Containment and equivalence of weighted automata: Probabilistic and max-plus cases

This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain

Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape

Series, Weighted Automata, Probabilistic Automata and Probability Distributions for Unranked Trees.

We study tree series and weighted tree automata over unranked trees. The message is that recognizable tree series for unranked trees can be defined and studied from recognizable tree series for binary representations of unranked trees. For this we prove results of Denis et al (2007) as follows. We extend hedge automata -- a class of tree automata for unranked trees -- to weighted hedge automata. We define weighted stepwise automata as weighted tree automata for binary representations of unranked trees. We show that recognizable tree series can be equivalently defined by weighted hedge automata or weighted stepwise automata. Then we consider real-valued tree series and weighted tree automata over the field of real numbers. We show that the result also holds for probabilistic automata -- weighted automata with normalisation conditions for rules. We also define convergent tree series and show that convergence properties for recognizable tree series are preserved via binary encoding. From Etessami and Yannakakis (2009), we present decidability results on probabilistic tree automata and algorithms for computing sums of convergent series. Last we show that streaming algorithms for unranked trees can be seen as slight transformations of algorithms on the binary representations

Minimization via duality

We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object

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

A Coalgebraic Approach to Reducing Finitary Automata

Compact representations of automata are important for efficiency. In this paper, we study methods to compute reduced automata, in which no two states accept the same language. We do this for finitary automata (FA), an abstract definition that encompasses probabilistic and weighted automata. Our procedure makes use of Milius' locally finite fixpoint. We present a reduction algorithm that instantiates to probabilistic and S-linear weighted automata (WA) for a large class of semirings. Moreover, we propose a potential connection between properness of a semiring and our provided reduction algorithm for WAs, paving the way for future work in connecting the reduction of automata to the properness of their associated coalgebras