Probabilistic regular graphs

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. We present here an algorithm to perform approximate verification of PCTL formulae. Moreover, we prove that the exact model-checking problem for PCTL on probabilistic regular graphs is undecidable, unless restricting to qualitative properties. Our results generalise those of EKM06, on probabilistic pushdown automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611

Almost Sure Productivity

We define Almost Sure Productivity (ASP), a probabilistic generalization of the productivity condition for coinductively defined structures. Intuitively, a probabilistic coinductive stream or tree is ASP if it produces infinitely many outputs with probability 1. Formally, we define almost sure productivity using a final coalgebra semantics of programs inspired from Kerstan and K\"onig. Then, we introduce a core language for probabilistic streams and trees, and provide two approaches to verify ASP: a sufficient syntactic criterion, and a reduction to model-checking pCTL* formulas on probabilistic pushdown automata. The reduction shows that ASP is decidable for our core language

Quantum counter automata

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.Comment: A revised version. 16 pages. A preliminary version of this paper appeared as A. C. Cem Say, Abuzer Yakary{\i}lmaz, and \c{S}efika Y\"{u}zsever. Quantum one-way one-counter automata. In R\={u}si\c{n}\v{s} Freivalds, editor, Randomized and quantum computation, pages 25--34, 2010 (Satellite workshop of MFCS and CSL 2010

Superiority of one-way and realtime quantum machines and new directions

In automata theory, the quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to the QFAs augmented with counters or stacks. Moreover, to our knowledge, there is no result related to QFAs having more than one input head. In this paper, we focus on such generalizations of QFAs whose input head(s) operate(s) in one-way or realtime mode and present many superiority of them to their classical counterparts. Furthermore, we propose some open problems and conjectures in order to investigate the power of quantumness better. We also give some new results on classical computation.Comment: A revised edition with some correction

New results on classical and quantum counter automata

We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.Comment: 21 page

Weighted One-Deterministic-Counter Automata

We introduce weighted one-deterministic-counter automata (ODCA). These are weighted one-counter automata (OCA) with the property of counter-determinacy, meaning that all paths labelled by a given word starting from the initial configuration have the same counter-effect. Weighted ODCAs are a strict extension of weighted visibly OCAs, which are weighted OCAs where the input alphabet determines the actions on the counter. We present a novel problem called the co-VS (complement to a vector space) reachability problem for weighted ODCAs over fields, which seeks to determine if there exists a run from a given configuration of a weighted ODCA to another configuration whose weight vector lies outside a given vector space. We establish two significant properties of witnesses for co-VS reachability: they satisfy a pseudo-pumping lemma, and the lexicographically minimal witness has a special form. It follows that the co-VS reachability problem is in P. These reachability problems help us to show that the equivalence problem of weighted ODCAs over fields is in P by adapting the equivalence proof of deterministic real-time OCAs by B\"ohm et al. This is a step towards resolving the open question of the equivalence problem of weighted OCAs. Furthermore, we demonstrate that the regularity problem, the problem of checking whether an input weighted ODCA over a field is equivalent to some weighted automaton, is in P. Finally, we show that the covering and coverable equivalence problems for uninitialised weighted ODCAs are decidable in polynomial time. We also consider boolean ODCAs and show that the equivalence problem for (non-deterministic) boolean ODCAs is in PSPACE, whereas it is undecidable for (non-deterministic) boolean OCAs.Comment: 36 pages, 11 figure