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In the Boehm theorem workshop on Crete island, Zoran Petric called Statman's ``Typical Ambiguity theorem'' typed Boehm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda…
The categorical models of the differential lambda-calculus are additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential…
Ten years ago, it was shown that nominal techniques can be used to design coalgebraic data types with variable binding, so that alpha-equivalence classes of infinitary terms are directly endowed with a corecursion principle. We introduce…
We give a new characterization of elementary and deterministic polynomial time computation in linear logic through the proofs-as-programs correspondence. Girard's seminal results, concerning elementary and light linear logic, achieve this…
We study a resource-sensitive fragment of the problem of extracting a logical discipline from a class of neural architectures by passing through categorization. The starting point is not a pre-existing logic but a category of zone-labelled…
A number of models of linear logic are based on or closely related to linear algebra, in the sense that morphisms are "matrices" over appropriate coefficient sets. Examples include models based on coherence spaces, finiteness spaces and…
We explore a proof language for intuitionistic multiplicative additive linear logic, incorporating the sup connective that introduces additive pairs with a probabilistic elimination, and sum and scalar products within the proof-terms. We…
This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently…
We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for…
Recent developments in the categorical foundations of universal algebra have given fresh impetus to an understanding of the lambda calculus coming from categorical logic: an interpretation is a semi-closed algebraic theory. Scott's…
We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…
First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant…
Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic called…
We present $\cal L$, an extension of Parigot's $\lambda\mu$-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction…
The lambda-PRK-calculus is a typed lambda-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend lambda-PRK to…
A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and…
In this thesis we give an algebraic characterization of the syntax and semantics of simply-typed languages. More precisely, we characterize simply-typed binding syntax equipped with reduction rules via a universal property, namely as the…
Formal mathematics and computer science proofs are formalized using Hilbert-Russell-style logical systems which are designed to not admit paradoxes and self-refencing reasoning. These logical systems are natural way to describe and reason…
We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott's original model of the pure lambda calculus. This calculus…
We show that any multiple-valued function can be represented by a linear lambda term typed in a second-order polymorphic type system, using two distinct styles. The first is a circuit style, which mimics combinational circuits in switching…