We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric, and we provide sufficient conditions on the predicates under which the model-checking problem is decidable. We then consider three particular automata models (finite automata, transducers and automata weighted by integers -sum-automata -) and instantiate the generic logic for each of them. We give tight complexity results for the three logics and the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. Consequently to our complexity results, we directly obtain that these classical properties can be decided in PTIME. IntroductionMotivations An important aspect of automata theory is the definition of automata subclasses with particular properties, of algorithmic interest for instance. As an example, the inclusion problem for non-deterministic finite automata is PSPACE-C but becomes PTIME if the automata are k-ambiguous for a fixed k [21].By automata theory, we mean automata in the general sense of finite state machines processing finite words. This includes what we call automata with outputs, which may also produce output values in a fixed monoid M = (D, ⊕, 0). In such an automaton, the transitions are extended with an (output) value in D, and the value of an accepting path is the sum (for ⊕) of all the values occurring along its transitions. Automata over finite words in Λ * and with outputs in M define subsets of Λ * × D as follows: to any input word w ∈ Λ * , we associate the set of values of all the accepting paths on w. For example, transducers are automata with outputs in a free monoid: they process input words and produce output words and therefore define binary relations of finite words [15].The many decidability properties of finite automata do not carry over to transducers, and many restrictions have been defined in the literature to recover decidability, or just to define subclasses relevant to particular applications. The inclusion problem for transducer is undecidable [13], but decidable for finitevalued transducers [23]. Another well-known subclass is that of the determinisable transducers [5], defining sequential functions of words. Finite-valuedness and determinisability are two properties decidable in PTIME, i.e., it is decidable in PTIME, given a transducer, whether it is finite-valued (resp. determinisable). As a second example of automata with outputs, we also consider sum-automata, i.e. automata with outputs in (Z, +, 0), which defines relations from words to Z. Properties such as functionality, determinisability, and k-valuedness (for a fixed k) are decidable in PTIME for sum-automata [11,10].In our e...
We propose a novel algorithm to decide the language inclusion between (nondeterministic) Büchi automata, a PSpace-complete problem. Our approach, like others before, leverage a notion of quasiorder to prune the search for a counterexample by discarding candidates which are subsumed by others for the quasiorder. Discarded candidates are guaranteed to not compromise the completeness of the algorithm. The novelty of our work lies in the quasiorder used to discard candidates. We introduce FORQs (family of right quasiorders) that we obtain by adapting the notion of family of right congruences put forward by Maler and Staiger in 1993. We define a FORQ-based inclusion algorithm which we prove correct and instantiate it for a specific FORQ, called the structural FORQ, induced by the Büchi automaton to the right of the inclusion sign. The resulting implementation, called Forklift, scales up better than the state-of-the-art on a variety of benchmarks including benchmarks from program verification and theorem proving for word combinatorics. Artifact:https://doi.org/10.5281/zenodo.6552870
Quantitative monitoring can be universal and approximate: For every finite sequence of observations, the specification provides a value and the monitor outputs a best-effort approximation of it. The quality of the approximation may depend on the resources that are available to the monitor. By taking to the limit the sequences of specification values and monitor outputs, we obtain precision-resource trade-offs also for limit monitoring. This paper provides a formal framework for studying such trade-offs using an abstract interpretation for monitors: For each natural number n, the aggregate semantics of a monitor at time n is an equivalence relation over all sequences of at most n observations so that two equivalent sequences are indistinguishable to the monitor and thus mapped to the same output. This abstract interpretation of quantitative monitors allows us to measure the number of equivalence classes (or “resource use”) that is necessary for a certain precision up to a certain time, or at any time. Our framework offers several insights. For example, we identify a family of specifications for which any resource-optimal exact limit monitor is independent of any error permitted over finite traces. Moreover, we present a specification for which any resource-optimal approximate limit monitor does not minimize its resource use at any time.
In this paper, we investigate the expressive power and the algorithmic properties of weighted expressions, which define functions from finite words to integers. First, we consider a slight extension of an expression formalism, introduced by Chatterjee. et. al. in the context of infinite words, by which to combine values given by unambiguous (max, +)-automata, using Presburger arithmetic. We show that important decision problems such as emptiness, universality and comparison are PSPACE-C for these expressions. We then investigate the extension of these expressions with Kleene star. This allows to iterate an expression over smaller fragments of the input word, and to combine the results by taking their iterated sum. The decision problems turn out to be undecidable, but we introduce the decidable and still expressive class of synchronised expressions.
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