Parameterized weighted containment

Guy Avni, Orna Kupferman

Research output: Contribution to journalArticlepeer-review


Partially specified systems and specifications are used in formal methods such as stepwise design and query checking. Existing methods consider a setting in which systems and their correctness are Boolean. In recent years, there has been growing interest and need for quantitative formal methods, where systems may be weighted and specifications may be multivalued.Weighted automata, which map input words to a numerical value, play a key role in quantitative reasoning. Technically, every transition in a weighted automaton A has a cost, and the value A assigns to a finite word w is the sum of the costs on the transitions traversed along the most expensive accepting run of A on w. We study parameterized weighted containment: given three weighted automata A, B, and C, with B being partial, the goal is to find an assignment to the missing costs in B so that we end up with B′ for which A ≤ B′ ≤ C, where ≤ is the weighted counterpart of containment. We also consider a one-sided version of the problem, where only A or only C is given in addition to B, and the goal is to find a minimal assignment with which A ≤ B′ or, respectively, a maximal one with which B′ ≤ C. We argue that both problems are useful in stepwise design of weighted systems as well as approximated minimization of weighted automata. We show that when the automata are deterministic, we can solve the problems in polynomial time. Our solution is based on the observation that the set of legal assignments to κ missing costs forms a κ-dimensional polytope. The technical challenge is to find an assignment in polynomial time even though the polytope is defined by means of exponentially many inequalities.We do so by developing a divide-and-conquer algorithm based on a separation oracle for polytopes. For nondeterministic automata, the weighted setting is much more complex, and in fact even nonparameterized containment is undecidable. We are able to show positive results for variants of the problems, where containment is replaced by simulation.

Original languageEnglish
Article number6
JournalACM Transactions on Computational Logic
Issue number1
StatePublished - 29 Dec 2014
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014 ACM.


  • Ellipsoid method
  • Partially specified systems
  • Quantitative verification
  • Weighted automata

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science
  • Logic
  • Computational Mathematics


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  • Parameterized weighted containment

    Avni, G. & Kupferman, O., 2013, Foundations of Software Science and Computation Structures - 16th Int. Conference, FOSSACS 2013, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2013, Proc.. p. 369-384 16 p. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); vol. 7794 LNCS).

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

    Open Access

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