Sergei Soloviev

The genericity theorem and parametricity in the polymorphic l-calculus

Abstract This paper focuses on how terms of the polymorphic λ-calculus, which may take types as inputs, depend on types. These terms are generally understood, in all models, to have an “essentially” constant meaning on input types. We show the proof theory of polymorphic λ-calculus suggests a clear syntactic description of this phenomenon. Namely, under a reasonable condition, we show that if two polymorphic functions agree on a single type, then they agree on all types (equivalently, types are generic inputs).

Coercion completion and conservativity in coercive subtyping

Abstract Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory such as Martin–Lofs type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.

Some Algorithmic and Proof-Theoretical Aspects of Coercive Subtyping

Coercive subtyping offers a conceptually simple but powerful framework to understand subtyping and subset relationships in type theory. In this paper we study some of its proof-theoretic and computational properties.

Remarks on isomorphisms of simple inductive types

We study isomorphisms of types in the system of simply-typed λ-calculus with inductive types and recursion operators. It is shown that in some cases (multiproducts, copies of types), it is possible to add new reductions in such a way that strong normalisation and confluence of the calculus are preserved, and the isomorphisms may be regarded as intensional w.r.t. a stronger equality relation.

A Deciding Algorithm for Linear Isomorphism of Types with Complexity O (n log2(n))

It is known, that ordinary isomorphisms (associativity and commutativity of “times”, isomorphisms for “times” unit and currying) provide a complete axiomatisation of isomorphism of types in multiplicative linear lambda calculus (isomorphism of objects in a free symmetric monoidal closed category). One of the reasons to consider linear isomorphism of types instead of ordinary isomorphism was that better complexity could be expected. Meanwhile, no upper bounds reasonnably close to linear were obtained. We describe an algorithm deciding if two types are linearly isomorphic with complexity O(nlog 2(n)).

Proof of a conjecture of S. Mac Lane

Abstract Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (this is the only case considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadous proof of the “abstract coherence theorem” found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture also proves the correctness of the modified description.

Typed lambda-terms in categorical attributed graph transformation

This paper deals with model transformation based on attributed graph rewriting. Our contribution investigates a single pushout approach for applying the rewrite rules. The computation of graph attributes is obtained through the use of typed λ -calculus with inductive types. In this paper we present solutions to cope with single pushout construction for the graph structure and the computations functions. As this rewrite system uses inductive types, the expr essiveness of attribute computations is facilitated and appears more efficient than the one based on Σ-algebras. Some examples showing the interest of our computation approach are described in this paper.

Inductive type schemas as functors

Parametric inductive types can be seen as functions taking type parameters as arguments and returning the instantiated inductive types. Given functions between parameters one can construct a function between the instantiated inductive types representing the change of parameters along these functions. It is well known that it is not a functor w.r.t. intensional equality based on standard reductions. We investigate a simple type system with inductive types and iteration and show by modular rewriting techniques that new reductions can be safely added to make this construction a functor, while the decidability of the internal conversion relation based on the strong normalization and confluence properties is preserved. Possible applications: new categorical and computational structures on λ-calculus, certified computation.

Computations in graph rewriting: inductive types and pullbacks in DPO approach

In this paper, we give a new formalism for attributed graph rewrites resting on category theory and type theory. Our main goal is to offer a single theoretical foundation that embeds the rewrite of structural parts of graphs and attribute computations which has more expressive power for attribute computations as well.

Coherence in SMCCs and equivalences on derivations in IMLL with unit

Abstract We study the coherence, that is the equality of canonical natural transformations in non-free symmetric monoidal closed categories ( smcc s). To this aim, we use proof theory for intuitionistic multiplicative linear logic ( imll ) with unit. The study of coherence in non-free smcc s is reduced to the study of equivalences on terms (representing morphisms) in the free category, which include the equivalences induced by the smcc structure. The free category is reformulated as the sequent calculus for imll with unit so that only equivalences on derivations in this system are to be considered. We establish that any equivalence induced by the equality of canonical natural transformations over a model can be axiomatized by some set of “critical” pairs of derivations. From this, we derive certain sufficient conditions for full coherence, and establish that the system of identities defining smcc s is not Post-complete: extending this system with an identity that does not hold in the free smcc does not in general cause the free smcc to collapse into a preorder. In order to give a larger context to these results, we study the equality of canonical morphisms in non-free symmetric monoidal categories, and establish that w.r.t. a broad subclass of smcc s, the equivalences induced by the equality of canonical natural transformations over a model coincide with the equivalences induced by the equality of canonical morphisms for all interpretations in that model.