2012

Editors: Luca Aceto, Andrzej Tarlecki, Lars Birkedal

This paper is concerned with the computational complexity of equivalence and minimisation for automata with transition weights in the field Q of rational numbers. We use polynomial identity testing and the Isolation Lemma to obtain complexity bounds, focussing on the class NC of problems within P solvable in polylogarithmic parallel time. For finite Q-weighted automata, we give a randomised NC procedure that either outputs that two automata are equivalent or returns a word on which they differ. We also give an NC procedure for deciding whether a given automaton is minimal, as well as a randomised NC procedure that minimises an automaton. We consider probabilistic automata with rewards, similar to Markov Decision Processes. For these automata we consider two notions of equivalence: expectation equivalence and distribution equivalence. The former requires that two automata have the same expected reward on each input word, while the latter requires that each input word induce the same distribution on rewards in each automaton. For both notions we give algorithms for deciding equivalence by reduction to equivalence of Q-weighted automata. Finally we show that the equivalence problem for Q-weighted visibly pushdown automata is logspace equivalent to the polynomial identity testing problem.

We present a framework for obtaining effective characterizations of simple fragments of future temporal logic (LTL) with the natural numbers as time domain. The framework is based on a form of strongly unambiguous automata, also known as prophetic automata or complete unambiguous Büchi automata and referred to as Carton-Michel automata in this paper. These automata enjoy strong structural properties, in particular, they separate the "finitary fraction" of a regular language of infinite words from its "infinitary fraction" in a natural fashion. Within our framework, we provide characterizations of several natural fragments of temporal logic, where, in some cases, no effective characterization had been known previously, and give lower and upper bounds for their computational complexity.

For endofunctors of varieties preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. The initial algebra consists of all well-pointed coalgebras that are well-founded in the sense of Osius and Taylor. And initial algebras are precisely the final well-founded coalgebras. Finally, the initial iterative algebra consists of all finite well-pointed coalgebras. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems.

We formalise a general concept of distributed systems as sequential components interacting asynchronously. We define a corresponding class of Petri nets, called LSGA nets, and precisely characterise those system specifications which can be implemented as LSGA nets up to branching ST-bisimilarity with explicit divergence.

Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry additional structure that containers alone do not account for. We introduce directed containers to capture the common situation where every position in a data-structure determines another data-structure, informally, the sub-data-structure rooted by that position. Some natural examples are non-empty lists and node-labelled trees, and data-structures with a designated position (zippers). While containers denote set functors via a fully-faithful functor, directed containers interpret fully-faithfully into comonads. But more is true: every comonad whose underlying functor is a container is represented by a directed container. In fact, directed containers are the same as containers that are comonads. We also describe some constructions of directed containers. We have formalized our development in the dependently typed programming language Agda.