The main result is a simple, incremental polynomial-time algorithm to correctly order rules in proof trees for combinations of K, K4, T and S4 necessity operators governed by a variety of interactions, assuming an encoding of negation using a scoped constant ⊥. This paper investigates the use of partial-order mechanisms to maintain constraints on the application of modal rules in proof search in restricted languages.
The key problem in modal deduction is reasoning about the world in a model (or scope in a proof) at which an inference rule is applied - a potentially hard problem. The results of this paper further the goal of building practical, efficient reasoning systems using modal logics. Modal logics offer natural, declarative representations for describing both the modular structure of logical specifications and the attitudes and behaviors of agents. Furthermore, we introduce a uniform concept for contextual reasoning, and sketch basic tacticals for the definition of tactics to organise the overall proof-search inside and across different reasoning levels.
In order to ensure soundness within a reasoning level, we use techniques developed in the context of matrix characterisation relying on the notion of indexed formulas. To structure reasoning levels the framework allows for an explicit representation of abstractions and proof-search refinement techniques. To this end we present a framework to explicitly support different reasoning levels.
Furthermore, proof construction is an engineering task and there is a need to support the design and application of proof-search engineering methods. For high reasoning levels the cognitive adequacy of representation and reasoning techniques is a major issue, while for lower reasoning levels the adequacy wrt. A major motivation of proof-planning is to bridge the gap between high-level, cognitively adequate reasoning for specific domains, and calculus-level reasoning to ensure soundness.
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