Automata with Modulo Counters and Nondeterministic Counter Bounds

We introduce and investigate Nondeterministically Bounded Modulo Counter Automata (NBMCA), which are two-way multi-head automata that comprise a constant number of modulo counters, where the counter bounds are nondeterministically guessed, and this is the only element of nondeterminism. NBMCA are tailored to recognising those languages that are characterised by the existence of a specific factorisation of their words, e. g., pattern languages. In this work, we subject NBMCA to a theoretically sound analysis

Automata with modulo counters and nondeterministic counter bounds

We introduce and investigate Nondeterministically Bounded Modulo Counter Automata (NBMCA), which are two-way one-head automata that comprise a constant number of modulo counters, where the counter bounds are nondeterministically guessed, and this is the only element of nondeterminism. NBMCA are tailored to recognising those languages that are characterised by the existence of a specific factorisation of their words, e. g., pattern languages. In this work, we subject NBMCA to a theoretically sound analysis

Reachability in Higher-Order-Counters

Higher-order counter automata (\HOCS) can be either seen as a restriction of higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of \HOCS: those that can test whether the topmost counter value is zero and those which cannot. We show that control-state reachability for level $k$ \HOCS with $0$-test is complete for \mbox{$(k-2)$}-fold exponential space; leaving out the $0$-test leads to completeness for \mbox{$(k-2)$}-fold exponential time. Restricting \HOCS (without $0$-test) to level $2$, we prove that global (forward or backward) reachability analysis is \PTIME-complete. This enhances the known result for pushdown systems which are subsumed by level $2$ \HOCS without $0$-test. We transfer our results to the formal language setting. Assuming that \PTIME \subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet's constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201

Bounded Counter Languages

We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters

On the Equivalence of Cellular Automata and the Tile Assembly Model

In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which every cell location is allowed to change state an infinite number of times) versus the static nature of the aTAM (in which tiles are static components that can never change or be removed once they attach to a growing assembly). We work with 2-dimensional systems in both models, and for our results we first define what it means for CA systems to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We use notions of simulate which are similar to those used in the study of intrinsic universality since they are in some sense strict, but also intuitively natural notions of simulation. We then demonstrate a particular nondeterministic CA which can be configured so that it can simulate any arbitrary aTAM system, and finally an aTAM tile set which can be configured so that it can be used to simulate any arbitrary nondeterministic CA system which begins with a finite initial configuration.Comment: In Proceedings MCU 2013, arXiv:1309.104

On the membership problem for pattern languages and related topics

In this thesis, we investigate the complexity of the membership problem for pattern languages. A pattern is a string over the union of the alphabets A and X, where X := {x_1, x_2, x_3, ...} is a countable set of variables and A is a finite alphabet containing terminals (e.g., A := {a, b, c, d}). Every pattern, e.g., p := x_1 x_2 a b x_2 b x_1 c x_2, describes a pattern language, i.e., the set of all words that can be obtained by uniformly substituting the variables in the pattern by arbitrary strings over A. Hence, u := cacaaabaabcaccaa is a word of the pattern language of p, since substituting cac for x_1 and aa for x_2 yields u. On the other hand, there is no way to obtain the word u' := bbbababbacaaba by substituting the occurrences of x_1 and x_2 in p by words over A. The problem to decide for a given pattern q and a given word w whether or not w is in the pattern language of q is called the membership problem for pattern languages. Consequently, (p, u) is a positive instance and (p, u') is a negative instance of the membership problem for pattern languages. For the unrestricted case, i.e., for arbitrary patterns and words, the membership problem is NP-complete. In this thesis, we identify classes of patterns for which the membership problem can be solved efficiently. Our first main result in this regard is that the variable distance, i.e., the maximum number of different variables that separate two consecutive occurrences of the same variable, substantially contributes to the complexity of the membership problem for pattern languages. More precisely, for every class of patterns with a bounded variable distance the membership problem can be solved efficiently. The second main result is that the same holds for every class of patterns with a bounded scope coincidence degree, where the scope coincidence degree is the maximum number of intervals that cover a common position in the pattern, where each interval is given by the leftmost and rightmost occurrence of a variable in the pattern. The proof of our first main result is based on automata theory. More precisely, we introduce a new automata model that is used as an algorithmic framework in order to show that the membership problem for pattern languages can be solved in time that is exponential only in the variable distance of the corresponding pattern. We then take a closer look at this automata model and subject it to a sound theoretical analysis. The second main result is obtained in a completely different way. We encode patterns and words as relational structures and we then reduce the membership problem for pattern languages to the homomorphism problem of relational structures, which allows us to exploit the concept of the treewidth. This approach turns out be successful, and we show that it has potential to identify further classes of patterns with a polynomial time membership problem. Furthermore, we take a closer look at two aspects of pattern languages that are indirectly related to the membership problem. Firstly, we investigate the phenomenon that patterns can describe regular or context-free languages in an unexpected way, which implies that their membership problem can be solved efficiently. In this regard, we present several sufficient conditions and necessary conditions for the regularity and context-freeness of pattern languages. Secondly, we compare pattern languages with languages given by so-called extended regular expressions with backreferences (REGEX). The membership problem for REGEX languages is very important in practice and since REGEX are similar to pattern languages, it might be possible to improve algorithms for the membership problem for REGEX languages by investigating their relationship to patterns. In this regard, we investigate how patterns can be extended in order to describe large classes of REGEX languages

History-Register Automata

Programs with dynamic allocation are able to create and use an unbounded number of fresh resources, such as references, objects, files, etc. We propose History-Register Automata (HRA), a new automata-theoretic formalism for modelling such programs. HRAs extend the expressiveness of previous approaches and bring us to the limits of decidability for reachability checks. The distinctive feature of our machines is their use of unbounded memory sets (histories) where input symbols can be selectively stored and compared with symbols to follow. In addition, stored symbols can be consumed or deleted by reset. We show that the combination of consumption and reset capabilities renders the automata powerful enough to imitate counter machines, and yields closure under all regular operations apart from complementation. We moreover examine weaker notions of HRAs which strike different balances between expressiveness and effectiveness.Comment: LMCS (improved version of FoSSaCS

Forward Analysis and Model Checking for Trace Bounded WSTS

We investigate a subclass of well-structured transition systems (WSTS), the bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete deterministic ones, which we claim provide an adequate basis for the study of forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth. Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered previously for the termination of forward analysis, boundedness is decidable. Boundedness turns out to be a valuable restriction for WSTS verification, as we show that it further allows to decide all $\omega$-regular properties on the set of infinite traces of the system