Programming with Quantum Communication

This work develops a formal framework for specifying, implementing, and analysing quantum communication protocols. We provide tools for developing simple proofs and analysing programs which involve communication, both via quantum channels and exhibiting the LOCC (local operations, classical communication) paradigm

Verification Conditions are Code

This paper presents a new theoretical result concerning Hoare Logic. It is shown here that the verification conditions which support a Hoare Logic program derivation are themselves sufficient to construct a correct implementation of the given pre-, post- condition specification. This property is mainly of theoretical interest, though it is possible that it may have some practical use, for example if predicative programming methodology is adopted. The result is shown to hold for both the original, partial correctness, Hoare logic, and also a variant for total correctness derivations

The Rooster and the Syntactic Bracket

We propose an extension of pure type systems with an algebraic presentation of inductive and co-inductive type families with proper indices. This type theory supports coercions toward from smaller sorts to bigger sorts via explicit type construction, as well as impredicative sorts. Type families in impredicative sorts are constructed with a bracketing operation. The necessary restrictions of pattern-matching from impredicative sorts to types are confined to the bracketing construct. This type theory gives an alternative presentation to the calculus of inductive constructions on which the Coq proof assistant is an implementation.Comment: To appear in the proceedings of the 19th International Conference on Types for Proofs and Program

Impredicative Encodings of (Higher) Inductive Types

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1

Programming Telepathy: Implementing Quantum Non-Locality Games

Quantum pseudo-telepathy is an intriguing phenomenon which results from the application of quantum information theory to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel. This paper provides a formal framework for specifying, implementing, and analysing quantum non-locality games

Equations for Hereditary Substitution in Leivant's Predicative System F: A Case Study

This paper presents a case study of formalizing a normalization proof for Leivant's Predicative System F using the Equations package. Leivant's Predicative System F is a stratified version of System F, where type quantification is annotated with kinds representing universe levels. A weaker variant of this system was studied by Stump & Eades, employing the hereditary substitution method to show normalization. We improve on this result by showing normalization for Leivant's original system using hereditary substitutions and a novel multiset ordering on types. Our development is done in the Coq proof assistant using the Equations package, which provides an interface to define dependently-typed programs with well-founded recursion and full dependent pattern- matching. Equations allows us to define explicitly the hereditary substitution function, clarifying its algorithmic behavior in presence of term and type substitutions. From this definition, consistency can easily be derived. The algorithmic nature of our development is crucial to reflect languages with type quantification, enlarging the class of languages on which reflection methods can be used in the proof assistant.Comment: In Proceedings LFMTP 2015, arXiv:1507.07597. www: http://equations-fpred.gforge.inria.fr

On the strength of proof-irrelevant type theories

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.Comment: 20 pages, Logical Methods in Computer Science, Long version of IJCAR 2006 pape

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

On choice rules in dependent type theory

In a dependent type theory satisfying the propositions as types correspondence together with the proofs-as-programs paradigm, the validity of the unique choice rule or even more of the choice rule says that the extraction of a computable witness from an existential statement under hypothesis can be performed within the same theory. Here we show that the unique choice rule, and hence the choice rule, are not valid both in Coquand\u2019s Calculus of Constructions with indexed sum types, list types and binary disjoint sums and in its predicative version implemented in the intensional level of the Minimalist Founda- tion. This means that in these theories the extraction of computational witnesses from existential statements must be performed in a more ex- pressive proofs-as-programs theory