What's decidable about weighted automata?

Shaull Almagor*, Udi Boker, Orna Kupferman

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

71 Scopus citations


Weighted automata map input words to numerical values. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. In the 90's, Krob studied the decidability of problems on rational series, which strongly relate to weighted automata. In particular, it follows from Krob's results that the universality problem (that is, deciding whether the values of all words are below some threshold) is decidable for weighted automata with weights in ℕ ∪ {∞}, and that the equality problem is undecidable when the weights are in ℤ ∪ {∞}. In this paper we continue the study of the borders of decidability in weighted automata, describe alternative and direct proofs of the above results, and tighten them further. Unlike the proofs of Krob, which are algebraic in their nature, our proofs stay in the terrain of state machines, and the reduction is from the halting problem of a two-counter machine. This enables us to significantly simplify Krob's reasoning and strengthen the results to apply already to a very simple class of automata: all the states are accepting, there are no initial nor final weights, and all the weights are from the set {∈-∈1,0,1}. The fact we work directly with automata enables us to tighten also the decidability results and to show that the universality problem for weighted automata with weights in ℕ ∪ {∞}, and in fact even with weights in ℚ≥0 ∪ {∞}, is PSPACE-complete. Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in both the front of equality vs. universality and the front of the ℕ ∪ {∞} vs. the ℤ ∪ {∞} domains.

Original languageAmerican English
Title of host publicationAutomated Technology for Verification and Analysis - 9th International Symposium, ATVA 2011, Proceedings
Number of pages10
StatePublished - 2011
Event9th International Symposium on Automated Technology for Verification and Analysis, ATVA 2011 - Taipei, Taiwan, Province of China
Duration: 11 Oct 201114 Oct 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6996 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference9th International Symposium on Automated Technology for Verification and Analysis, ATVA 2011
Country/TerritoryTaiwan, Province of China


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