Computer Science

The classical subset construction for non-deterministic automata can be generalized to other side-effects captured by a monad. The key insight is that both the state space of the determinized automaton and its semantics---languages over an alphabet---have a common algebraic structure: they are Eilenberg-Moore algebras for the powerset monad. In this paper we study the reverse question to determinization. We will present a construction to associate succinct automata to languages based on different algebraic structures. For instance, for classical regular languages the construction will transform a deterministic automaton into a non-deterministic one, where the states represent the join-irreducibles of the language accepted by a (potentially) larger deterministic automaton. Other examples will yield alternating automata, automata with symmetries, CABA-structured automata, and weighted automata.

We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a Büchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible.

Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify these using category theory in order to ease correctness proofs and guide the design of new algorithms. In this paper, we extend CALF to cover learning of algebraic structures that may not have a coalgebraic presentation. Furthermore, we provide a detailed algorithmic account of an abstract version of the popular L* algorithm, which was missing from CALF. We instantiate the abstract theory to a large class of Set functors, by which we recover for the first time practical tree automata learning algorithms from an abstract framework and at the same time obtain new algorithms to learn algebras of quotiented polynomial functors.

A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a zigzag word . A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each n≥1 in one of three ways: it grows forward [ t(1) t(2) ⋯ t(n)] , backward [ t(n) ⋯ t(2) t(1) ], or just occurs once [ t ]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth Θ( n k ) iff it is a zigzag word of depth k . As corollaries, we obtain that the morphic words with growth O(n) are exactly the ultimately periodic words, and the morphic words with growth O( n 2 ) are exactly the multilinear words.

We conduct a systematic study of asynchronous models of distributed computing consisting of identical finite-state devices that cooperate in a network to decide if the network satisfies a given graph-theoretical property. Models discussed in the literature differ in the detection capabilities of the agents residing at the nodes of the network (detecting the set of states of their neighbors, or counting the number of neighbors in each state), the notion of acceptance (acceptance by halting in a particular configuration, or by stable consensus), the notion of step (synchronous move, interleaving, or arbitrary timing), and the fairness assumptions (non-starving, or stochastic-like). We study the expressive power of the combinations of these features, and show that the initially twenty possible combinations fit into seven equivalence classes. The classification is the consequence of several equi-expressivity results with a clear interpretation. In particular, we show that acceptance by halting configuration only has non-trivial expressive power if it is combined with counting, and that synchronous and interleaving models have the same power as those in which an arbitrary set of nodes can move at the same time. We also identify simple graph properties that distinguish the expressive power of the seven classes.

In this work we use a framework of finite-state automata constructions based on equivalences over words to provide new insights on the relation between well-known methods for computing the minimal deterministic automaton of a language.

We describe a mathematical framework for equational reasoning about infinite families of string diagrams which is amenable to computer automation. The framework is based on context-free families of string diagrams which we represent using context-free graph grammars. We model equations between infinite families of diagrams using rewrite rules between context-free grammars. Our framework represents equational reasoning about concrete string diagrams and context-free families of string diagrams using double-pushout rewriting on graphs and context-free graph grammars respectively. We prove that our representation is sound by showing that it respects the concrete semantics of string diagrammatic reasoning and we show that our framework is appropriate for software implementation by proving the membership problem is decidable.

A regular language L is said to be prime, if it is not the product of two non-trivial languages. Martens et al. settled the exact complexity of deciding primality for deterministic finite automata in 2010. For finite languages, Mateescu et al. and Wieczorek suspect the NP - completeness of primality, but no actual bounds are given. Using techniques of Martens et al., we prove the NP lower bound and give a Π P 2 upper bound for deciding primality of finite languages given as deterministic finite automata.

We propose a new symbolic trace semantics for register automata (extended finite state machines) which records both the sequence of input symbols that occur during a run as well as the constraints on input parameters that are imposed by this run. Our main result is a generalization of the classical Myhill-Nerode theorem to this symbolic setting. Our generalization requires the use of three relations to capture the additional structure of register automata. Location equivalence ≡ l captures that symbolic traces end in the same location, transition equivalence ≡ t captures that they share the same final transition, and a partial equivalence relation ≡ r captures that symbolic values v and v ′ are stored in the same register after symbolic traces w and w ′ , respectively. A symbolic language is defined to be regular if relations ≡ l , ≡ t and ≡ r exist that satisfy certain conditions, in particular, they all have finite index. We show that the symbolic language associated to a register automaton is regular, and we construct, for each regular symbolic language, a register automaton that accepts this language. Our result provides a foundation for grey-box learning algorithms in settings where the constraints on data parameters can be extracted from code using e.g. tools for symbolic/concolic execution or tainting. We believe that moving to a grey-box setting is essential to overcome the scalability problems of state-of-the-art black-box learning algorithms.

Specification synthesis is the process of deriving a model from the input-output traces of a system. It is used extensively in test design, reverse engineering, and system identification. One type of the resulting artifact of this process for cyber-physical systems is hybrid automata. They are intuitive, precise, tool independent, and at a high level of abstraction, and can model systems with both discrete and continuous variables. In this paper, we propose a new technique for synthesizing hybrid automaton from the input-output traces of a non-linear cyber-physical system. Similarity detection in non-linear behaviors is the main challenge for extracting such models. We address this problem by utilizing the Dynamic Time Warping technique. Our approach is passive, meaning that it does not need interaction with the system during automata synthesis from the logged traces; and online, which means that each input/output trace is used only once in the procedure. In other words, each new trace can be used to improve the already synthesized automaton. We evaluated our algorithm in two industrial and simulated case studies. The accuracy of the derived automata show promising results.