We give a first-principles description of the context semantics of Gonthier, Abadi, and Lévy, a computer-science analogue of Girard’s geometry of interaction. We explain how this denotational semantics models λ-calculus, and more generally multiplicative-exponential linear logic (MELL), by explaining the call-by-name (CBN) coding of the λcalculus, and proving the correctness of readback, where ...

The discrete logarithm problem forms the basis of numerous cryptographic systems. The most eeective attack on the discrete logarithm problem in the multiplicative group of a nite eld is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller 23] has given a brief heuristic argument as to why no such method can exist. IN this note we give a deta...

We reconsider work by Bellin and Scott in the 1990s on R. Milner and S. Abramsky’s encoding of linear logic in the π-calculus and give an account of efforts to establish a tight connection between the structure of proofs and of the cut elimination process in multiplicative linear logic, on one hand, and the input-output behaviour of the processes that represent them, on the other, resulting in ...

The notion of multilinear quantum Lie operation appears naturally in connection with a different attempts to generalize the Lie algebras. There are a number of reasons why the generalizations are necessary. First of all this is the demands for a ”quantum algebra” which was formed in the papers by Ju.I. Manin, V.G. Drinfeld, S.L. Woronowicz, G. Lusztig, L.D. Faddeev, and many others. These deman...

In the seminal work [12] Retoré introduced Pomset logic, an extension of linear logic with a self-dual noncommutative connective. Pomset logic is defined by means of proof-nets, later a deep inference system BV [8] was designed for this extension, but equivalence of system has not been proven up to now. As for a sequent calculus formulation, it has not been known for either of these logics, and...

Using Serre’s adelic interpretation of the cohomology, we develop “differential and integral calculus” on an algebraic curve X over an algebraically closed constant field k of characteristic zero, define an algebraic analogs of additive and multiplicative multi-valued functions on X, and prove corresponding generalized residue theorem and A. Weil reciprocity law. Using the representation theory...

Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expre...

Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cut-elimination. We give a simple and annotation-free display calculus for FILL which satisfies Belnap’s generic cutelimi...

Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cut-elimination. We give a simple and annotation-free display calculus for FILL which satisfies Belnap’s generic cut-elim...