Integer Vector Addition Systems with States

This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ZVASS) and extensions and restrictions thereof. A ZVASS comprises a finite-state controller with a finite number of counters ranging over the integers. Although it is folklore that reachability in ZVASS is NP-complete, it turns out that despite their naturalness, from a complexity point of view this class has received little attention in the literature. We fill this gap by providing an in-depth analysis of the computational complexity of the aforementioned decision problems. Most interestingly, it turns out that while the addition of reset operations to ordinary VASS leads to undecidability and Ackermann-hardness of reachability and coverability, respectively, they can be added to ZVASS while retaining NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure

Quantum, Stochastic, and Pseudo Stochastic Languages with Few States

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.Comment: A new version with new results. Previous version: Arseny M. Shur, Abuzer Yakaryilmaz: Quantum, Stochastic, and Pseudo Stochastic Languages with Few States. UCNC 2014: 327-33

A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory

The submonoid and rational subset membership problems for graph groups

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids. We also show that the rational subset membership problem is decidable for a graph group if and only if the independence graph is a transitive forest, answering a question of Kambites, Silva, and the second author. Finally we prove that for certain amalgamated free products and HNN-extensions the rational subset and submonoid membership problems are recursively equivalent. In particular, this applies to finitely generated groups with two or more ends that are either torsion-free or residually finite