Pumping lemmas for weighted automata

Russian Mathematics, 2010

We introduce weighted automata over in…nite words with Muller acceptance condition and we show that their behaviors coincide with the semantics of weighted restricted MSO-sentences. Furthermore, we establish an equivalence property of weighted Muller and weighted Büchi automata over certain semirings.

2020

Recently, weighted ω-pushdown automata have been introduced by Droste, Ésik, Kuich. This new type of automaton has access to a stack and models quantitative aspects of infinite words. Here, we consider a simple version of those automata. The simple ω-pushdown automata do not use -transitions and have a very restricted stack access. In previous work, we could show this automaton model to be expressively equivalent to context-free ω-languages in the unweighted case. Furthermore, semiring-weighted simple ω-pushdown automata recognize all ω-algebraic series. Here, we consider ω-valuation monoids as weight structures. As a first result, we prove that for this weight structure and for simple ω-pushdown automata, Büchi-acceptance and Muller-acceptance are expressively equivalent. In our second result, we derive a Nivat theorem for these automata stating that the behaviors of weighted ω-pushdown automata are precisely the projections of very simple ω-series restricted to ω-context-free lang...

Theoretical Computer Science

This report contains the program and the abstracts of lectures delivered at the workshop "Weighted Automata-Theory and Applications" which took place at Leipzig University, March 27-31, 2006. This workshop covered all aspects of weighted automata, ranging from the theory of formal power series to quantum automata and applications e.g. in digital image processing or model-checking of probabilistic systems. The workshop was attended by 45 participants from countries. Two tutorials were given by J. Albert (Würzburg), J. Gruska (Brno).

IFAC-PapersOnLine, 2018

Max,+)-automata are weighted automata over the (max,+) semiring. In this paper we investigate simulation like equivalences between (max,+)-automata. Since (max,+)-automata are nondeterministic (weighted) automata, there exist extensions of bisimilarity properties that are weaker than equality of their weighted languages (formal power series). The main advantage of bisimulation like properties is that they can be checked in polynomial time, while equality (as well as inequality) of formal power series is undecidable. We show that a form of weak simulation can be used as a sufficient condition for comparing the formal power series.

Computer Science – Theory and Applications, 2006

We show that two equivalent K-automata are conjugate to a third one, when K is equal to B, N, Z, or any (skew) field and that the same holds true for functional tranducers as well.

Linear Algebra and its Applications, 2012

We investigate the continuity of the ω-functions and real functions defined by weighted finite automata (WFA). We concentrate on the case of average preserving WFA. We show that every continuous ω-function definable by some WFA can be defined by an average preserving WFA and then characterize minimal average preserving WFA whose ω-function or ω-function and real function are continuous. We obtain several algorithmic reductions for WFA-related decision problems. In particular, we show that deciding whether the ω-function and real function of an average preserving WFA are both continuous is computationally equivalent to deciding stability of a set of matrices. We also present a method for constructing WFA that compute continuous real functions.

Theoretical Computer Science, 2006

We define a weighted monadic second order logic for trees where the weights are taken from a commutative semiring. We prove that a restricted version of this logic characterizes the class of formal tree series which are accepted by weighted bottom-up finite state tree automata. The restriction on the logic can be dropped if additionally the semiring is locally finite. This generalizes corresponding classical results of Thatcher, Wright, and Doner for tree languages and it extends recent results of Droste and Gastin [Weighted automata and weighted logics, informal power series on words to formal tree series.

Springer eBooks, 2014

Real and complex number computations Decision problems for automata over uncountable structures In a recent work, Gandhi, Khoussainov, and Liu [7] introduced and studied a generalized model of finite automata able to work over arbitrary structures. As one relevant area of research for this model the authors identify studying such automata over particular structures such as real and algebraically closed fields. In this paper we start investigations into this direction. We prove several structural results about sets accepted by such automata, and analyze decidability as well as complexity of several classical questions about automata in the new framework. Our results show quite a diverse picture when compared to the well known results for finite automata over finite alphabets.

Annals of Pure and Applied Logic, 2009

We extend some of the classical connections between automata and logic due to Büchi (1960) [5] and McNaughton and Papert (1971) [12] to languages of finitely varying functions or ''signals''. In particular, we introduce a natural class of automata for generating finitely varying functions called ST-NFA's, and show that it coincides in terms of language definability with a natural monadic second-order logic interpreted over finitely varying functions Rabinovich (2002) [15]. We also identify a ''counter-free'' subclass of ST-NFA's which characterise the first-order definable languages of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic (MTL) Chevalier et al. (2006, 2007) [6,7].

Mathematical Systems Theory, 1971