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.

scholarly journals Coinductive Algorithms for Büchi Automata

2021 ◽  
Vol 180 (4) ◽  
pp. 351-373
Author(s):  
Denis Kuperberg ◽  
Laureline Pinault ◽  
Damien Pous
Keyword(s):  
State Space ◽  
State Of The Art ◽  
The State ◽  
Specific Structure ◽  
Büchi Automata ◽  
Deterministic Automata ◽  
Buchi Automata ◽  
Language Equivalence ◽  
Art Techniques

We propose a new algorithm for checking language equivalence of non-deterministic Büchi automata. We start from a construction proposed by Calbrix, Nivat and Podelski, which makes it possible to reduce the problem to that of checking equivalence of automata on finite words. Although this construction generates large and highly non-deterministic automata, we show how to exploit their specific structure and apply state-of-the art techniques based on coinduction to reduce the state-space that has to be explored. Doing so, we obtain algorithms which do not require full determinisation or complementation.

scholarly journals Synthesizing Good-Enough Strategies for LTLf Specifications

2021 ◽  
Author(s):  
Yong Li ◽  
Andrea Turrini ◽  
Moshe Y. Vardi ◽  
Lijun Zhang
Keyword(s):  
Temporal Logic ◽  
Linear Temporal Logic ◽  
Synthesis Algorithm ◽  
Infinite Words ◽  
Büchi Automata ◽  
Winning Strategies ◽  
Deterministic Automata ◽  
Computationally Expensive ◽  
Buchi Automata

We consider the problem of synthesizing good-enough (GE)-strategies for linear temporal logic (LTL) over finite traces or LTLf for short. The problem of synthesizing GE-strategies for an LTL formula φ over infinite traces reduces to the problem of synthesizing winning strategies for the formula (∃Oφ)⇒φ where O is the set of propositions controlled by the system. We first prove that this reduction does not work for LTLf formulas. Then we show how to synthesize GE-strategies for LTLf formulas via the Good-Enough (GE)-synthesis of LTL formulas. Unfortunately, this requires to construct deterministic parity automata on infinite words, which is computationally expensive. We then show how to synthesize GE-strategies for LTLf formulas by a reduction to solving games played on deterministic Büchi automata, based on an easier construction of deterministic automata on finite words. We show empirically that our specialized synthesis algorithm for GE-strategies outperforms the algorithms going through GE-synthesis of LTL formulas by orders of magnitude.

scholarly journals BÜCHI COMPLEMENTATION MADE TIGHTER

2006 ◽  
Vol 17 (04) ◽  
pp. 851-867 ◽  
Author(s):  
EHUD FRIEDGUT ◽  
ORNA KUPFERMAN ◽  
MOSHE Y. VARDI
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Blow Up ◽  
Careful Analysis ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Theoretical Point ◽  
Infinite Words ◽  
Büchi Automata ◽  
Buchi Automata

The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems is reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2n blow-up that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. From a theoretical point of view, the problem is considered solved since 1988, when Safra came up with a determinization construction for Büchi automata, leading to a 2O(n log n) complementation construction, and Michel came up with a matching lower bound. A careful analysis, however, of the exact blow-up in Safra's and Michel's bounds reveals an exponential gap in the constants hiding in the O( ) notations: while the upper bound on the number of states in Safra's complementary automaton is n2n, Michel's lower bound involves only an n! blow up, which is roughly (n/e)n. The exponential gap exists also in more recent complementation constructions. In particular, the upper bound on the number of states in the complementation construction of Kupferman and Vardi, which avoids determinization, is (6n)n. This is in contrast with the case of automata on finite words, where the upper and lower bounds coincides. In this work we describe an improved complementation construction for nondeterministic Büchi automata and analyze its complexity. We show that the new construction results in an automaton with at most (0.96n)n states. While this leaves the problem about the exact blow up open, the gap is now exponentially smaller. From a practical point of view, our solution enjoys the simplicity of the construction of Kupferman and Vardi, and results in much smaller automata.

