A coalgebraic perspective on linear weighted automata

Weighted automata are a generalization of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterize weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterize weighted language equivalence. Relying on the second characterization, we show three different procedures for computing weighted language equivalence. The first one consists in a generalizion of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages

Nested Weighted Limit-Average Automata of Bounded Width

While weighted automata provide a natural framework to express quantitative properties, many basic properties like average response time cannot be expressed with weighted automata. Nested weighted automata extend weighted automata and consist of a master automaton and a set of slave automata that are invoked by the master automaton. Nested weighted automata are strictly more expressive than weighted automata (e.g., average response time can be expressed with nested weighted automata), but the basic decision questions have higher complexity (e.g., for deterministic automata, the emptiness question for nested weighted automata is PSPACE-hard, whereas the corresponding complexity for weighted automata is PTIME). We consider a natural subclass of nested weighted automata where at any point at most a bounded number k of slave automata can be active. We focus on automata whose master value function is the limit average. We show that these nested weighted automata with bounded width are strictly more expressive than weighted automata (e.g., average response time with no overlapping requests can be expressed with bound k=1, but not with non-nested weighted automata). We show that the complexity of the basic decision problems (i.e., emptiness and universality) for the subclass with k constant matches the complexity for weighted automata. Moreover, when k is part of the input given in unary we establish PSPACE-completeness

Minimisation in Logical Form

Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially observable deterministic finite automata, linear weighted automata viewed as coalgebras over finite-dimensional vector spaces, and belief automata, which are coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowski's double-reversal minimisation algorithm for deterministic finite automata was described categorically and its correctness explained via the duality between reachability and observability. This work includes generalisations of Brzozowski's algorithm to Moore and weighted automata over commutative semirings. In this paper we propose a general categorical framework within which such minimisation algorithms can be understood. The goal is to provide a unifying perspective based on duality. Our framework consists of a stack of three interconnected adjunctions: a base dual adjunction that can be lifted to a dual adjunction between coalgebras and algebras and also to a dual adjunction between automata. The approach provides an abstract understanding of reachability and observability. We illustrate the general framework on range of concrete examples, including deterministic Kripke frames, weighted automata, topological automata (belief automata), and alternating automata

Polynomial Time Decidability of Weighted Synchronization under Partial Observability

We consider weighted automata with both positive and negative integer weights on edges and study the problem of synchronization using adaptive strategies that may only observe whether the current weight-level is negative or nonnegative. We show that the synchronization problem is decidable in polynomial time for deterministic weighted automata

Deterministic Weighted Automata under Partial Observability

Weighted automata is a basic tool for specification in quantitative verification, which allows to express quantitative features of analysed systems such as resource consumption. Quantitative specification can be assisted by automata learning as there are classic results on Angluin-style learning of weighted automata. The existing work assumes perfect information about the values returned by the target weighted automaton. In assisted synthesis of a quantitative specification, knowledge of the exact values is a strong assumption and may be infeasible. In our work, we address this issue by introducing a new framework of partially-observable deterministic weighted automata, in which weighted automata return intervals containing the computed values of words instead of the exact values. We study the basic properties of this framework with the particular focus on the challenges o

Non-deterministic Weighted Automata on Random Words

We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables. The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose an approximation algorithm for the probabilistic questions, which works in exponential time in the automaton and polynomial time in the Markov chain. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric

Automata Minimization: a Functorial Approach

In this paper we regard languages and their acceptors - such as deterministic or weighted automata, transducers, or monoids - as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: A) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. B) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. C) We show how this framework can be instantiated to unify several phenomena in automata theory, starting with determinization, minimization and syntactic algebras. We provide explanations of Choffrut's minimization algorithm for subsequential transducers and of Brzozowski's minimization algorithm in this setting.Comment: journal version of the CALCO 2017 paper arXiv:1711.0306