Abstract
We propose a syntactic representation of reversible logic circuits in their entirety, based on Feynman’s control interpretation of Toffoli’s reversible gate set. A pair of interacting proof calculi for reasoning about these circuits is presented, based on classical propositional logic and monoidal structure, and a natural order-theoretic structure is developed, demonstrated equivalent to Boolean algebras, and extended categorically to form a sound and complete semantics for this system. We show that all strong equivalences of reversible logic circuits are provable in the system, derive an equivalent equational theory, and describe its main applications in the verification of both reversible circuits and template-based reversible circuit rewriting systems.
The authors acknowledge support from the Danish Council for Independent Research\(\mid \) Natural Sciences under the Foundations of Reversible Computing project, and partial support from COST Action IC1405 Reversible Computation.
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Notes
- 1.
We use the notation \(\varDelta _\mathbb {N}\) for the discrete category specifically to avoid confusion with the ordinal category \(\omega \), which some authors denote \(\mathbb {N}\).
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Axelsen, H.B., Glück, R., Kaarsgaard, R. (2016). A Classical Propositional Logic for Reasoning About Reversible Logic Circuits. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_4
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