arXiv subjects

Niccolò Veltri

Publications and source records attributed to Niccolò Veltri.

16 recordsLinked to original sources

arXiv2026

Doctrinal Semantics of Directed First-Order Logic

We present a first-order logic equipped with an "asymmetric" directed notion of equality, which can be thought of as rewrites between terms, allowing for types to be interpreted as preorders. The logic is equipped with a precise syntactic system of polarities, inspired by dinaturality, that keeps track of the occurrence of variables (positive/negative/both). We use this to give a characterization of directed equality as a relative left adjoint, generalizing the idea by Lawvere of equality as left adjoint; intuitively, the relativeness is used to capture the syntactic restriction that avoids symmetry of equality. The semantics of this logic and its system of variances is captured categorically using the notion of directed doctrine, which we prove sound and complete with respect to the syntax. Moreover, we prove that the classical fragment of our directed logic is complete with respect to a standard semantics in preorders.

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arXiv2026

Di- is for Directed: First-Order Directed Type Theory via Dinaturality

We show how dinaturality plays a central role in the interpretation of directed type theory where types are interpreted as (1-)categories and directed equality is represented by $\hom$-functors. We present a general elimination principle based on dinaturality for directed equality which very closely resembles the $J$-rule used in Martin-Löf type theory, and we highlight which syntactical restrictions are needed to interpret this rule in the context of directed equality. We then use these rules to characterize directed equality as a left relative adjoint to a functor between (para)categories of dinatural transformations which contracts together two variables appearing naturally with a single dinatural one, with the relative functor imposing the syntactic restrictions needed. We then argue that the quantifiers of such a directed type theory should be interpreted as ends and coends, which dinaturality allows us to present in adjoint-like correspondences to a weakening functor. Using these rules we give a formal interpretation to Yoneda reductions and (co)end calculus, and we use logical derivations to prove the Fubini rule for quantifier exchange, the adjointness property of Kan extensions via (co)ends, exponential objects of presheaves, and the (co)Yoneda lemma. We show transitivity (composition), congruence (functoriality), and transport (coYoneda) for directed equality by closely following the same approach of Martin-Löf type theory, with the notable exception of symmetry. We formalize our main theorems in Agda.

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arXiv2025

Derivatives for Containers in Univalent Foundations

Containers conveniently represent a wide class of inductive data types. Their derivatives compute representations of types of one-hole contexts, useful for implementing tree-traversal algorithms. In the category of containers and cartesian morphisms, derivatives of discrete containers (whose positions have decidable equality) satisfy a universal property. Working in Univalent Foundations, we extend the derivative operation to untruncated containers (whose shapes and positions are arbitrary types). We prove that this derivative, defined in terms of a set of isolated positions, satisfies an appropriate universal property in the wild category of untruncated containers and cartesian morphisms, as well as basic laws with respect to constants, sums and products. A chain rule exists, but is in general non-invertible. In fact, a globally invertible chain rule is inconsistent in the presence of non-set types, and equivalent to a classical principle when restricted to set-truncated containers. We derive a rule for derivatives of smallest fixed points from the chain rule, and characterize its invertibility. All of our results are formalized in Cubical Agda.

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arXiv2024

Semi-Substructural Logics with Additives

This work concerns the proof theory of (left) skew monoidal categories and their variants (e.g. closed monoidal, symmetric monoidal), continuing the line of work initiated in recent years by Uustalu et al. Skew monoidal categories are a weak version of Mac Lane's monoidal categories, where the structural laws are not required to be invertible, they are merely natural transformations with a specific orientation. Sequent calculi which can be modelled in such categories can be identified as deductive systems for restricted substructural fragments of intuitionistic linear logic. These calculi enjoy cut elimination and admit a focusing strategy, sharing resemblance with Andreoli's normalization technique for linear logic. The focusing procedure is useful for solving the coherence problem of the considered categories with skew structure. Here we investigate possible extensions of the sequent calculi of Uustalu et al. with additive connectives. As a first step, we extend the sequent calculus with additive conjunction and disjunction, corresponding to studying the proof theory of skew monoidal categories with binary products and coproducts satisfying a left-distributivity condition. We introduce a new focused sequent calculus of derivations in normal form, which employs tag annotations to reduce non-deterministic choices in bottom-up proof search. The focused sequent calculus and the proof of its correctness have been formalized in the Agda proof assistant. We also discuss extensions of the logic with additive units, a form of skew exchange and linear implication.

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arXiv2022

Bicategories in Univalent Foundations

We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories in a modular fashion, we develop displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. We demonstrate the applicability of this notion, and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-categories is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.

