Automating Agential Reasoning: Proof-Calculi and Syntactic Decidability for STIT Logics

Abstract

This work provides proof-search algorithms and automated counter-model extraction for a class of STIT logics. With this, we answer an open problem concerning syntactic decision procedures and cut-free calculi for STIT logics. A new class of cut-free complete labelled sequent calculi G3Ldm m n , for multi-agent STIT with at most n-many choices, is introduced. We refine the calculi G3Ldm m n through the use of propagation rules and demonstrate the admissibility of their structural rules, resulting in the auxiliary calculi Ldm m n L. In the single-agent case, we show that the refined calculi Ldm m n L derive theorems within a restricted class of (forestlike) sequents, allowing us to provide proof-search algorithms that decide single-agent STIT logics. We prove that the proof-search algorithms are correct and terminate.

On the correspondence between nested calculi and semantic systems for intuitionistic logics

“…This refinement process allows us to capture the best of both worlds: we invoke the general results of the labelled setting to obtain satisfactory labelled calculi for a class of logics, and via refinement, transform the systems into nested calculi better suited for applications. Similar ideas and relationships have been discussed in the literature [13,17,16,20,21], where refined calculi (which can be considered nested calculi) were derived from labelled calculi for modal, intuitionistic, and related logics. (NB.…”

“…First, it is relatively straightforward to transform the semantics of a logic into a calculus; in fact, this process has been shown to be automatable [5]. Second, the approach is exceptionally modular-allowing for the addition or deletion of rules to immediately obtain calculi for weaker or stronger logics-and is applicable to a wide variety of logics [6,17,24,27]. Last, labelled calculi consistently possess fundamental proof-theoretic properties such as invertibility of rules, admissibility of structural rules, and cut-admissibility-with fairly general results provided for large classes of modal, intuitionistic, and related logics [6,17,24,27].…”

Alternative Axiomatization for Logics of Agency in a G3 Calculus

“…The treatment of stit modalities in terms of proof theory remains relatively scant, notable exceptions being (Wansing 2006(Wansing , 2017Wansing 2018, 2019), which all involve a tableaux approach. Approaches utilizing a sequent calculus have been developed recently, in particular (using a simplification of a tableux) van Berkel and ; Lyon and van Berkel (2019) and Negri and Pavlović (2020) (which builds the calculus based on the semantics from Belnap et al 2001). The system from the latter will be used in this paper, since in addition to the usual range of desirable proof-theoretic properties (like the admissibility of contraction and cut), it also offers a structural proof of multi-agent decidability and, even though it focuses on dstit, provides a uniform basis for the treatment of multiple stit modalities.…”

Syntactic Cut-Elimination for Intuitionistic Fuzzy Logic via Linear Nested Sequents

“…Such calculi are effective tools for designing automated reasoning procedures and for proving certain (meta-)logical properties of a logic. For example, analytic calculi have been leveraged to provide decidability procedures for logics [11], to prove that logics interpolate [18], for counter-model extraction [21], and to understand the computational content of proofs [23].…”