Computational isomorphisms in classical logic: (Extended Abstract)

We prove that any pair of derivations, without structural rules, of F ) G and G ) F , where F , G are rst-order formulas `without any qualities', in a constrained classical sequent calculus LK p , denes a computational isomorphism up to an equivalence on derivations based upon reversibility properties of logical rules. This result gives a rationale behind the success of Girard's denotational semantics for classical logic, in which all standard `linear' boolean equations are satised. 1 Introduction 1.1 A patch of paradise to be broadened In recent work [1] devoted to the proof theory of classical logic, we embarked on the project of overcoming the obstacles that prevent cut from being a decent binary operation on the set of classical sequent derivations. To clarify what we mean by decency, let us have a look at the world of simply typed -calculus, which, seen from a normalization-as-computation point of view, is something close to a patch of paradise. danos@logique..

Computational isomorphisms in classical logic

All standard ‘linear’ boolean equations are shown to be computationally realized within a suitable classical sequent calculus LKη p. Specifically, LKη p can be equipped with a cut-elimination compatible equivalence on derivations based upon reversibility properties of logical rules. So that any pair of derivations, without structural rules, of F ⇒ G and G ⇒ F, where F, G are first-order formulas ‘without any qualities’, defines a computational isomorphism

Classical BI: Its Semantics and Proof Theory

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure

The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs

. We construct the exponential graph of a proof ß in (second order) linear logic, an artefact displaying the interdependencies of exponentials in ß. Within this graph superfluous exponentials are defined, the removal of which is shown to yield a correct proof ß . with essentially the same set of reductions. Applications to intuitionistic and classical proofs are given by means of reduction-preserving embeddings into linear logic. The last part of the paper puts things the other way round, and defines families of linear logics in which exponential dependencies are ruled by a given graph. We sketch some work in progress and possible applications. 1 Introduction An exponential "!", "?" in a linear proof is superfluous if we can remove it and obtain a proof that (1) is still correct, and (2) has the same dynamics as the original one. If we can get rid of an exponential in a linear proof, we know that the subproof introducing it (by a L? or a R! rule) will endure no non-linear handling (..

A New Deconstructive Logic: Linear Logic

The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's ¯, FD ([10, 12, 29, 33]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' ([24, 25]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructive purposes'..

A New Deconstructive Logic: Linear Logic

The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi..

On the Jordan-Holder Decomposition of Proof Nets

Having defined a notion of homology for paired graphs, M'etayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net G there exists a Jordan-Holder decomposition of H 0 (G). This decomposition is determined by a certain enumeration of the pairs in G. We correct his proof of this fact and show that there exists a 1-1 correspondence between these Jordan-Holder decompositions of H 0 (G) and the possible `construction-orders' of the par-net underlying G. 1 Introduction M'etayer ([Ma, Mb]) introduced a notion of homology for paired graphs. He showed that by means of the homology groups it is possible to characterize a proper subclass of paired graphs, a subclass that corresponds precisely to Girard's proof nets for multiplicative linear logic ([G]), thus arriving at a new, algebraical, equivalent of the well known correction criteria that characterize the inductively defined class of proof nets within the class of the more general proof struc..