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arXiv2022

Normalization by Evaluation for the Lambek Calculus

The syntactic calculus of Lambek is a deductive system for the multiplicative fragment of intuitionistic non-commutative linear logic. As a fine-grained calculus of resources, it has many applications, mostly in formal computational investigations of natural languages. This paper introduces a calculus of beta-eta-long normal forms for derivations in the Lambek calculus with multiplicative unit and conjunction among its logical connectives. Reduction to normal form follows the normalization by evaluation (NbE) strategy: (i) evaluate a derivation in a Kripke model of Lambek calculus; (ii) extract normal forms from the obtained semantic values. The implementation of the NbE algorithm requires the presence of a strong monad in the Kripke interpretation of positive formulae, in analogy with the extension of intuitionistic propositional logic with falsity and disjunction. The NbE algorithm can also be instantiated with minor variations to calculi for related substructural logics, such as multiplicative and dual intuitionistic linear logic (MILL and DILL).

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arXiv2022

Proof Theory of Skew Non-Commutative MILL

Monoidal closed categories naturally model NMILL, non-commutative multiplicative intuitionistic linear logic: the monoidal unit and tensor interpret the multiplicative verum and conjunction; the internal hom interprets linear implication. In recent years, the weaker notion of (left) skew monoidal closed category has been proposed by Ross Street, where the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation. A question arises: is it possible to find a logic which is naturally modelled by skew monoidal closed categories? We answer positively by introducing a cut-free sequent calculus for a skew version of NMILL that is a presentation of the free skew monoidal closed category. We study the proof-theoretic semantics of the sequent calculus by identifying a calculus of derivations in normal form, obtained from an adaptation of Andreoli's focusing technique to the skew setting. The resulting focused sequent calculus peculiarly employs a system of tags for keeping track of new formulae appearing in the antecedent and appropriately reducing non-deterministic choices in proof search. Focusing solves the coherence problem for skew monoidal closed categories by exhibiting an effective procedure for deciding equality of maps in the free such category.

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arXiv2021

Inductive and Coinductive Predicate Liftings for Effectful Programs

We formulate a framework for describing behaviour of effectful higher-order recursive programs. Examples of effects are implemented using effect operations, and include: execution cost, nondeterminism, global store and interaction with a user. The denotational semantics of a program is given by a coinductive tree in a monad, which combines potential return values of the program in terms of effect operations. Using a simple test logic, we construct two sorts of predicate liftings, which lift predicates on a result type to predicates on computations that produce results of that type, each capturing a facet of program behaviour. Firstly, we study inductive predicate liftings which can be used to describe effectful notions of total correctness. Secondly, we study coinductive predicate liftings, which describe effectful notions of partial correctness. The two constructions are dual in the sense that one can be used to disprove the other. The predicate liftings are used as a basis for an endogenous logic of behavioural properties for higher-order programs. The program logic has a derivable notion of negation, arising from the duality of the two sorts of predicate liftings, and it generates a program equivalence which subsumes a notion of bisimilarity. Appropriate definitions of inductive and coinductive predicate liftings are given for a multitude of effect examples. The whole development has been fully formalized in the Agda proof assistant.

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arXiv2021

Constructing Higher Inductive Types as Groupoid Quotients

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. Then we construct a biinitial object in the bicategory of algebras in groupoids, which gives the desired algebra. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. We present several examples of HITs which are definable using our notion of signature. In particular, we show that each signature gives rise to a HIT corresponding to the freely generated algebraic structure over it. We also start the development of universal algebra in 1-types. We show that the bicategory of algebras has PIE limits, i.e. products, inserters and equifiers, and we prove a version of the first isomorphism theorem for 1-types. Finally, we give an alternative characterization of the foundamental groups of some HITs, exploiting our construction of HITs via the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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arXiv2021

Proof Theory of Partially Normal Skew Monoidal Categories

The skew monoidal categories of Szlachányi are a weakening of monoidal categories where the three structural laws of left and right unitality and associativity are not required to be isomorphisms but merely transformations in a particular direction. In previous work, we showed that the free skew monoidal category on a set of generating objects can be concretely presented as a sequent calculus. This calculus enjoys cut elimination and admits focusing, i.e. a subsystem of canonical derivations, which solves the coherence problem for skew monoidal categories. In this paper, we develop sequent calculi for partially normal skew monoidal categories, which are skew monoidal categories with one or more structural laws invertible. Each normality condition leads to additional inference rules and equations on them. We prove cut elimination and we show that the calculi admit focusing. The result is a family of sequent calculi between those of skew monoidal categories and (fully normal) monoidal categories. On the level of derivability, these define 8 weakenings of the (unit,tensor) fragment of intuitionistic non-commutative linear logic.

