Abstract
The vast majority of modal theorem provers implement modal tableau, or backwards proof search in (cut-free) sequent calculi. The design of suitable calculi is highly non-trivial, and employs nested sequents, labelled sequents and/or specifically designated transitional formulae. Theorem provers for first-order logic, on the other hand, are by and large based on resolution. In this paper, we present a resolution system for preference-based modal logics, specifically Burgess’ system . Our main technical results are soundness and completeness. Conceptually, we argue that resolution-based systems are not more difficult to design than cut-free sequent calculi but their purely syntactic nature makes them much better suited for implementation in automated reasoning systems.
The first author was partially supported by FWF START Y544-N23.
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The next two proofs were automatically generated by a prototype prover which implements the calculus given in this paper. Only clauses needed in the refutation are shown. Also the inference rule is always applied together with , so clauses are already in simplified form. First, as part of the proof of Lemma 7, we show that for \(\varphi ,\lnot \psi \in b\), we have that \(\mathsf {Prefer}(a,b,B) \) and \(\varphi \Rightarrow \psi \in (a,A)\) is contradictory.
The following refutation is part of the proof of Lemma 8, where we show that, for \(\varphi , \psi \in b\), we have that \(\lnot ((b \vee \bigvee B) \Rightarrow \bigvee B)\) and \(\lnot (\varphi \Rightarrow \psi ) \in a\) is not -consistent
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Nalon, C., Pattinson, D. (2018). A Resolution-Based Calculus for Preferential Logics. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_33
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