List-Decoding Homomorphism Codes with Arbitrary Codomains

The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case. The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}. The limitations on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018). We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list

Contextuality in foundations and quantum computation

Contextuality is a key concept in quantum theory. We reveal just how important it is by demonstrating that quantum theory builds on contextuality in a fundamental way: a number of key theorems in quantum foundations can be given a unifi ed presentation in the topos approach to quantum theory, which is based on contextuality as the common underlying principle. We review existing results and complement them by providing contextual reformulations for Stinespring's and Bell's theorem. Both have a number of consequences that go far beyond the evident confirmation of the unifying character of contextuality in quantum theory. Complete positivity of quantum channels is already encoded in contexts, nonlocality arises from a notion of composition of contexts, and quantum states can be singled out among more general non-signalling correlations over the composite context structure by a notion of time orientation in subsystems, thus solving a much discussed open problem in quantum information theory. We also discuss nonlocal correlations under the generalisation to orthomodular lattices and provide generalised Bell inequalities in this setting. The dominant role of contextuality in quantum foundations further supports a recent hypothesis in quantum computation, which identifi es contextuality as the resource for the supposed quantum advantage over classical computers. In particular, within the architecture of measurement-based quantum computation, the resource character of nonlocality and contextuality exhibits rather clearly. We study contextuality in this framework and generalise the strong link between contextuality and computation observed in the qubit case to qudit systems. More precisely, we provide new proofs of contextuality as well as a universal implementation of computation in this setting, while emphasising the crucial role played by phase relations between measurement eigenstates. Finally, we suggest a fine-grained measure for contextuality in the form of the number of qubits required for implementation in the non-adaptive, deterministic case.Open Acces

Fibred computational effects

We study the interplay between dependent types and computational effects, two important areas of modern programming language research. On the one hand, dependent types underlie proof assistants such as Coq and functional programming languages such as Agda, Idris, and F*, providing programmers a means for encoding detailed specifications of program behaviour using types. On the other hand, computational effects, such as exceptions, nondeterminism, state, I/O, probability, etc., are integral to all widely-used programming languages, ranging from imperative languages, such as C, to functional languages, such as ML and Haskell. Separately, dependent types and computational effects both come with rigorous mathematical foundations, dependent types in the effect-free setting and computational effects in the simply typed setting. Their combination, however, has received much less attention and no similarly exhaustive theory has been developed. In this thesis we address this shortcoming by providing a comprehensive treatment of the combination of these two fields, and demonstrating that they admit a mathematically elegant and natural combination. Specifically, we develop a core effectful dependently typed language, eMLTT, based on Martin-L¨of’s intensional type theory and a clear separation between (effect-free) values and (possibly effectful) computations familiar from simply typed languages such as Levy’s Call-By-Push-Value and Egger et al.’s Enriched Effect Calculus. A novel feature of our language is the computational S-type, which we use to give a uniform treatment of type-dependency in sequential composition. In addition, we define and study a class of category-theoretic models, called fibred adjunction models, that are suitable for defining a sound and complete interpretation of eMLTT. Specifically, fibred adjunction models naturally combine standard category-theoretic models of dependent types (split closed comprehension categories) with those of computational effects (adjunctions). We discuss and study various examples of these models, including a domain-theoretic model so as to extend eMLTT with general recursion. We also investigate a dependently typed generalisation of the algebraic treatment of computational effects by showing how to extend eMLTT with fibred algebraic effects and their handlers. In particular, we specify fibred algebraic effects using a dependently typed generalisation of Plotkin and Pretnar’s effect theories, enabling us to capture precise notions of computation such as state with location-dependent store types and dependently typed update monads. For handlers, we observe that their conventional term-level definition leads to unsound program equivalences becoming derivable in languages that include a notion of homomorphism, such as eMLTT. To solve this problem, we propose a novel type-based treatment of handlers via a new computation type, the user-defined algebra type, which pairs a value type (the carrier) with a family of value terms (the operations). This type internalises Plotkin and Pretnar’s insight that handlers denote algebras for a given equational theory of computational effects. We demonstrate the generality of our type-based treatment of handlers by showing that their conventional term-level presentation can be routinely derived, and this treatment provides a useful mechanism for reasoning about effectful computations. Finally, we show that these extensions of eMLTT can be soundly interpreted in a fibred adjunction model based on the families of sets fibration and models of Lawvere theories

Glossarium BITri 2016 : Interdisciplinary Elucidation of Concepts, Metaphors, Theories and Problems Concerning Information

222 p.Terms included in this glossary recap some of the main concepts, theories, problems and metaphors concerning INFORMATION in all spheres of knowledge. This is the first edition of an ambitious enterprise covering at its completion all relevant notions relating to INFORMATION in any scientific context. As such, this glossariumBITri is part of the broader project BITrum, which is committed to the mutual understanding of all disciplines devoted to information across fields of knowledge and practic

Interdisciplinary elucidation of concepts, metaphors, theories and problems concerning INFORMATION

Terms included in this glossary recap some of the main concepts, theories, problems and metaphors concerning INFORMATION in all spheres of knowledge. This is the first edition of an ambitious enterprise covering at its completion all relevant notions relating to INFORMATION in any scientific context. As such, this glossariumBITri is part of the broader project BITrum, which is committed to the mutual understanding of all disciplines devoted to information across fields of knowledge and practice. This glossary pretends to make explicit the conflicts and agreements among use and meaning of terms related to information phenomena. Information is approached from opposing paradigms and also from competing and cooperating disciplines. Both in science and in ordinary life, conceptual, ethical, technical and societal problems regard information in an essential way. This glossary does not endorse or presuppose any paradigm or any theory, but rather locates into a public, explicit and commonly understandable space some of the crucial assumptions dividing informational concepts, theories, problems and metaphors. Moreover, we purport to embrace all distinct paradigms with a critical and comprehensive attitude. The glossary is the result of an original methodology, which places any entrance under the responsibility of its editor. Authors possibly distinct from the editor contribute to different articles with texts, comments or discussions. Since authors come from many distinct fields of knowledge, each article should reflect many perspectival but rigorous approaches. The glossary is an open work: the number and contents of all its entrances are updated and submitted to revision by editors and authors. For this reason, this first edition is only a first step in the active development of this collaborative methodology. Any interested reader wishing to contribute, may contact the general editors. This glossary is most indebted to the enthusiasm and work of José María Díaz Nafría. The editorial team, authors and correctors thank the Universidad de León and Caja España for their support to this initiative