Implicit computational complexity studies theory and applications of machine-independent approaches to computational complexity, with particular emphasis on approaches based on mathematical logic. Although related one to the other, all the proposed systems have been studied with different, often unrelated methodologies, and few results are known about relative intentional expressive power. For example, it is not clear at all whether combining different systems into one language would break the correspondence to complexity classes. Similarly, it is not usually the case that a system can be extended with new features preserving its quantitative properties. What is needed, are some unifying semantic frameworks for the analysis of resource consumptions of programming languages and proof systems. Such frameworks should be powerful enough to capture a wide amount of systems and programming language, but simple enough to be useful in studying quantitative properties of programs and proofs.
Abstract semantic framework s for implicit computational complexity will be investigated. Particular emphasis will be placed on approaches based on game semantics, geometry of interaction and coherence semantics. The major expected result is the definition of an abstract mathematical structure in which heterogeneous systems inside implicit computational complexity can be interpreted. Crucially, the model must be sufficiently rich to reflect the quantitative behaviour of the interpreted object, in contrast to usual denotational semantics. The investigation will proceed in two steps. First of all, game models and geometry of interaction models will be developed in such a way that a large class of subsystems of linear logic (including the so called light logics) can be interpreted, studied and compared. Secondly, the semantics is extended in order to model languages outside linear logic, starting from higher-order recursion.
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