Conservative extensions of the λ-calculus for the computational interpretation of sequent calculus

This thesis offers a study of the Curry-Howard correspondence for a certain fragment (the canonical fragment) of sequent calculus based on an investigation of the relationship between cut elimination in that fragment and normalisation. The output of this study may be summarised in a new assignment θ, to proofs in the canonical fragment, of terms from certain conservative extensions of the λ-calculus. This assignment, in a sense, is an optimal improvement over the traditional assignment φ, in that it is an isomorphism both in the sense of sound bijection of proofs and isomorphism of normalisation procedures.First, a systematic definition of calculi of cut-elimination for the canonical fragment is carried out. We study various right protocols, i.e. cut-elimination procedures which give priority to right permutation. We pay particular attention to the issue of what parts of the procedure are to be implicit, that is, performed by meta-operators in the style of natural deduction. Next, a comprehensive study of the relationship between normalisation and these calculi of cut-elimination is done, producing several new insight of independent interest, particularly concerning a generalisation of Prawitz’s mapping of normal natural deduction proofs into sequent calculus.This study suggests the definition of conservative extensions of natural deduc­tion (and λ-calculus) based on the idea of a built-in distinction between applica­tive term and application, and also between head and tail application. These extensions offer perfect counterparts to the calculi in the canonical fragment, as established by the mentioned mapping θ . Conceptual rearrangements in proof- theory deriving from these extensions of natural deduction are discussed.Finally, we argue that, computationally, both the canonical fragment and natural deduction (in the extended sense introduced here) correspond to extensions of the λ-calculus with applicative terms; and that what distinguishes them is the way applicative terms are structured. In the canonical fragment, the head application of an applicative term is “focused” . This, in turn, explains the following observation: some reduction rules of calculi in the canonical fragment may be interpreted as transition rules for abstract call-by-name machines

Deep Inference and Symmetry in Classical Proofs

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems

Revisiting the correspondence between cut-elimination and normalisation

Cut-free proofs in Herbelin's sequent calculus are in 1-1 correspondence with normal natural deduction proofs. For this reason Herbelin's sequent calculus has been considered a privileged middle-point between L-systems and natural deduction. However, this bijection does not extend to proofs containing cuts and Herbelin observed that his cut-elimination procedure is not isomorphic to $\beta$-reduction. In this paper we equip Herbelin's system with rewrite rules which, at the same time: (1) complete in a sense the cut elimination procedure firstly proposed by Herbelin; and (2) perform the intuitionistic "fragment'' of the tq-protocol - a cut-elimination procedure for classical logic defined by Danos, Joinet and Schellinx. Moreover we identify the subcalculus of our system which is isomorphic to natural deduction, the isomorphism being with respect not only to proofs but also to normalisation. Our results show, for the implicational fragment of intuitionistic logic, how to embed natural deduction in the much wider world of sequent calculus and what a particular cut-elimination procedure normalisation is.Fundação para a Ciência e a Tecnologia (FCT)

From X to Pi; Representing the Classical Sequent Calculus in the Pi-calculus

We study the Pi-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor "arrow". We encode the circuits of the calculus X into this variant of Pi, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies that all proofs in LK have a representation in Pi.Comment: International Workshop on Classical Logic and Computation (CL&C'08), Reykjavik, Iceland, July 200

The dagger lambda calculus

We present a novel lambda calculus that casts the categorical approach to the study of quantum protocols into the rich and well established tradition of type theory. Our construction extends the linear typed lambda calculus with a linear negation of "trivialised" De Morgan duality. Reduction is realised through explicit substitution, based on a symmetric notion of binding of global scope, with rules acting on the entire typing judgement instead of on a specific subterm. Proofs of subject reduction, confluence, strong normalisation and consistency are provided, and the language is shown to be an internal language for dagger compact categories.Comment: In Proceedings QPL 2014, arXiv:1412.810

Generic Modal Cut Elimination Applied to Conditional Logics

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature

Dual-Context Calculi for Modal Logic

We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089