In this work we introduce new generalised quantifiers which allow us to express the Rabin-Mostowski index of automata. Our main results study expressive power and decidability of the monadic second-order (MSO) logic extended with these quantifiers. We study these problems in the realm of both ω-words and infinite trees. As it turns out, the pictures in these two cases are very different. In the case of ω-words the new quantifiers can be effectively expressed in pure MSO logic. In contrast, in the case of infinite trees, addition of these quantifiers leads to an undecidable formalism. To realise index-quantifier elimination, we consider the extension of MSO by game quantifiers. As a tool, we provide a specific quantifier-elimination procedure for them. Moreover, we introduce a novel construction of transducers realising strategies in ω-regular games with monadic parameters.

We focus on formulae ∃X.φ(Y, X) of monadic second-order logic over the full binary tree, such that the witness X is a well-founded set. The ordinal rank rank(X) < ω₁ of such a set X measures its depth and branching structure. We search for the least upper bound for these ranks, and discover the following dichotomy depending on the formula φ. Let η_φ be the minimal ordinal such that, whenever an instance Y satisfies the formula, there is a witness X with rank(X) ≤ η_φ. Then η_φ is either strictly smaller than ω² or it reaches the maximal possible value ω₁. Moreover, it is decidable which of the cases holds. The result has potential for applications in a variety of ordinal-related problems, in particular it entails a result about the closure ordinal of a fixed-point formula.

The Rabin tree theorem yields an algorithm to solve the satisfiability problem for monadic second-order logic over infinite trees. Here we solve the probabilistic variant of this problem. Namely, we show how to compute the probability that a randomly chosen tree satisfies a given formula. We additionally show that this probability is an algebraic number. This closes a line of research where similar results were shown for formalisms weaker than the full monadic second-order logic.

Daniel Simson left us unexpectedly on the 16th of April 2022.He served as editor of Fundamenta Informaticae since 2011.An eminent mathematician, he made a lasting contribution to modern algebra, in particular by his work on Grothendieck categories.Since the last two decades, Simson showed a vivid interest in mathematical challenges of computer science.His own work concentrated on symbolic algorithms issuing from algebra and spectral analysis of graphs, but he also animated a group of young mathematicians working in the area.His role in Fundamenta Informaticae was invaluable for his unlimited competence in mathematics, continuous readiness to help, and perfect manners.For several generations of Polish mathematicians, Professor Daniel Simson embodied the highest values of academic work.

Calude, Jain, Khoussainov, Li, and Stephan (2017) proposed a quasi-polynomial-time algorithm solving parity games. After this breakthrough result, a few other quasi-polynomial-time algorithms were introduced; none of them is easy to understand. Moreover, it turns out that in practice they operate very slowly. On the other side there is Zielonka’s recursive algorithm, which is very simple, exponential in the worst case, and the fastest in practice. We combine these two approaches: we propose a small modification of Zielonka’s algorithm, which ensures that the running time is at most quasi-polynomial. In effect, we obtain a simple algorithm that solves parity games in quasi-polynomial time. We also hope that our algorithm, after further optimizations, can lead to an algorithm that shares the good performance of Zielonka’s algorithm on typical inputs, while reducing the worst-case complexity on difficult inputs.

We study guidable parity automata over infinite trees introduced by Colcombet and Löding, which form an expressively complete subclass of all non-deterministic tree automata. We show that, for any non-deterministic automaton, an equivalent guidable automaton with the smallest possible index can be effectively found. Moreover, if an input automaton is of a special kind, i.e. it is deterministic or game automaton then a guidable automaton with an optimal index can be deterministic (respectively game) automaton as well. Recall that the problem whether an equivalent non-deterministic automaton with the smallest possible index can be effectively found is open, and a positive answer is known only in the case when an input automaton is a deterministic, or more generally, a game automaton.

University in Poland, 4-6 July 2017.It was inaugurated in 2000 at the 14th Czech and Slovak International Conference on Number Theory in Liptovski Jan, Slovak Republic, with a special crypto session and since then has been organized in a selected Central European country every year.The aim of the CECC is to gather people involved in cryptology.The talks cover a wide range of topics including symmetric or asymmetric algorithms and protocols, as well as practical and theoretical aspects of computer security and computational number theory.This edition gathered over 50 persons from Europe and all the world.11 articles related to applied cryptography were published in the Int.J. Electronics and Telecom.v. 64 n. 2, (2018) and nine more theoretical papers were submitted for the special conference volume of Fundamenta Informaticae.They were then subjected to a 3 round review process by the journal experts.Ultimately, 5 papers were selected to be published in this special conference volume.The issues of these works include the following topics: encryptions, hashing, randomness, curves and compression.We wish to express our deep appreciation to the authors for their contributions and to the reviewers for their careful, insightful and constructive reviews.

