Formal Languages in Dynamical Systems

We treat here the interrelation between formal languages and those dynamical systems that can be described by cellular automata (CA). There is a well-known injective map which identifies any CA-invariant subshift with a central formal language. However, in the special case of a symbolic dynamics, i.e. where the CA is just the shift map, one gets a stronger result: the identification map can be extended to a functor between the categories of symbolic dynamics and formal languages. This functor additionally maps topological conjugacies between subshifts to empty-string-limited generalized sequential machines between languages. If the periodic points form a dense set, a case which arises in a commonly used notion of chaotic dynamics, then an even more natural map to assign a formal language to a subshift is offered. This map extends to a functor, too. The Chomsky hierarchy measuring the complexity of formal languages can be transferred via either of these functors from formal languages to symbolic dynamics and proves to be a conjugacy invariant there. In this way it acquires a dynamical meaning. After reviewing some results of the complexity of CA-invariant subshifts, special attention is given to a new kind of invariant subshift: the trapped set, which originates from the theory of chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Maximal Sharing in the Lambda Calculus with letrec

Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec. We introduce a notion of `maximal compactness' for lambda-letrec-terms among all terms with the same infinite unfolding. Instead of defined purely syntactically, this notion is based on a graph semantics. lambda-letrec-terms are interpreted as first-order term graphs so that unfolding equivalence between terms is preserved and reflected through bisimilarity of the term graph interpretations. Compactness of the term graphs can then be compared via functional bisimulation. We describe practical and efficient methods for the following two problems: transforming a lambda-letrec-term into a maximally compact form; and deciding whether two lambda-letrec-terms are unfolding-equivalent. The transformation of a lambda-letrec-term $L$ into maximally compact form $L_0$ proceeds in three steps: (i) translate L into its term graph $G = [[ L ]]$; (ii) compute the maximally shared form of $G$ as its bisimulation collapse $G_0$; (iii) read back a lambda-letrec-term $L_0$ from the term graph $G_0$ with the property $[[ L_0 ]] = G_0$. This guarantees that $L_0$ and $L$ have the same unfolding, and that $L_0$ exhibits maximal sharing. The procedure for deciding whether two given lambda-letrec-terms $L_1$ and $L_2$ are unfolding-equivalent computes their term graph interpretations $[[ L_1 ]]$ and $[[ L_2 ]]$, and checks whether these term graphs are bisimilar. For illustration, we also provide a readily usable implementation.Comment: 18 pages, plus 19 pages appendi

Computability and Tiling Problems

In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles $S$ has total planar tilings, which we denote $TILE$, or whether it has infinite connected but not necessarily total tilings, $WTILE$ (short for `weakly tile'). We show that both $TILE \equiv_m ILL \equiv_m WTILE$, and thereby both $TILE$ and $WTILE$ are $\Sigma^1_1$-complete. We also show that the opposite problems, $\neg TILE$ and $SNT$ (short for `Strongly Not Tile') are such that $\neg TILE \equiv_m WELL \equiv_m SNT$ and so both $\neg TILE$ and $SNT$ are both $\Pi^1_1$-complete. Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets $PTile$ of periodic tilings, and $ATile$ of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form $(\Sigma^1_1 \wedge \Pi^1_1)$. We also present results for finite versions of these tiling problems. We then move on to consider the Weihrauch reducibility for a general total tiling principle $CT$ as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, $C_{\omega^\omega}$. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to $C_{\omega^\omega}$. Finally, we give a prototile set of 15 prototiles that can encode any Elementary Cellular Automaton (ECA). We make use of an unusual tile set, based on hexagons and lozenges that we have not see in the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure

Computability and Tiling Problems

In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles S has total planar tilings, which we denote TILE, or whether it has infinite connected but not necessarily total tilings, WTILE (short for ‘weakly tile’). We show that both TILE ≡m ILL ≡m WTILE, and thereby both TILE and WTILE are Σ11-complete. We also show that the opposite problems, ¬TILE and SNT (short for ‘Strongly Not Tile’) are such that ¬TILE ≡m WELL ≡m SNT and so both ¬TILE and SNT are both Π11-complete. Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets PTile of periodic tilings, and ATile of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form (Σ1 1 ∧Π1 1). We also present results for finite versions of these tiling problems. We then move on to consider the Weihrauch reducibility for a general total tiling principle CT as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, Cωω. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to Cωω. Finally, we give a prototile set of 15 prototiles that can encode any Elementary CellularAutomaton(ECA). We make use of an unusual tileset, based on hexagons and lozenges that we have not seen in the literature before, in order to achieve this