question therefore arises: What happens to the security of the above reduction when computationally secure commitments are used instead of perfect ones?<br /> <br />In this paper, we address the security of 1-2 QOT when computationally binding string commitments are available. In particular, we analyse the security of a primitive called <em>Quantum Measurement Commitment</em> when it is constructed from unconditionally concealing but computationally binding commitments. As measuring a quantum state induces an irreversible collapse, we describe a QMC as an instance of ``computational collapse of a quantum state''. In a QMC a state appears to be collapsed to a polynomial time observer who cannot extract full information about the state without breaking a computational assumption.<br /> <br />We reduce the security of QMC to a <em>weak</em> binding criteria for the string commitment. We also show that <em>secure</em> QMCs implies QOT using a straightforward variant of the reduction above.

scholarly journals Certifying Inexpressibility

2021 ◽  
pp. 385-405
Author(s):  
Orna Kupferman ◽  
Salomon Sickert
Keyword(s):  
Winning Strategy ◽  
Expressive Power ◽  
The State ◽  
Infinite Words ◽  
Current State ◽  
Büchi Automata ◽  
Deterministic Automata ◽  
Buchi Automata

AbstractDifferent classes of automata on infinite words have different expressive power. Deciding whether a given language$$L \subseteq \varSigma ^\omega $$L⊆Σωcan be expressed by an automaton of a desired class can be reduced to deciding a game between Prover and Refuter: in each turn of the game, Refuter provides a letter in$$\varSigma $$Σ, and Prover responds with an annotation of the current state of the run (for example, in the case of Büchi automata, whether the state is accepting or rejecting, and in the case of parity automata, what the color of the state is). Prover wins if the sequence of annotations she generates is correct: it is an accepting run iff the word generated by Refuter is inL. We show how a winning strategy for Refuter can serve as a simple and easy-to-understand certificate to inexpressibility, and how it induces additional forms of certificates. Our framework handles all classes of deterministic automata, including ones with structural restrictions like weak automata. In addition, it can be used for refutingseparationof two languages by an automaton of the desired class, and for finding automata thatapproximateLand belong to the desired class.

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