Revisiting Counter-model Generation for Minimal Implicational Logic
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The LMT^→ sequent calculus was introduced in Santos (2016). This paper presents a Termination proof and a new (more direct) Completeness proof for it. LMT^→ is aimed to be used for proof search in Propositional Minimal Implicational Logic (M^→), in a bottom-up approach. Termination of the calculus is guaranteed by a rule application strategy that stresses all the possible combinations. For an initial formula α, proofs in LMT^→ have an upper bound of |α| × 2^|α| + 1 + 2 × log_2|α|, which together with the system strategy ensure decidability. LMT^→ has the property to allow extractability of counter-models from failed proof searches (bicompleteness), i.e., the attempt proof tree of an expanded branch produces a Kripke model that falsifies the initial formula.
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