There Is No Best \(\beta \)-Normalization Strategy for Higher-Order Reasoners
The choice of data structures for the internal representation of terms in logical frameworks and higher-order theorem provers is a crucial low-level factor for their performance. We propose a representation of terms based on a polymorphically typed nameless spine data structure in conjunction with perfect term sharing and explicit substitutions.
In related systems the choice of a \(\beta \)-normalization method is usually statically fixed and cannot be adjusted to the input problem at runtime. The predominant strategies are hereby implementation specific adaptions of leftmost-outermost normalization. We introduce several different \(\beta \)-normalization strategies and empirically evaluate their performance by reduction step measurement on about 7000 heterogeneous problems from different (TPTP) domains.
Our study shows that there is no generally best \(\beta \)-normalization strategy and that for different problem domains, different best strategies can be identified. The evaluation results suggest a problem-dependent choice of a preferred \(\beta \)-normalization strategy for higher-order reasoning systems.
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