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We introduce the Delta-framework, LF-Delta, a dependent type theory based on the Edinburgh Logical Framework LF, extended with the strong proof-functional connectives, i.e. strong intersection, minimal relevant implication and strong union. Strong proof-functional connectives take into account the shape of logical proofs, thus reflecting polymorphic features of proofs in formulae. This is in contrast to classical or intuitionistic connectives where the meaning of a compound formula depends only on the truth value or the provability of its subformulae. Our framework encompasses a wide range of type disciplines. Moreover, since relevant implication permits to express subtyping, LF-Delta subsumes also Pfenning's refinement types. We discuss the design decisions which have led us to the formulation of LF-Delta, study its metatheory, and provide various examples of applications. Our strong proof-functional type theory can be plugged in existing common proof assistants.

In 2005, Abramsky introduced various linear/affine combinatory algebras of partial involutions over a suitable formal language, to discuss reversible computation in a game-theoretic setting. These algebras arise as instances of the general paradigm explored by Haghverdi (Abramsky's Programme), which amounts to defining a lambda-algebra starting from a GoI Situation in a traced symmetric monoidal category. We investigate this construction from the point of view of the model theory of lambda-calculus. We focus on the strictly linear and affine parts of Abramsky's Affine Combinatory Algebras, sketching how to encompass the full algebra. The gist of our approach is that the GoI interpretation of a term based on involutions is dual to the principal type of the term, w.r.t. the type discipline for a linear/affine lambda-calculus. In the general case the type discipline and the calculus need to be extended, resp., with intersection, !-types, and !-abstractions. Our analysis unveils three conceptually independent, but ultimately equivalent, accounts of application in the lambda-calculus: beta-reduction, the GoI application of involutions based on symmetric feedback (Girard's Execution Formula), and unification of principal types. Thus we provide an answer, in the strictly affine case, to the question raised in [1] of characterising the partial involutions arising from bi-orthogonal pattern matching automata, which are denotations of affine combinators, and we point to the answer to the full question. Furthermore, we prove that the strictly linear combinatory algebra of partial involutions is a strictly linear lambda-algebra, albeit not a combinatory model, while both the strictly affine combinatory algebra and the full affine combinatory algebra are not. To check all the equations involved in the definition of affine lambda-algebra, we implement in Erlang application of involutions., Comment: 43 pages (22+21 of Appendix)

We extend the constructive dependent type theory of the Logical Framework $\mathsf{LF}$ with monadic, dependent type constructors indexed with predicates over judgements, called Locks. These monads capture various possible proof attitudes in establishing the judgment of the object logic encoded by an $\mathsf{LF}$ type. Standard examples are factoring-out the verification of a constraint or delegating it to an external oracle, or supplying some non-apodictic epistemic evidence, or simply discarding the proof witness of a precondition deeming it irrelevant. This new framework, called Lax Logical Framework, $\mathsf{LLF}_{\cal P}$, is a conservative extension of $\mathsf{LF}$, and hence it is the appropriate metalanguage for dealing formally with side-conditions in rules or external evidence in logical systems. $\mathsf{LLF}_{\cal P}$ arises once the monadic nature of the lock type-constructor, ${\cal L}^{\cal P}_{M,\sigma}[\cdot]$, introduced by the authors in a series of papers, together with Marina Lenisa, is fully exploited. The nature of the lock monads permits to utilize the very Lock destructor, ${\cal U}^{\cal P}_{M,\sigma}[\cdot]$, in place of Moggi's monadic $let_T$, thus simplifying the equational theory. The rules for ${\cal U}^{\cal P}_{M,\sigma}[\cdot]$ permit also the removal of the monad once the constraint is satisfied. We derive the meta-theory of $\mathsf{LLF}_{\cal P}$ by a novel indirect method based on the encoding of $\mathsf{LLF}_{\cal P}$ in $\mathsf{LF}$. We discuss encodings in $\mathsf{LLF}_{\cal P}$ of call-by-value $\lambda$-calculi, Hoare's Logic, and Fitch-Prawitz Naive Set Theory., Comment: Accepted for publication in LMCS

In recent years, logical frameworks and tile logic have been separately proposed by our research groups, respectively in Udine and in Pisa, as suitable metalanguages with higher-order features for encoding and studying nominal calculi. This paper discusses the main features of the two approaches, tracing differences and analogies on the basis of two case studies: late π-calculus and lazy simply typed Λ-calculus.