Tensor of Quantitative Equational Theories

Authors Giorgio Bacci , Radu Mardare, Prakash Panangaden, Gordon Plotkin

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Giorgio Bacci
  • Department of Computer Science, Aalborg University, Denmark
Radu Mardare
  • Department of Computer & Information Sciences, University of Strathclyde, Glasgow, UK
Prakash Panangaden
  • School of Computer Science, McGill University, Montreal, Canada
Gordon Plotkin
  • LFCS, School of Informatics, University of Edinburgh, UK

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Giorgio Bacci, Radu Mardare, Prakash Panangaden, and Gordon Plotkin. Tensor of Quantitative Equational Theories. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We develop a theory for the commutative combination of quantitative effects, their tensor, given as a combination of quantitative equational theories that imposes mutual commutation of the operations from each theory. As such, it extends the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We show that under certain assumptions on the quantitative theories the free monad that arises from the tensor of two theories is the categorical tensor of the free monads on the theories. As an application, we provide the first algebraic axiomatizations of labelled Markov processes and Markov decision processes. Apart from the intrinsic interest in the axiomatizations, it is pleasing they are obtained compositionally by means of the sum and tensor of simpler quantitative equational theories.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Quantitative equational theories
  • Tensor
  • Monads
  • Quantitative Effects


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