Abstract
Based on tableau systems, we in this chapter prove decidability results for hybrid logic using tableau systems. The chapter is structured as follows. In the first section of the chapter we sketch the basics of tableau systems. In the second section we give a tableau-based decision procedure for a very expressive hybrid logic including the universal modality. In the third section we show how the decision procedure of the second section can be modified such that simpler tableau-based decision procedures (that is, without loop-checks) are obtained for a weaker hybrid logic where the universal modality is not included. In the fourth section we reformulate the tableau systems of the second and the third sections as Gentzen systems and we discuss how to reformulate the decision procedures. In the fifth section we discuss the results. The results of the second, fourth, and fifth sections of this chapter are taken from Bolander and Braüner (2006). The material in the third section is new (but the tableau systems considered in the third section are obtained by directly modifying the tableau system given in the second section, inspired by a tableau-based decision procedure given in Bolander and Blackburn (2007)).
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Notes
- 1.
This terminology is used in a somewhat different sense than is common: Our destructive rules preserve information in the sense that if a conclusion of a destructive rule has a model, then this model is a model for the premise of the rule as well, that is, no models are included (note that this is opposite of soundness which says that no models are excluded). In the usual sense destructive rules are rules that do not preserve information, see Fitting (1972a).
- 2.
An occurrence of a satisfaction statement \(@_{a} \phi\) or the negation of a satisfaction statement \(\neg @_{a} \phi\) in a tableau can be seen as a formula φ together with a pair consisting of the representation of a possible world (the nominal a) and the representation of a truth-value (depending on whether the satisfaction statement is negated or not). Note in this connection that in the possible worlds semantics, the semantic value assigned to a formula is a function from possible worlds to truth-values, and set-theoretically, such a function is a set of pairs of possible worlds and truth-values (called the graph of the function). Hence, the pairs of nominals and representations of truth-values associated with formulas in the tableau system can be considered representations of elements of functions constituting semantic values. Thus, the tableau rules step by step build up semantic values of the formulas involved, similar to the way in which the accessibility relation step by step is built up (there is a difference however; the accessibility relation can be any relation, but the semantic value of a formula has to be a function, that is, a relation where no element of the domain is related to more than one element of the codomain, and this is exactly what is required of an open branch in a tableau, namely that no satisfaction statement is related to more than one truth-value).
- 3.
This was pointed out to the author by Jens Ulrik Hansen.
- 4.
This was pointed out to the author by Thomas Bolander.
- 5.
It follows that if the conclusion sequent of a rule is valid, then the premise sequents of the rule are valid as well (this is opposite of soundness)
- 6.
Essentially, backtracking is not needed since the premise sequents of a rule are valid if the conclusion sequent is valid, hence, no information is lost when a rule is applied.
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Braüner, T. (2011). Tableaus and Decision Procedures for Hybrid Logic. In: Hybrid Logic and its Proof-Theory. Applied Logic Series, vol 37. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0002-4_3
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