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Higher-Order Causal Theories Are Models of BV-Logic

Authors Will Simmons , Aleks Kissinger

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Author Details

Will Simmons
  • Department of Computer Science, University of Oxford, Oxford, UK
  • Cambridge Quantum, Terrington House, 13-15 Hills Road, Cambridge, UK
Aleks Kissinger
  • Department of Computer Science, University of Oxford, Oxford, UK


The authors would like to thank Alessio Guglielmi for posing the question of modelling BV-logic within Caus[𝒞], as well as Chris Barrett, Lutz Straßburger, and members of the quantum group at University of Oxford for useful discussions.

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Will Simmons and Aleks Kissinger. Higher-Order Causal Theories Are Models of BV-Logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 80:1-80:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[𝒞] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmi’s BV logic. Furthermore, we obtain a natural interpretation for the sequential product as "A can signal to B, but not vice-versa", which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for 𝒞 recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Linear logic
  • Theory of computation → Categorical semantics
  • Causality
  • linear logic
  • categorical logic
  • probabilistic coherence spaces
  • quantum channels


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