On Strong Normalization in Proof-Graphs for Propositional Logic

Traditional proof theory of Propositional Logic deals with proofs whose size can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The use of proof-graphs, instead of trees or lists, for representing proofs is getting popular among proof-theoreticians. Proof-graphs serve as a way to study complexity of propositional proofs and to provide more efficient theorem provers, concerning size of propositional proofs. Fpl-graphs were initially developed for minimal implicational logic representing proofs through references rather than copy. Thus, formulas and sub-deductions preserved in the graph structure, can be shared deleting unnecessary sub-deductions resulting in the reduced proof. In this work, we consider full minimal propositional logic and show how to reduce (eliminating maximal formulas) these representations such that strong normalization theorem can be proved by simply counting the number of maximal formulas in the original derivation. In proof-graphs, the main reason for obtaining the strong normalization property using such simple complexity measure is a direct consequence of the fact that each formula occurs only once in the proof-graph and the case of the hidden maximum formula that usually occurs in the tree-form derivation is already represented in the fpl-graph.