A combinatorial study of soundness and normalization in n-graphs

Abstract

N-Graphs is a multiple conclusion natural deduction with proofs as directed graphs, motivated by the idea of proofs as geometric objects and aimed towards the study of the geometry of Natural Deduction systems. Following that line of research, this work revisits the system under a purely combinatorial perspective, determining geometrical conditions on the graphs of proofs to explain its soundness criterion and proof growth during normalization. Applying recent developments in the fields of proof graphs, proof-nets and N-Graphs itself, we propose a linear time algorithm for proof verification of the full system, a result that can be related to proof-nets solutions from Murawski (2000) and Guerrini (2011), and a normalization procedure based on the notion of sub-N-Graphs, introduced by Carvalho, in 2014. We first present a new soundness criterion for meta-edges, along with the extension of Carvalho’s sequentization proof for the full system. For this criterion we define an algorithm for proof verification that uses a DFS-like search to find invalid cycles in a proof-graph. Since the soundness criterion in proof graphs is analogous to the proof-nets procedure, the algorithm can also be extended to check proofs in the multiplicative linear logic without units (MLL−) with linear time complexity. The new normalization proposed here combines a modified version of Alves’ (2009) original beta and permutative reductions with an adaptation of Carbone’s duplication operation on sub-N-Graphs. The procedure is simpler than the original one and works as an extension of both the normalization defined by Prawitz and the combinatorial study developed by Carbone, i.e. normal proofs enjoy the separation and subformula properties and have a structure that can represent how patterns lying in normal proofs can be recovered from the graph of the original proof with cuts.