The question of developing a computational interpretation of J.-Y. Girard's (1987) linear logic and obtaining efficient decision algorithms for this logic, based on the bottom-up approach, is addressed. The approach taken is to start with the simplest natural fragment of linear logic and then expand it step-by-step. The smallest natural Horn fragment of Girard's linear logic is considered, and it is proved that this fragment is NP-complete. As a corollary, an affirmative solution for the problem of whether the multiplicative fragment of Girard's linear logic is NP-complete is obtained. Then a complete computational interpretation for Horn fragments enriched by two additive connectives and by the storage operator is given. Within the framework of this interpretation, it becomes possible to explicitly formalize and clarify the computational aspects of the fragments of linear logic in question and establish exactly the complexity level of these fragments.
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