scholarly journals A coalgebraic take on regular and $\omega$-regular behaviours

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Tomasz Brengos
Keyword(s):  
Tree Automata ◽  
Probabilistic Automata ◽  
Acceptance Condition

We present a general coalgebraic setting in which we define finite and infinite behaviour with B\"uchi acceptance condition for systems whose type is a monad. The first part of the paper is devoted to presenting a construction of a monad suitable for modelling (in)finite behaviour. The second part of the paper focuses on presenting the concepts of a (coalgebraic) automaton and its ($\omega$-) behaviour. We end the paper with coalgebraic Kleene-type theorems for ($\omega$-) regular input. The framework is instantiated on non-deterministic (B\"uchi) automata, tree automata and probabilistic automata.

Progress measures, immediate determinacy, and a subset construction for tree automata

1992 ◽  
Vol 21 (430) ◽  
Author(s):  
Nils Klarlund
Keyword(s):  
Blow Up ◽  
Fundamental Result ◽  
Tree Automata ◽  
Infinite Games ◽  
Infinite Words ◽  
Graph Theoretic ◽  
The Past ◽  
Double Exponential ◽  
Acceptance Condition ◽  
Better Than

<p>Using the concept of progress measure, we give a new proof of Rabin's fundamental result that the languages defined by tree automata are closed under complementation.</p><p>To do this we show that for certain infinite games based on tree automata, an <em>immediate determinacy</em> property holds for the player who is trying to win according to a Rabin acceptance condition. Immediate determinacy is stronger than the <em> forgetful determinacy</em> of Gurevich and Harrington, which depends on more information about the past, but applies to another class of games.</p><p>Next, we show a graph theoretic duality theorem for winning conditions. Finally, we present an extended version of Safra's determinization construction. Together, these ingredients and the determinacy of Borel games yield a straightforward recipe for complementing tree automata.</p><p>Our construction is almost optimal, i.e. the state space blow-up is essentially exponential --- thus roughly the same as for automata on finite or infinite words.</p><p>To our knowledge, no prior constructions have been better than double exponential.</p>

On the Strength of Unambiguous Tree Automata

2018 ◽  
Vol 29 (05) ◽  
pp. 911-933
Author(s):  
Henryk Michalewski ◽  
Michał Skrzypczak
Keyword(s):  
Index Formula ◽  
Separation Procedure ◽  
Tree Automata ◽  
Descriptive Complexity ◽  
Infinite Trees ◽  
Acceptance Condition ◽  
Deterministic Automata ◽  
The One ◽  
Büchi Automaton ◽  
Parity Condition

This work is a study of the class of non-deterministic automata on infinite trees that are unambiguous i.e. have at most one accepting run on every tree. The motivating question asks if the fact that an automaton is unambiguous implies some drop in the descriptive complexity of the language recognised by the automaton. As it turns out, such a drop occurs for the parity index and does not occur for the weak parity index.More precisely, given an unambiguous parity automaton [Formula: see text] of index [Formula: see text], we show how to construct an alternating automaton [Formula: see text] which accepts the same language, but is simpler in terms of the acceptance condition. In particular, if [Formula: see text] is a Büchi automaton ([Formula: see text]) then [Formula: see text] is a weak alternating automaton. In general, [Formula: see text] belongs to the class [Formula: see text], what implies that it is simultaneously of alternating index [Formula: see text] and of the dual index [Formula: see text]. The transformation algorithm is based on a separation procedure of Arnold and Santocanale (2005).In the case of non-deterministic automata with the weak parity condition, we provide a separation procedure analogous to the one used above. However, as illustrated by examples, this separation procedure cannot be used to prove a complexity drop in the weak case, as there is no such drop.

Projection for Büchi Tree Automata with Constraints between Siblings

2020 ◽  
Vol 31 (06) ◽  
pp. 749-775
Author(s):  
Patrick Landwehr ◽  
Christof Löding
Keyword(s):  
Decision Procedure ◽  
Order Logic ◽  
Tree Automata ◽  
Boolean Operations ◽  
Regular Tree ◽  
Infinite Trees ◽  
Acceptance Condition ◽  
Emptiness Problem ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We consider an extension of tree automata on infinite trees that can use equality and disequality constraints between direct subtrees of a node. Recently, it has been shown that the emptiness problem for these kind of automata with a parity acceptance condition is decidable and that the corresponding class of languages is closed under Boolean operations. In this paper, we show that the class of languages recognizable by such tree automata with a Büchi acceptance condition is closed under projection. This construction yields a new algorithm for the emptiness problem, implies that a regular tree is accepted if the language is non-empty (for the Büchi condition), and can be used to obtain a decision procedure for an extension of monadic second-order logic with predicates for subtree comparisons.

Multiple Regularity and Binary ETOL-Systems1

1981 ◽  
Vol 4 (1) ◽  
pp. 19-34
Author(s):  
Ryszard Danecki
Keyword(s):  
Equivalence Problem ◽  
Tree Automata ◽  
Closure Properties ◽  
Multiple Tree

Closure properties of binary ETOL-languages are investigated by means of multiple tree automata. Decidability of the equivalence problem of deterministic binary ETOL-systems is proved.

Some Structural Properties of Systolic Tree Automata

1989 ◽  
Vol 12 (4) ◽  
pp. 571-585
Author(s):  
E. Fachini ◽  
A. Maggiolo Schettini ◽  
G. Resta ◽  
D. Sangiorgi
Keyword(s):  
Structural Properties ◽  
Stability Problem ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient

We prove that the classes of languages accepted by systolic automata over t-ary trees (t-STA) are always either equal or incomparable if one varies t. We introduce systolic tree automata with base (T(b)-STA), a subclass of STA with interesting properties of modularity, and we give a necessary and sufficient condition for the equivalence between a T(b)-STA and a t-STA, for a given base b. Finally, we show that the stability problem for T(b)-ST A is decidible.

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.

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