We show that the separation property fails for the classes Σ n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Σ 2 (i.e., for co-Büchi sets). The non-separation result is also adapted to the analogous classes induced by weak alternating automata.To prove our main result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Π n and fails for Σ n -classes. The construction invented for words turns out to be useful for trees via a suitable game.It remains open if the separation property holds for all classes Π n of the index hierarchy for tree automata. To give a positive answer it would be enough to show the reduction property of the dual classes—a method well-known in descriptive set theory. We show that it cannot work here, because the reduction property fails for all classes in the index hierarchy.

Report on the Ackermann Award 2013.

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game.

The Central European Conferences on Cryptology (CECC) were inaugurated in 2000 at the 14th Czech and Slovak International Conference on Number Theory in Liptovský Ján, Slovak Republic with a special crypto session.Since 2001 the CECC have been organized every year in a selected Central European country: in the Slovak Republic (three times), in Hungary and the Czech Republic (twice each), and in Poland and Austria (once each).The jubilee 10th CECC was held in Poland in 2010.The conference was organized by the Stefan Banach

An infinite binaryword can be identified with a branch in the full binary tree. We consider sets of branches definable in monadic second-order logic over the tree, where we allow some extra monadic predicates on the nodes. We show that this class equals to the Boolean combinations of sets in the Borel class Σ $^0_2$ over the Cantor discontinuum. Note that the last coincides with the Borel complexity of ω-regular languages.

Beauty is truth, truth beauty -that is all Ye know on earth,

The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these sets (which amount to one up to an obvious renaming) are complete in the class of co-analytic sets. We show that they cannot be separated by any Borel set, hence {\em a fortiori\/} by any weakly definable set of trees. This settles a case left open by L. Santocanale and A. Arnold, who have thoroughly investigated the separation property within the $\mu $-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type $\Sigma^{\mu }_{n} $, starting from level $n=3$, while our result settles the missing case $n=2$.

The Horn /spl mu/-calculus is a logic programming language allowing arbitrary nesting of least and greatest fixed points. The Horn /spl mu/-programs can naturally express safety and liveness properties for reactive systems. We extend the set-based analysis of classical logic programs by mapping arbitrary /spl mu/-programs into "uniform" /spl mu/-programs. Our two main results are that uniform /spl mu/-programs express regular sets of trees and that emptiness for uniform /spl mu/-programs is EXPTIME-complete. Hence we have a nontrivial decidable relaxation for the Horn /spl mu/-calculus. In a different reading, the results express a kind of robustness of the notion of regularity: alternating Rabin tree automata preserve the same expressiveness and algorithmic complexity if we extend them with pushdown transition rules (in the same way Buchi extended word automata to canonical systems).

Abstract We show that no formula of first order logic using linear ordering and the logical relationy= 2xcan define the property that the size of a finite model is divisible by 3. This answers a long-standing question which may be of relevance to certain open problems in circuit complexity.

It is shown that no formula of first-order logic using linear ordering and the logical relation y=2x can define the property that the size of a finite model is divisible by 3. This answers a long-standing question that may be of relevance to certain open problems in circuit complexity.>

We analyze the computational feasibility of checking temporal integrity constraints formulated in some sublanguages of first-order temporal logic. Our results illustrate the impact of the quantification on the complexity of this problem. The presence of a single quantifier in the scope of a temporal operator makes the problem undecidable. On the other hand, if no quantifiers are in the scope of a temporal operator and all the quantifiers are universal, temporal integrity checking can be done in exponential time.

The author characterizes Rabin definability (see M.O. Rabin, 1969) of properties of infinite trees of fixed-point definitions based on the basic operations of a standard powerset algebra of trees and involving the least and greatest fixed-point operators as well as the finite union operator and functional composition. A strict connection is established between a hierarchy resulting from alternating the least and greatest fixed-point operators and the hierarchy induced by Rabin indices of automata. The characterization result is actually proved on a more general level, namely, for arbitrary powerset algebra, where the concept of Rabin automaton is replaced by the more general concept of infinite grammar.>