In this thesis, we study a semiring structure with a product based on theresolution rule of logic programming. This mathematical object was introducedinitially in the setting of the geometry of interaction program in order to modelthe cut-elimination procedure of linear logic. It provides us with an algebraicand abstract setting, while being presented in a syntactic and concrete way, inwhich a theoretical study of computation can be carried on.We will review first the interactive interpretation

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... of proof theory withinthis semiring via the categorical axiomatization of the geometry of interactionapproach. This interpretation establishes a way to translate functional programsinto a very simple form of logic programs.Secondly, complexity theory problematics will be considered: while thenilpotency problem in the semiring we study is undecidable in general, it willappear that certain restrictions allow for characterizations of (deterministicand non-deterministic) logarithmic space and (deterministic) polynomial timecomputation.