This work presents a computational interpretation of the construction process for cyclic linear logic and non-commutative logic sequential proofs. We assume a proof construction paradigm, based on a normalisation procedure known as focussing, which efficiently manages the non-determinism of the construction. Similarly to the commutative case, a new formulation of focussing for NL is used to introduce a general constraint-based technique in order to dealwith partial information during proof construction. In particular, the procedure develops through construction steps propagating constraints in (...) intermediate objects called abstract proofs. (shrink)

This essay proposes a procedural interpretation of negative information in terms of split negation as procedural prohibition. Information frames and models are introduced, with negation defined as the implication of bottom, 0. A method for extracting the procedures prohibited by complex formulas is outlined, and the relationship between types of prohibited procedures is identified. Definitions of negation types in terms of the implication of 0 on an informational interpretation have been criticized. This criticism turns on the definitions creating a purportedly (...) unnatural asymmetry between positive and negative information. It is demonstrated below that a strong asymmetry between positive and negative information is in fact the case. As such, an asymmetry between positive and negative information is natural, and something that we should want an informational interpretation of negation to preserve. (shrink)

Permutative logic is a non-commutative conservative extension of linear logic suggested by some investigations on the topology of linear proofs. In order to syntactically reflect the fundamental topological structure of orientable surfaces with boundary, permutative sequents turn out to be shaped like q-permutations. Relaxation is the relation induced on q-permutations by the two structural rules divide and merge; a decision procedure for relaxation has been already provided by stressing some standard achievements in theory of permutations. In these pages, we provide (...) a parallel procedure in which the problem at issue is approached from the point of view afforded by geometry of 2-manifolds and solved by making specific surfaces interact. (shrink)