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arXiv2021

Deductive Systems and Coherence for Skew Prounital Closed Categories

In this paper, we develop the proof theory of skew prounital closed categories. These are variants of the skew closed categories of Street where the unit is not represented. Skew closed categories in turn are a weakening of the closed categories of Eilenberg and Kelly where no structural law is required to be invertible. The presence of a monoidal structure in these categories is not required. We construct several equivalent presentations of the free skew prounital closed category on a given set of generating objects: a categorical calculus (Hilbert-style system), a cut-free sequent calculus and a natural deduction system corresponding to a variant of planar (= non-commutative linear) typed lambda-calculus. We solve the coherence problem for skew prounital closed categories by showing that the sequent calculus admits focusing and presenting two reduction-free normalization procedures for the natural deduction calculus: normalization by evaluation and hereditary substitutions. Normal natural deduction derivations (beta-eta-long forms) are in one-to-one correspondence with derivations in the focused sequent calculus. Unexpectedly, the free skew prounital closed category on a set satisfies a left-normality condition which makes it lose its skew aspect. This pitfall can be avoided by considering the free skew prounital closed category on a skew multicategory instead. The latter has a presentation as a cut-free sequent calculus for which it is easy to see that the left-normality condition generally fails. The whole development has been fully formalized in the dependently typed programming language Agda.

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arXiv2020

Ticking clocks as dependent right adjoints: Denotational semantics for clocked type theory

Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract form of step-indexing. CloTT has previously been shown to enjoy a number of syntactic properties including strong normalisation, canonicity and decidability of the equational theory. In this paper we present a denotational semantics for CloTT useful, e.g., for studying future extensions of CloTT with constructions such as path types. The main challenge for constructing this model is to model the notion of ticks on a clock used in CloTT for coinductive reasoning about coinductive types. We build on a category previously used to model guarded recursion with multiple clocks. In this category there is an object of clocks but no object of ticks, and so tick-assumptions in a context can not be modelled using standard tools. Instead we model ticks using dependent right adjoint functors, a generalisation of the category theoretic notion of adjunction to the setting of categories with families. Dependent right adjoints are known to model Fitch-style modal types, but in the case of CloTT, the modal operators constitute a family indexed internally in the type theory by clocks. We model this family using a dependent right adjoint on the slice category over the object of clocks. Finally we show how to model the tick constant of CloTT using a semantic substitution. This work improves on a previous model by the first two named authors which not only had a flaw but was also considerably more complicated.

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arXiv2020

The Sequent Calculus of Skew Monoidal Categories

Szlachányi's skew monoidal categories are a well-motivated variation of monoidal categories in which the unitors and associator are not required to be natural isomorphisms, but merely natural transformations in a particular direction. We present a sequent calculus for skew monoidal categories, building on the recent formulation by one of the authors of a sequent calculus for the Tamari order (skew semigroup categories). In this calculus, antecedents consist of a stoup (an optional formula) followed by a context, and the connectives behave like in the standard monoidal sequent calculus except that the left rules may only be applied in stoup position. We prove that this calculus is sound and complete with respect to existence of maps in the free skew monoidal category, and moreover that it captures equality of maps once a suitable equivalence relation is imposed on derivations. We then identify a subsystem of focused derivations and establish that it contains exactly one canonical representative from each equivalence class. This coherence theorem leads directly to simple procedures for deciding equality of maps in the free skew monoidal category and for enumerating any homset without duplicates. Finally, and in the spirit of Lambek's work, we describe the close connection between this proof-theoretic analysis and Bourke and Lack's recent characterization of skew monoidal categories as left representable skew multicategories. We have formalized this development in the dependently typed programming language Agda.

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arXiv2020

Terminal Coalgebras and Non-wellfounded Sets in Homotopy Type Theory

Non-well-founded material sets have been modelled in Martin-L\"of type theory by Lindstr\"om using setoids. In this paper we construct models of non-wellfounded material sets in Homotopy Type Theory (HoTT) where equality is interpreted as the identity type. The first model satisfies Scott's Anti-Foundation Axiom (SAFA) and dualises the construction of iterative sets. The second model satisfies Aczel's Anti-Foundation Axiom (AFA), and is constructed by adaption of Aczel-Mendler's terminal coalgebra theorem to type theory, which requires propositional resizing. In an bid to extend coalgebraic theory and anti-foundation axioms to higher type levels, we formulate generalisations of AFA and SAFA, and construct a hierarchy of models which satisfies the SAFA generalisations. These generalisations build on the framework of Univalent Material Set Theory, previously developed by two of the authors. Since the model constructions are based on M-types, the paper also includes a characterisation of the identity type of M-types as indexed M-types. Our results are formalised in the proof-assistant Agda.

math.LO↗

arXiv2018

Bisimulation as path type for guarded recursive types

In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.

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arXiv2016

Variations on Noetherianness

In constructive mathematics, several nonequivalent notions of finiteness exist. In this paper, we continue the study of Noetherian sets in the dependently typed setting of the Agda programming language. We want to say that a set is Noetherian, if, when we are shown elements from it one after another, we will sooner or later have seen some element twice. This idea can be made precise in a number of ways. We explore the properties and connections of some of the possible encodings. In particular, we show that certain implementations imply decidable equality while others do not, and we construct counterexamples in the latter case. Additionally, we explore the relation between Noetherianness and other notions of finiteness.

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