• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2021-01-05
Martín Hötzel Escardó

We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, and algebraic injectivity is defined by a given section of the restriction map along any embedding. Under propositional resizing axioms, the main results

更新日期：2021-01-05
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-12-29
Thomas Streicher

I recall how Martin Hofmann and I found the groupoid model of type theory in the early 1990s.

更新日期：2020-12-29
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-12-18

The present work achieves a mathematical, in particular syntax-independent, formulation of dynamics and intensionality of computation in terms of games and strategies. Specifically, we give game semantics of a higher-order programming language that distinguishes programmes with the same value yet different algorithms (or intensionality) and the hiding operation on strategies that precisely corresponds

更新日期：2020-12-18
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-11-24
James Wallbridge

We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth

更新日期：2020-11-25
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-10-27
Hans Kleine Büning; P. Wojciechowski; K. Subramani

In this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that

更新日期：2020-11-21
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-10-19
Alain Finkel; Jean Goubault-Larrecq

We define representations for downward-closed subsets of a rich family of well-quasi-orders, and more generally for closed subsets of an even richer family of Noetherian topological spaces. This includes the cases of finite words, of multisets, of finite trees, notably. Those representations are given as finite unions of ideals, or more generally of irreducible closed subsets. All the representations

更新日期：2020-11-21
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-11-09

We give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory Q, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere

更新日期：2020-11-21
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-10-28
Kevin Coulembier; Ross Street; Michel van den Bergh

Given a monoidal category $\mathcal C$ with an object J, we construct a monoidal category $\mathcal C[{J^ \vee }]$ by freely adjoining a right dual ${J^ \vee }$ to J. We show that the canonical strong monoidal functor $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual

更新日期：2020-10-30
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-10-26
Felix Cherubini; Egbert Rijke

Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization

更新日期：2020-10-30
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-10-14
Simon Boulier; Nicolas Tabareau

Model categories constitute the major context for doing homotopy theory. More recently, homotopy type theory (HoTT) has been introduced as a context for doing syntactic homotopy theory. In this paper, we show that a slight generalization of HoTT, called interval type theory (⫿TT), allows to define a model structure on the universe of all types, which, through the model interpretation, corresponds to

更新日期：2020-10-15
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-10-08
Mauricio Ayala-Rincón; Philippe Balbiani

In the days of its foundation, the field of science covered by UNIF – a series of annual international workshops on unification – was still in its infancy. With the advent of automated reasoning, term rewriting, logic programming, natural language processing, and program analysis, the areas of computer science concerned by unification were seething with excitement. With the coming out of researches

更新日期：2020-10-08
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-09-16
Ajay Kumar Eeralla; Christopher Lynch

We consider the problem of the unification modulo an equational theory associativity and commutativity (ACh), which consists of a function symbol h that is homomorphic over an associative–commutative operator +. Since the unification modulo ACh theory is undecidable, we define a variant of the problem called bounded ACh unification. In this bounded version of ACh unification, we essentially bound the

更新日期：2020-10-08
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-09-02
Auke B. Booij

Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known. We overcome this by considering real numbers equipped with additional structure, which we call a locator. With this structure, it is possible, for instance, to

更新日期：2020-09-02
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Math. Struct. Comput. Sci. (IF 0.647) Pub Date : 2020-04-07

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
• 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
• 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
• 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
• 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
• 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
• 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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