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

Boundedness in languages of infinite words

We define a new class of languages of $\omega$-words, strictly extending $\omega$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $\omega$-regular expressions where two new variants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression $(a^Bb)^\omega$ represents the language of infinite words over the letters $a,b$ where there is a common bound on the number of consecutive letters $a$. The expression $(a^Sb)^\omega$ represents a similar language, but this time the distance between consecutive $b$'s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending B\"uchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent---either $L^B$ or $L^S$---is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets

Digraph Complexity Measures and Applications in Formal Language Theory

We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.Comment: 19 pages, 1 figur

From LTL and Limit-Deterministic B\"uchi Automata to Deterministic Parity Automata

Controller synthesis for general linear temporal logic (LTL) objectives is a challenging task. The standard approach involves translating the LTL objective into a deterministic parity automaton (DPA) by means of the Safra-Piterman construction. One of the challenges is the size of the DPA, which often grows very fast in practice, and can reach double exponential size in the length of the LTL formula. In this paper we describe a single exponential translation from limit-deterministic B\"uchi automata (LDBA) to DPA, and show that it can be concatenated with a recent efficient translation from LTL to LDBA to yield a double exponential, \enquote{Safraless} LTL-to-DPA construction. We also report on an implementation, a comparison with the SPOT library, and performance on several sets of formulas, including instances from the 2016 SyntComp competition

Distributed Graph Automata and Verification of Distributed Algorithms

Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants of our automata for which the emptiness problem is decidable. As an application, we suggest how suitable graph automata might be useful in formal verification of distributed algorithms, using Floyd-Hoare logic.Comment: 26 pages, 6 figures, includes a condensed version of the author's Master's thesis arXiv:1404.6503. (This version of the article (v2) is identical to the previous one (v1), except for minor changes in phrasing.

Partially Ordered Two-way B\"uchi Automata

We introduce partially ordered two-way B\"uchi automata and characterize their expressive power in terms of fragments of first-order logic FO[<]. Partially ordered two-way B\"uchi automata are B\"uchi 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 B\"uchi automata coincide with the first-order fragment Sigma2. Our main contribution is that deterministic partially ordered two-way B\"uchi automata are expressively complete for the first-order fragment Delta2. As an intermediate step, we show that deterministic partially ordered two-way B\"uchi 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 B\"uchi automata.Comment: The results of this paper were presented at CIAA 2010; University of Stuttgart, Computer Scienc