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  • Implicative algebras: a new foundation for realizability and forcing
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-07-03
    Alexandre Miquel

    We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show

    更新日期:2020-07-13
  • Nilpotent types and fracture squares in homotopy type theory
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-07-08
    Luis Scoccola

    We develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of fibrations with fiber an Eilenberg–Mac Lane space are proven. We also construct fracture squares for localizations away from sets of numbers. All of our proofs are constructive.

    更新日期:2020-07-13
  • Convenient antiderivatives for differential linear categories
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-07-03
    Jean-Simon Pacaud Lemay

    Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations

    更新日期:2020-07-13
  • Denotational semantics for guarded dependent type theory
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-05-08
    Aleš Bizjak; Rasmus Ejlers Møgelberg

    We present a new model of guarded dependent type theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with and reason about coinductive types. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants

    更新日期:2020-06-23
  • Encodings of Turing machines in linear logic
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-06-10
    James Clift; Daniel Murfet

    The Sweedler semantics of intuitionistic differential linear logic takes values in the category of vector spaces, using the cofree cocommutative coalgebra to interpret the exponential and primitive elements to interpret the differential structure. In this paper, we explicitly compute the denotations under this semantics of an interesting class of proofs in linear logic, introduced by Girard: the encodings

    更新日期:2020-06-23
  • Cofree coalgebras and differential linear logic
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-06-10
    James Clift; Daniel Murfet

    We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras is also a model of differential linear logic, and that the Cartesian closed category of cofree coalgebras is a model of the simply typed differential λ-calculus.

    更新日期:2020-06-23
  • Indexed type theories
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-05-22
    Valery Isaev

    In this paper, we define indexed type theories which are related to indexed (∞-)categories in the same way as (homotopy) type theories are related to (∞-)categories. We define several standard constructions for such theories including finite (co)limits, arbitrary (co)products, exponents, object classifiers, and orthogonal factorization systems. We also prove that these constructions are equivalent

    更新日期:2020-05-22
  • Rewriting with generalized nominal unification
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-05-22
    Yunus Kutz; Manfred Schmidt-Schauß

    We consider matching, rewriting, critical pairs and the Knuth–Bendix confluence test on rewrite rules in a nominal setting extended by atom-variables. We utilize atom-variables instead of atoms to formulate and rewrite rules on constrained expressions, which is an improvement of expressiveness over previous approaches. Nominal unification and nominal matching are correspondingly extended. Rewriting

    更新日期:2020-05-22
  • Higher-order pattern generalization modulo equational theories
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-05-20
    David M. Cerna; Temur Kutsia

    We consider anti-unification for simply typed lambda terms in theories defined by associativity, commutativity, identity (unit element) axioms and their combinations and develop a sound and complete algorithm which takes two lambda terms and computes their equational generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of such generalizations contains

    更新日期:2020-05-20
  • Open Petri nets
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-04-07
    John C. Baez; Jade Master

    The reachability semantics for Petri nets can be studied using open Petri nets. For us, an “open” Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category Open(Petri), which becomes symmetric monoidal under disjoint union

    更新日期:2020-04-07
  • SMT-based verification of data-aware processes: a model-theoretic approach
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-04-03
    Diego Calvanese; Silvio Ghilardi; Alessandro Gianola; Marco Montali; Andrey Rivkin

    In recent times, satisfiability modulo theories (SMT) techniques gained increasing attention and obtained remarkable success in model-checking infinite-state systems. Still, we believe that whenever more expressivity is needed in order to specify the systems to be verified, more and more support is needed from mathematical logic and model theory. This is the case of the applications considered in this

    更新日期:2020-04-03
  • Witness algebra and anyon braiding
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-03-16
    Andreas Blass; Yuri Gurevich

    Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of

    更新日期:2020-03-16
  • Computing knowledge in equational extensions of subterm convergent theories
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-03-02
    Serdar Erbatur; Andrew M. Marshall; Christophe Ringeissen

    We study decision procedures for two knowledge problems critical to the verification of security protocols, namely the intruder deduction and the static equivalence problems. These problems can be related to particular forms of context matching and context unification. Both problems are defined with respect to an equational theory and are known to be decidable when the equational theory is given by

    更新日期:2020-03-02
  • Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-01-21
    Silvio Ghilardi; Luigi Santocanale

    Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms

    更新日期:2020-01-21
  • Homotopy type-theoretic interpretations of constructive set theories
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2019-11-29
    Cesare Gallozzi

    We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation

    更新日期:2019-11-29
  • Extensions of unificationmodulo ACUI
    Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2019-11-11
    Franz Baader; Pavlos Marantidis; Antoine Mottet; Alexander Okhotin

    The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close

    更新日期:2019-11-11
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