scholarly journals The Determinacy of Context-Free Games

2013 ◽  
Vol 78 (4) ◽  
pp. 1115-1134 ◽  
Author(s):  
Olivier Finkel
Keyword(s):  
Set Theory ◽  
Winning Strategy ◽  
Large Cardinal ◽  
Büchi Automata ◽  
Büchi Automaton ◽  
Buchi Automata ◽  
Context Free

AbstractWe prove that the determinacy of Gale-Stewart games whose winning sets are accepted by realtime 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge ofω-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automatonand a Büchi automatonsuch that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge gameW(L(),L()); (2) There exists a model of ZFC in which the Wadge gameW(L(),L()) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge gameW(L(),L()).

scholarly journals From LTL to unambiguous Büchi automata via disambiguation of alternating automata

2021 ◽  
Author(s):  
Simon Jantsch ◽  
David Müller ◽  
Christel Baier ◽  
Joachim Klein
Keyword(s):  
Model Checking ◽  
Markov Chains ◽  
Important Application ◽  
High Complexity ◽  
The Core ◽  
Büchi Automata ◽  
Deterministic Automata ◽  
Tool Set ◽  
Buchi Automata ◽  
Alternating Automata

AbstractDue to the high complexity of translating linear temporal logic (LTL) to deterministic automata, several forms of “restricted” nondeterminism have been considered with the aim of maintaining some of the benefits of deterministic automata, while at the same time allowing more efficient translations from LTL. One of them is the notion of unambiguity. This paper proposes a new algorithm for the generation of unambiguous Büchi automata (UBA) from LTL formulas. Unlike other approaches it is based on a known translation from very weak alternating automata (VWAA) to NBA. A notion of unambiguity for alternating automata is introduced and it is shown that the VWAA-to-NBA translation preserves unambiguity. Checking unambiguity of VWAA is determined to be PSPACE-complete, both for the explicit and symbolic encodings of alternating automata. The core of the LTL-to-UBA translation is an iterative disambiguation procedure for VWAA. Several heuristics are introduced for different stages of the procedure. We report on an implementation of our approach in the tool and compare it to an existing LTL-to-UBA implementation in the tool set. Our experiments cover model checking of Markov chains, which is an important application of UBA.

scholarly journals TYPENESS FOR ω-REGULAR AUTOMATA

2006 ◽  
Vol 17 (04) ◽  
pp. 869-883 ◽  
Author(s):  
ORNA KUPFERMAN ◽  
GILA MORGENSTERN ◽  
ANIELLO MURANO
Keyword(s):  
Blow Up ◽  
The Other ◽  
Complete Picture ◽  
Infinite Words ◽  
Other Hand ◽  
Büchi Automata ◽  
Acceptance Condition ◽  
Buchi Automata

We introduce and study three notions of typeness for automata on infinite words. For an acceptance-condition class γ (that is, γ is weak, Büchi, co-Büchi, Rabin, or Streett), deterministic γ-typeness asks for the existence of an equivalent γ-automaton on the same deterministic structure, nondeterministic γ-typeness asks for the existence of an equivalent γ-automaton on the same structure, and γ-powerset-typeness asks for the existence of an equivalent γ-automaton on the (deterministic) powerset structure – one obtained by applying the subset construction. The notions are helpful in studying the complexity and complication of translations between the various classes of automata. For example, we prove that deterministic Büchi automata are co-Büchi type; it follows that a translation from deterministic Büchi to deterministic co-Büchi automata, when exists, involves no blow up. On the other hand, we prove that nondeterministic Büchi automata are not co-Büchi type; it follows that a translation from a nondeterministic Büchi to nondeterministic co-Büchi automata, when exists, should be more complicated than just redefining the acceptance condition. As a third example, by proving that nondeterministic co-Büchi automata are Büchi-powerset type, we show that a translation of nondeterministic co-Büchi to deterministic Büchi automata, when exists, can be done applying the subset construction. We give a complete picture of typeness for the weak, Büchi, co-Büchi, Rabin, and Streett acceptance conditions, and discuss its usefulness.

