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 associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.

We build a logical system named a conjunctive sequent calculus which is a conjunctive fragment of the classical propositional sequent calculus in the sense of proof theory. We prove that a special class of formulae of a consistent conjunctive sequent calculus forms a bounded complete continuous domain without greatest element (for short, a proper BC domain), and each proper BC domain can be obtained in this way. More generally, we present conjunctive consequence relations as morphisms between consistent conjunctive sequent calculi and build a category which is equivalent to that of proper BC domains with Scott-continuous functions. A logical characterization of purely syntactic form for proper BC domains is obtained.

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 in constraint solving and admissibility of inference rules and with the breaking out of applications, such as type checking, query answering, and cryptographic protocol analysis, the development of unification was not long in going at full speed.

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 maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic, and hence a computational theory of non-linear partial differential equations and their solutions based on variational calculus.

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 ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

We develop a technique for representing and preserving cardinals in realizability models, and we apply this technique to define a realizability model of Zorn’s lemma restricted to an ordinal.

In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.

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. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves internally right orthogonal to the object of clocks. In the case of dependent types, this translates to a lifting condition similar to the one found in homotopy theoretic models of type theory, but here with an additional requirement of uniqueness of lifts. Since the universes defined by the standard Hofmann–Streicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. We show how to model these universes in such a way that inclusions of clock contexts give rise to inclusions of universes commuting with type operations on the nose.

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 that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.

We construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

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 paper: we study verification over a general model of relational, data-aware processes, to assess (parameterized) safety properties irrespectively of the initial database (DB) instance. Toward this goal, we take inspiration from array-based systems and tackle safety algorithmically via backward reachability. To enable the adoption of this technique in our rich setting, we make use of the model-theoretic machinery of model completion, which surprisingly turns out to be an effective tool for verification of relational systems and represents the main original contribution of this paper. In this way, we pursue a twofold purpose. On the one hand, we isolate three notable classes for which backward reachability terminates, in turn witnessing decidability. Two of such classes relate our approach to conditions singled out in the literature, whereas the third one is genuinely novel. On the other hand, we are able to exploit SMT technology in implementations, building on the well-known MCMT (Model Checker Modulo Theories) model checker for array-based systems and extending it to make all our foundational results fully operational. All in all, the present contribution is deeply rooted in the long-standing tradition of the application of model theory in computer science. In particular, this paper applies these ideas in an original mathematical context and shows how these techniques can be used for the first time to empower algorithmic techniques for the verification of infinite-state systems based on arrays, so as to make such techniques applicable to the timely, challenging settings of data-aware processes.

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 of step functions of Turing machines. Along the way we prove some useful technical results about linear independence of denotations of Church numerals and binary integers.

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.

Sobriety, well-filteredness, and monotone convergence are three of the most important properties of topological spaces extensively studied in domain theory. Some other weak forms of sobriety and well-filteredness have also been investigated by some authors. In this paper, we introduce the notion of Θ-fine spaces, which provides a unified approach to such properties. In addition, this general approach leads to the definitions of some new topological properties.

Is the current formulation of multiset theory, which is based on sets and multiplicities of their elements, adequate? We exhibit both mathematical and metamathematical reasons which should cause one to rethink the definition. Some problems with multiset theory in its accepted formulation concern even the basic operations of union, intersection, and complement; others, more deeply rooted, concern Cartesian products, relations, or morphisms. We compare current definitions and conclude that the problems of multiset theory need to be resolved at the fundamental level of sets and mappings (or equivalent constructs) with multiplicities introduced only as a secondary concept. As a consequence, we propose to define multisets as families. A mapping establishes the connection to the familiar theory of multisets. Without losing anything, our proposal is simple and provides for an elegant mathematical theory.

Network models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by a monoid. A feature of the ordinary construction of network models is that it imposes commutativity relations between all edge components. Because of this, it cannot be used to model networks with bounded degree. In this paper, we construct the free network model on a given monoid, which can model networks with bounded degree. To do this, we generalize Green’s graph products of groups to pointed categories which are finitely complete and cocomplete.

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 we explore are computable, in the sense that we exhibit algorithms that decide inclusion, and compute finite unions and finite intersections. The origin of this work lies in the need for computing finite representations of sets of successors of the downward closure of one state, or more generally of a downward-closed set of states, in a well-structured transition system, and this is where we start: we define adequate notions of completions of well-quasi-orders, and more generally, of Noetherian spaces. For verification purposes, we argue that the required completions must be ideal completions, or more generally sobrifications, that is, spaces of irreducible closed subsets.

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 of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.

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 to the (small-step) operational semantics (or dynamics) of the language. Categorically, our games and strategies give rise to a cartesian closed bicategory, and our game semantics forms an instance of a bicategorical generalisation of the standard interpretation of functional programming languages in cartesian closed categories. This work is intended to be a step towards a mathematical foundation of intensional and dynamic aspects of logic and computation; it should be applicable to a wide range of logics and computations.