The ZX-calculus is complete for stabilizer quantum mechanics

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer quantum mechanics, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.Comment: 26 page

Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems

We describe the first results of a project of analyzing in which theories formal proofs can be ex- pressed. We use this analysis as the basis of interoperability between proof systems.Comment: In Proceedings PxTP 2017, arXiv:1712.0089

Session types revisited

Session types are a formalism used to model structured communication-based programming. A binary session type describes communication by specifying the type and direction of data exchanged between two parties. When session types and session processes are added to the syntax of standard π-calculus they give rise to additional separate syntactic categories. As a consequence, when new type features are added, there is duplication of effort in the theory: the proofs of properties must be checked both on standard types and on session types. We show that session types are encodable into standard π- types, relying on linear and variant types. Besides being an expressivity result, the encoding (i) removes the above redundancies in the syntax, and (ii) the properties of session types are derived as straightforward corollaries, exploiting the corresponding properties of standard π-types. The robustness of the encoding is tested on a few extensions of session types, including subtyping, polymorphism and higher-order communications

Integrating a Global Induction Mechanism into a Sequent Calculus

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.Comment: 16 page

A Spatial-Epistemic Logic for Reasoning about Security Protocols

Reasoning about security properties involves reasoning about where the information of a system is located, and how it evolves over time. While most security analysis techniques need to cope with some notions of information locality and knowledge propagation, usually they do not provide a general language for expressing arbitrary properties involving local knowledge and knowledge transfer. Building on this observation, we introduce a framework for security protocol analysis based on dynamic spatial logic specifications. Our computational model is a variant of existing pi-calculi, while specifications are expressed in a dynamic spatial logic extended with an epistemic operator. We present the syntax and semantics of the model and logic, and discuss the expressiveness of the approach, showing it complete for passive attackers. We also prove that generic Dolev-Yao attackers may be mechanically determined for any deterministic finite protocol, and discuss how this result may be used to reason about security properties of open systems. We also present a model-checking algorithm for our logic, which has been implemented as an extension to the SLMC system.Comment: In Proceedings SecCo 2010, arXiv:1102.516

Reversing Single Sessions

Session-based communication has gained a widespread acceptance in practice as a means for developing safe communicating systems via structured interactions. In this paper, we investigate how these structured interactions are affected by reversibility, which provides a computational model allowing executed interactions to be undone. In particular, we provide a systematic study of the integration of different notions of reversibility in both binary and multiparty single sessions. The considered forms of reversibility are: one for completely reversing a given session with one backward step, and another for also restoring any intermediate state of the session with either one backward step or multiple ones. We analyse the costs of reversing a session in all these different settings. Our results show that extending binary single sessions to multiparty ones does not affect the reversibility machinery and its costs