Automata theory in nominal sets

We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts

Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid G induced by G, and representations of G. Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for G to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the "out-degrees of vertices". From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.Comment: 69 page

String rewriting for Double Coset Systems

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration. Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio

Combinatorial models of expanding dynamical systems

We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.Comment: The new version differs substantially from the first one. Many parts are moved to other (mostly future) papers, the main open question of the first version is solve

Separation and Renaming in Nominal Sets

Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. We show that these automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions

O-Minimal Hybrid Reachability Games

In this paper, we consider reachability games over general hybrid systems, and distinguish between two possible observation frameworks for those games: either the precise dynamics of the system is seen by the players (this is the perfect observation framework), or only the starting point and the delays are known by the players (this is the partial observation framework). In the first more classical framework, we show that time-abstract bisimulation is not adequate for solving this problem, although it is sufficient in the case of timed automata . That is why we consider an other equivalence, namely the suffix equivalence based on the encoding of trajectories through words. We show that this suffix equivalence is in general a correct abstraction for games. We apply this result to o-minimal hybrid systems, and get decidability and computability results in this framework. For the second framework which assumes a partial observation of the dynamics of the system, we propose another abstraction, called the superword encoding, which is suitable to solve the games under that assumption. In that framework, we also provide decidability and computability results