Deciding the value 1 problem for probabilistic leaktight automata

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton, are there words accepted with probability arbitrarily close to 1? This problem was proved undecidable recently; to overcome this, several classes of probabilistic automata of different nature were proposed, for which the value 1 problem has been shown decidable. In this paper, we introduce yet another class of probabilistic automata, called leaktight automata, which strictly subsumes all classes of probabilistic automata whose value 1 problem is known to be decidable. We prove that for leaktight automata, the value 1 problem is decidable (in fact, PSPACE-complete) by constructing a saturation algorithm based on the computation of a monoid abstracting the behaviours of the automaton. We rely on algebraic techniques developed by Simon to prove that this abstraction is complete. Furthermore, we adapt this saturation algorithm to decide whether an automaton is leaktight. Finally, we show a reduction allowing to extend our decidability results from finite words to infinite ones, implying that the value 1 problem for probabilistic leaktight parity automata is decidable

On cascade products of answer set programs

Describing complex objects by elementary ones is a common strategy in mathematics and science in general. In their seminal 1965 paper, Kenneth Krohn and John Rhodes showed that every finite deterministic automaton can be represented (or "emulated") by a cascade product of very simple automata. This led to an elegant algebraic theory of automata based on finite semigroups (Krohn-Rhodes Theory). Surprisingly, by relating logic programs and automata, we can show in this paper that the Krohn-Rhodes Theory is applicable in Answer Set Programming (ASP). More precisely, we recast the concept of a cascade product to ASP, and prove that every program can be represented by a product of very simple programs, the reset and standard programs. Roughly, this implies that the reset and standard programs are the basic building blocks of ASP with respect to the cascade product. In a broader sense, this paper is a first step towards an algebraic theory of products and networks of nonmonotonic reasoning systems based on Krohn-Rhodes Theory, aiming at important open issues in ASP and AI in general.Comment: Appears in Theory and Practice of Logic Programmin

Modified Deep Pushdown Automata

Tato práce představuje dvě nové modifikace hlubokých zásobníkových automatů - bezestavové hluboké zásobníkové automaty a paralelní bezestavové hluboké zásobníkové automaty.V teoretické části jsou zavedeny formální definice a také je zde zkoumána síla těchto automatů. V pratické části je ukázána na jednoduchém příkladu implementace těchto automatů.This thesis introduce two new modifications of deep pushdown automata - stateless deep pushdown automata and parallel deep pushdown automata. In theoretical part of this thesis is formal definition and research into power of these modification. Practical part consists of an implementation of simple automata program.

What is known about the Value 1 Problem for Probabilistic Automata?

The value 1 problem is a decision problem for probabilistic automata over finite words: are there words accepted by the automaton with arbitrarily high probability? Although undecidable, this problem attracted a lot of attention over the last few years. The aim of this paper is to review and relate the results pertaining to the value 1 problem. In particular, several algorithms have been proposed to partially solve this problem. We show the relations between them, leading to the following conclusion: the Markov Monoid Algorithm is the most correct algorithm known to (partially) solve the value 1 problem

Pushing undecidability of the isolation problem for probabilistic automata

This short note aims at proving that the isolation problem is undecidable for probabilistic automata with only one probabilistic transition. This problem is known to be undecidable for general probabilistic automata, without restriction on the number of probabilistic transitions. In this note, we develop a simulation technique that allows to simulate any probabilistic automaton with one having only one probabilistic transition

IST Austria Technical Report

We consider probabilistic automata on infinite words with acceptance defined by parity conditions. We consider three qualitative decision problems: (i) the positive decision problem asks whether there is a word that is accepted with positive probability; (ii) the almost decision problem asks whether there is a word that is accepted with probability 1; and (iii) the limit decision problem asks whether for every ε > 0 there is a word that is accepted with probability at least 1 − ε. We unify and generalize several decidability results for probabilistic automata over infinite words, and identify a robust (closed under union and intersection) subclass of probabilistic automata for which all the qualitative decision problems are decidable for parity conditions. We also show that if the input words are restricted to lasso shape words, then the positive and almost problems are decidable for all probabilistic automata with parity conditions