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Semantic Techniques for Cut-Elimination in Higher Order Logic.

This paper is part of an ongoing effort to examine the role of extensionality in higher-order logic and provide tools for analyzing higher-order calculi. In an earlier paper, we have presented eight classes of higher order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we have developed a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of higher-order calculi with respect to these model classes. This framework, employs a strong saturation criterion which prevents analysis of, e.g., the deductive power of machine-oriented calculi. In this paper we extend our saturated abstract consistency approach and obtain analogous model existence results without assuming saturation. For this, we replace the saturation conditions by a set of weaker acceptability conditions which are sufficient to prove model existence. We further show that saturation can be as hard to prove as cut elimination. We apply our extended abstract consistency approach to show completeness of five different sequent calculi (with varying strength regarding Boolean and functional extensionality reasoning) with respect to five of our eight model classes. We conclude that cut-elimination holds for each of these five calculi.