Logic Beyond Formulas

Abstract

In this paper we present a proof system that operates on graphs instead of formulas. We begin our quest with the well-known correspondence between formulas and cographs, which are undirected graphs that do not have P 4 (the fourvertex path) as vertex-induced subgraph; and then we drop that condition and look at arbitrary (undirected) graphs. The consequence is that we lose the tree structure of the formulas corresponding to the cographs. Therefore we cannot use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic (MLL) with mix, meaning that if a graph is a cograph and provable in our system, then it is also provable in MLL+mix.

Ideograph: A Language for Expressing and Manipulating Structured Data

“…There is work on representing formulas in multiplicative linear logic as undirected graphs [2,25]. Our type system is closely related to these when we do not use exponentials or second-order propositional quantifiers.…”

“…There are two key differences between our work and interaction nets: (1) In interaction nets, each symbol (roughly our "node") has a principal port, which is used in reduction; in our work, nodes do not have privileged ports, and reduction proceeds exclusively by substituting definitions (of types or terms) for their occurrences. (2) In interaction nets, the set of symbols and their associated ports must be fixed ahead of time; in our work, the symbol set is not fixed, with occurrences of symbols potentially adding fresh symbols to the set.…”