Tensor of quantitative equational theories

Giorgio Bacci, Radu Mardare, Prakash Panangaden, Gordon D. Plotkin

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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.
Original languageEnglish
Number of pages29
Publication statusPublished - 3 Sep 2021
Event9th Conference on Algebra and Coalgebra in Computer Science - Salburg, Austria
Duration: 31 Aug 20213 Sep 2021
Conference number: 9th


Conference9th Conference on Algebra and Coalgebra in Computer Science
Abbreviated titleCALCO 2021


  • quantitative equational thories
  • tensor
  • monads
  • quantitative effects


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