ON WEIGHTED BÜCHI AUTOMATA WITH ORDER-COMPLETE WEIGHTS

2007 ◽  
Vol 17 (02) ◽  
pp. 235-260 ◽  
Author(s):  
MANFRED DROSTE ◽  
ULRIKE PÜSCHMANN
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Finite Automata ◽  
Infinite Words ◽  
Büchi Automata ◽  
Buchi Automata ◽  
Weighted Finite Automata ◽  
Formal Power

We investigate Büchi automata with weights for the transitions. Assuming that the weights are taken in a suitable ordered semiring, we show how to define the behaviors of these automata on infinite words. Our main result shows that the formal power series arising in this way are precisely the ones which can be constructed using ω-rational operations. This extends the classical Kleene–Schützenberger result for weighted finite automata to the case of infinite words and generalizes Büchi's theorem on languages of infinite words. We also derive versions of our main result for non-complete semirings and for other automata models.

scholarly journals Ambiguity, Weakness, and Regularity in Probabilistic Büchi Automata

2020 ◽  
pp. 522-541
Author(s):  
Christof Löding ◽  
Anton Pirogov
Keyword(s):  
Natural Generalization ◽  
Regular Languages ◽  
Natural Class ◽  
Regularity Problem ◽  
Infinite Words ◽  
Büchi Automata ◽  
Buchi Automata

AbstractProbabilistic Büchi automata are a natural generalization of PFA to infinite words, but have been studied in-depth only rather recently and many interesting questions are still open. PBA are known to accept, in general, a class of languages that goes beyond the regular languages. In this work we extend the known classes of restricted PBA which are still regular, strongly relying on notions concerning ambiguity in classical $$\omega $$ ω -automata. Furthermore, we investigate the expressivity of the not yet considered but natural class of weak PBA, and we also show that the regularity problem for weak PBA is undecidable.

scholarly journals PARTIALLY ORDERED TWO-WAY BÜCHI AUTOMATA

2011 ◽  
Vol 22 (08) ◽  
pp. 1861-1876 ◽  
Author(s):  
MANFRED KUFLEITNER ◽  
ALEXANDER LAUSER
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
Inclusion Problem ◽  
Intermediate Step ◽  
First Order ◽  
Partially Ordered ◽  
Büchi Automata ◽  
Emptiness Problem ◽  
Small Model ◽  
Buchi Automata

We introduce partially ordered two-way Büchi automata and characterize their expressive power in terms of fragments of first-order logic FO[<]. Partially ordered two-way Büchi automata are Büchi automata which can change the direction in which the input is processed with the constraint that whenever a state is left, it is never re-entered again. Nondeterministic partially ordered two-way Büchi automata coincide with the first-order fragment Σ2. Our main contribution is that deterministic partially ordered two-way Büchi automata are expressively complete for the first-order fragment Δ2. As an intermediate step, we show that deterministic partially ordered two-way Büchi automata are effectively closed under Boolean operations. A small model property yields coNP-completeness of the emptiness problem and the inclusion problem for deterministic partially ordered two-way Büchi automata.

On the Development of Islamic Jurisprudence

1999 ◽  
Vol 16 (1) ◽  
Author(s):  
Murad Wilfried Hofmann
Keyword(s):  
The State ◽  
Current State ◽  
Islamic Jurisprudence ◽  
The Status ◽  
Sensitive Issues ◽  
Status Of Women ◽  
Islam And Democracy

This article examines the state of Islamic jurisprudence with regard to many sensitive issues, such as the status of women and minorities in Islam, Islam and Democracy, hudud punishments. The author explores the current state of Islamic discourse on jurisprudence and identifies three approaches-traditional, secular and reformist. The paper explores the positions of the traditional ulama and the reformist muj­tahids on the mentioned topics and finds the reformist position more sensible and closer to the position of ihe Qur'an and Sunnah. This paper while advocating neo-ijtihad, makes an impressive case for the merit???? and Islamic credibility of the reformist jurisprudence.

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