Abstract
Hybrid-dynamic first-order logic is a kind of modal logic obtained by enriching many-sorted first-order logic with features that are common to hybrid and to dynamic logics. This provides us with a logical system with an increased expressive power thanks to a number of distinctive attributes: first, the possible worlds of Kripke structures, as well as the nominals used to identify them, are endowed with an algebraic structure; second, we distinguish between rigid symbols, which have the same interpretation across possible worlds – and thus provide support for the standard rigid quantification in modal logic – and flexible symbols, whose interpretation may vary; third, we use modal operators over dynamic-logic actions, which are defined as regular expressions over binary nominal relations. In this context, we propose a general notion of hybrid-dynamic Horn clause and develop a proof calculus for the Horn-clause fragment of hybrid-dynamic first-order logic. We investigate soundness and compactness properties for the syntactic entailment system that corresponds to this proof calculus, and prove a Birkhoff-completeness result for hybrid-dynamic first-order logic.
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Notes
- 1.
This last attribute is meant to indicate the fact that users have control over the symbols that should be interpreted the same across the worlds of a Kripke structure.
- 2.
By symbol we usually refer to sorts as well, not only to operation/relation symbols.
- 3.
Note that, by Fact 1, if the arity \( ar \) is rigid, then the sets
coincide. - 4.
In general, by a \(\varDelta [X]\)-expansion of (W, M) we understand a \(\varDelta [X]\)-model \((W', M')\) that interprets all symbols in \(\varDelta \) in the same way as (W, M).
- 5.
By the definition of reducts, \((W', M')\) and
have the same possible worlds. - 6.
This means that \(h_{w, s}(a_{1}) = h_{w, s}(a_{2})\) for all \(a_{1}, a_{2} \in M_{w, s}\) such that \(a_{1} \equiv _{w, s} a_{2}\).
- 7.
Note that we use the diagrammatic notation for function composition.
coincide.
have the same possible worlds.References
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Găină, D., Ţuţu, I. (2019). Birkhoff Completeness for Hybrid-Dynamic First-Order Logic. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_16
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