Silent Transitions in Automata with Storage

We consider the computational power of silent transitions in one-way automata with storage. Specifically, we ask which storage mechanisms admit a transformation of a given automaton into one that accepts the same language and reads at least one input symbol in each step. We study this question using the model of valence automata. Here, a finite automaton is equipped with a storage mechanism that is given by a monoid. This work presents generalizations of known results on silent transitions. For two classes of monoids, it provides characterizations of those monoids that allow the removal of \lambda-transitions. Both classes are defined by graph products of copies of the bicyclic monoid and the group of integers. The first class contains pushdown storages as well as the blind counters while the second class contains the blind and the partially blind counters.Comment: 32 pages, submitte

SMT-based Verification of LTL Specifications with Integer Constraints and its Application to Runtime Checking of Service Substitutability

An important problem that arises during the execution of service-based applications concerns the ability to determine whether a running service can be substituted with one with a different interface, for example if the former is no longer available. Standard Bounded Model Checking techniques can be used to perform this check, but they must be able to provide answers very quickly, lest the check hampers the operativeness of the application, instead of aiding it. The problem becomes even more complex when conversational services are considered, i.e., services that expose operations that have Input/Output data dependencies among them. In this paper we introduce a formal verification technique for an extension of Linear Temporal Logic that allows users to include in formulae constraints on integer variables. This technique applied to the substitutability problem for conversational services is shown to be considerably faster and with smaller memory footprint than existing ones

Sciduction: Combining Induction, Deduction, and Structure for Verification and Synthesis

Even with impressive advances in automated formal methods, certain problems in system verification and synthesis remain challenging. Examples include the verification of quantitative properties of software involving constraints on timing and energy consumption, and the automatic synthesis of systems from specifications. The major challenges include environment modeling, incompleteness in specifications, and the complexity of underlying decision problems. This position paper proposes sciduction, an approach to tackle these challenges by integrating inductive inference, deductive reasoning, and structure hypotheses. Deductive reasoning, which leads from general rules or concepts to conclusions about specific problem instances, includes techniques such as logical inference and constraint solving. Inductive inference, which generalizes from specific instances to yield a concept, includes algorithmic learning from examples. Structure hypotheses are used to define the class of artifacts, such as invariants or program fragments, generated during verification or synthesis. Sciduction constrains inductive and deductive reasoning using structure hypotheses, and actively combines inductive and deductive reasoning: for instance, deductive techniques generate examples for learning, and inductive reasoning is used to guide the deductive engines. We illustrate this approach with three applications: (i) timing analysis of software; (ii) synthesis of loop-free programs, and (iii) controller synthesis for hybrid systems. Some future applications are also discussed

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

Deciding Conditional Termination

We address the problem of conditional termination, which is that of defining the set of initial configurations from which a given program always terminates. First we define the dual set, of initial configurations from which a non-terminating execution exists, as the greatest fixpoint of the function that maps a set of states into its pre-image with respect to the transition relation. This definition allows to compute the weakest non-termination precondition if at least one of the following holds: (i) the transition relation is deterministic, (ii) the descending Kleene sequence overapproximating the greatest fixpoint converges in finitely many steps, or (iii) the transition relation is well founded. We show that this is the case for two classes of relations, namely octagonal and finite monoid affine relations. Moreover, since the closed forms of these relations can be defined in Presburger arithmetic, we obtain the decidability of the termination problem for such loops.Comment: 61 pages, 6 figures, 2 table

Synthesising Graphical Theories

In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category. In particular, categorical quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of quantum theory into abstract structural properties, expressed in the form of diagrammatic identities. One way we search for these properties is to start with a concrete model (e.g. a set of linear maps or finite relations) and start composing generators into diagrams and looking for graphical identities. Naively, we could automate this procedure by enumerating all diagrams up to a given size and check for equalities, but this is intractable in practice because it produces far too many equations. Luckily, many of these identities are not primitive, but rather derivable from simpler ones. In 2010, Johansson, Dixon, and Bundy developed a technique called conjecture synthesis for automatically generating conjectured term equations to feed into an inductive theorem prover. In this extended abstract, we adapt this technique to diagrammatic theories, expressed as graph rewrite systems, and demonstrate its application by synthesising a graphical theory for studying entangled quantum states.Comment: 10 pages, 22 figures. Shortened and one theorem adde