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Implicational relevance logic is 2-ExpTime-complete. (English) Zbl 1370.03029
This paper establishes that provability in the pure implication fragment $$\mathbf{R}_\to$$ of the relevance logic R is 2-ExpTime-complete. This draws crucially on the result of S. Demri et al. [J. Comput. Syst. Sci. 79, No. 1, 23–38 (2013; Zbl 1260.68264)] that the covering and boundedness problems for branching vector addition systems with states (BVASS) are themselves 2-ExpTime-complete. The author shows that expansive BVASSs can simulate proofs in $$\mathbf{R}_\to$$, or its sequent formulation $$\mathit{L}\mathbf{R}_\to$$, and that their reachability reduces to coverability. This provides a decision procedure for $$\mathbf{R}_\to$$ in 2-ExpTime. As BVASS coverability can be reduced to provability in $$\mathbf{R}_\to$$, this provides its 2-ExpTime-hardness. The latter result draws on a variant sequent calculus for $$\mathbf{R}_\to$$, viz. $$\mathit{F}\mathbf{R}_\to$$, for focusing proofs (cf. [J.-M. Andreoli, J. Log. Comput. 2, No. 3, 297–347 (1992; Zbl 0764.03020)] for linear logic).

##### MSC:
 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q25 Analysis of algorithms and problem complexity
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