Elementary Complexity and Geometry of Interaction (extended Abstract)

2 Istituto per le Applicazioni del Calcolo \Mauro Picone" (Roma), C.N.R. Abstract. We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following Gir95a]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (Execution) to more programs of the algebra than just those coming from proofs, we deene a variant of Execution (called Weak Execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that Weak Execution coincides with standard Execution on programs coming from proofs. Geometry of interaction (goi) was introduced by Girard ((Gir88a]) as a semantics of computation which: on the one hand, contrary to denotational semantics interprets explicitly the dynamics of computation and handles nite objects; on the other hand expresses this dynamic by more mathematical means than syntactical rewriting. The execution operation is the mathematical tool inside the model used to interpret the cut-elimination process. This operation is not always deened and suucient conditions have been given which ensure termination of the computation : in the case of second-order Linear Logic ((Gir88a, Gir95a]) and of untyped lambda-calculus MR91]), operators coming from the syntax do satisfy such conditions (a nilpotency condition for instance in the case of LL). Various frameworks have been used to describe goi models: bounded operators on Hilbert spaces ((Gir88a, DR95]), partial applications ((Dan90, Reg92]) and clauses ((Gir95a]). This latter point of view is the one we adopt here. Elementary Linear Logic (ELL), as Light Linear Logic (LLL), is a variant of Linear Logic in which the rules introducing exponentials have been modiied (cf. Gir95b]) in order to limit the size explosion of proofs during normalization. It is obtained by removing the two principles : !A ` A and !A ` !!A; contraction and weakening are kept unchanged. We consider here a version of ELL without additive connectives and where introduction of the modality ! is handled through a (multi-)functorialpromotion rule (called t-promotion, see Ped96]), which ooers the advantage of having simple proof-nets. A proof-net has two main parameters: its size (say the number of edges) and its depth (maximal nesting of the boxes it contains). The number of steps of its normalization is bounded by a function of the size which is elementary: the expression of this …