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

Series which are both max-plus and min-plus rational are unambiguous

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series

The Equivalence, Unambiguity and Sequentiality Problems of Finitely Ambiguous Max-Plus Tree Automata are Decidable

We show that the equivalence, unambiguity and sequentiality problems are decidable for finitely ambiguous max-plus tree automata

Finite-Valued Weighted Automata

Any weighted automaton (WA) defines a relation from finite words to values: given an input word, its set of values is obtained as the set of values computed by each accepting run on that word. A WA is k-valued if the relation it defines has degree at most k, i.e., every set of values associated with an input word has cardinality at most k. We investigate the class of quantitative languages defined by k-valued automata, for all parameters k. We consider several measures to associate values with runs: sum, discounted-sum, and more generally values in groups. We define a general procedure which decides, given a bound k and a WA over a group, whether this automaton is k-valued. We also show that any k-valued WA over a group, under some general conditions, can be decomposed as a union of k unambiguous WA. While inclusion and equivalence are undecidable problems for arbitrary sum-automata, we show, based on this decomposition, that they are decidable for k-valued sum-automata, and k-valued discounted sum-automata over inverted integer discount factors. We finally show that the quantitative Church problem is undecidable for k-valued sum-automata, even given as finite unions of deterministic sum-automata

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton

Finite Sequentiality of Unambiguous Max-Plus Tree Automata

We show the decidability of the finite sequentiality problem for unambiguous max-plus tree automata. A max-plus tree automaton is called unambiguous if there is at most one accepting run on every tree. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton

Towards Formalizing QoS of Web Services with Weighted Automata

Web services (WSs) are used more and more as components of distributed applications with a goal to resolve complex tasks that simple services cannot. This use of WSs is connected to the emergence of languages like WS-BPEL which allows describing the external behaviour of WSs on top of the service interfaces. The use of WSs as components of distributed applications implies the possibility to change a failing service for another which can do at least the same things as the replaced service. The composition issues are also of particular interest to WSs users. Different solutions have been proposed during the last years to check such properties, but, to our knowledge, none of them takes QoS aspects into account. This paper introduces underpinnings and a tool for verifying WSs substitutivity and well-formed composition while considering WSs costs such as the execution time of the different operations provided by WSs

How to Tackle Integer Weighted Automata Positivity

International audienceThis paper is dedicated to candidate abstractions to capture relevant aspects of the integer weighted automata. The expected effect of applying these abstractions is studied to build the deterministic reachability graphs allowing us to semi-decide the positivity problem on these automata. Moreover, the papers reports on the implementations and experimental results, and discusses other encodings

Determinisation of Finitely-Ambiguous Copyless Cost Register Automata

Cost register automata (CRA) are machines reading an input word while computing values using write-only registers: values from registers are combined using the two operations, as well as the constants, of a semiring. Particularly interesting is the subclass of copyless CRAs where the content of a register cannot be used twice for updating the registers. Originally deterministic, non-deterministic variant of CRA may also be defined: the semantics is then obtained by combining the values of all accepting runs with the additive operation of the semiring (as for weighted automata). We show that finitely-ambiguous copyless non-deterministic CRAs (i.e. the ones that admit a bounded number of accepting runs on every input word) can be effectively transformed into an equivalent copyless (deterministic) CRA, without requiring any specific property on the semiring. As a corollary, this also shows that regular look-ahead can effectively be removed from copyless CRAs

A Burnside Approach to the Termination of Mohri’s Algorithm for Polynomially Ambiguous Min-Plus-Automata

We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm