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We show that the principal types of the closed terms of the affine fragment of $\lambda$-calculus, with respect to a simple type discipline, are structurally isomorphic to their interpretations, as partial involutions, in a natural Geometry of Interaction model \`a la Abramsky. This permits to explain in elementary terms the somewhat awkward notion of linear application arising in Geometry of Interaction, simply as the resolution between principal types using an alternate unification algorithm. As a consequence, we provide an answer, for the purely affine fragment, to the open problem raised by Abramsky of characterising those partial involutions which are denotations of combinatory terms.

We explore the possibility of extending Mardare et al. quantitative algebras to the structures which naturally emerge from Combinatory Logic and the lambda-calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results which clearly delineate to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras.

The Lax Logical Framework, LLFP, was introduced, by a team including the last two authors, to provide a conceptual framework for integrating different proof development tools, thus allowing for external evidence and for postponing, delegating, or factoring-out side conditions. In particular, LLFP allows for reducing the number of times a proof-irrelevant check is performed. In this paper we give a shallow, actually definitional, implementation of LLFP in Coq, i.e. we use Coq both as host framework and oracle for LLFP. This illuminates the principles underpinning the mechanism of Lock-types and also suggests how to possibly extend Coq with the features of LLFP. The derived proof editor is then put to use for developing case-studies on an emerging paradigm, both at logical and implementation level, which we call fast and loose reasoning following Danielsson et alii [6]. This paradigm trades off efficiency for correctness and amounts to postponing, or running in parallel, tedious or computationally demanding checks, until we are really sure that the intended goal can be achieved. Typical examples are branch-prediction in CPUs and optimistic concurrency control., Comment: In Proceedings LFMTP 2019, arXiv:1910.08712

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

We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLFP, is the canonical version of the system LLFP, presented earlier by the authors. The second system, CLLFP?, features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value lambda-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms., Comment: In Proceedings LFMTP 2015, arXiv:1507.07597

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science., Comment: 30 pages

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 of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic., Comment: 35 pages, published in Logical Methods in Computer Science