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Beyond formulas-as-cographs: an extension of Boolean logic to arbitrary graphs

We propose a graph-based extension of Boolean logic called Boolean Graph Logic (BGL). Construing formula trees as the cotrees of cographs, we may state semantic notions such as evaluation and entailment in purely graph-theoretic terms, whence we recover the definition of BGL. Naturally, it is conservative over usual Boolean logic. Our contributions are the following: (1) We give a natural semantics of BGL based on Boolean relations, i.e. it is a multivalued semantics, and show adequacy of this semantics for the corresponding notions of entailment. (2) We show that the complexity of evaluation is NP-complete for arbitrary graphs (as opposed to ALOGTIME-complete for formulas), while entailment is $Π^p_2$-complete (as opposed to coNP-complete for formulas). (3) We give a 'recursive' algorithm for evaluation by induction on the modular decomposition of graphs. (Though this is not polynomial-time, cf. point (2) above). (4) We characterise evaluation in a game-theoretic setting, in terms of both static and sequentical strategies, extending the classical notion of positional game forms beyond cographs. (5) We give an axiomatisation of BGL, inspired by deep-inference proof theory, and show soundness and completeness for the corresponding notions of entailment. One particular feature of the graph-theoretic setting is that it escapes certain no-go theorems such as a recent result of Das and Strassburger, that there is no linear axiomatisation of the linear fragment of Boolean logic (equivalently the multiplicative fragment of Japaridze's Computability Logic or Blass' game semantics for Mutliplicative Linear Logic).