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On the Axiomatizability of Quantitative Algebras

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Original languageEnglish
Title of host publication2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
Number of pages12
ISBN (Electronic)978-1-5090-3018-7
StatePublished - 18 Aug 2017


Quantitative algebras (QAs) are algebras over metric spaces defined by quantitative equational theories as introduced by us in 2016. They provide the mathematical foundation for metric semantics of probabilistic, stochastic and other quantitative systems. This paper considers the issue of axiomatizability of
QAs. We investigate the entire spectrum of types of quantitative equations that can be used to axiomatize theories: (i) simple quantitative equations; (ii) Horn clauses with no more than c equations between variables as hypotheses, where c is a cardinal and (iii) the most general case of Horn clauses. In each case we characterize the class of QAs and prove variety/quasivariety theorems that extend and generalize classical results from model theory for algebras and first-order structures.

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