### Tableaux calculi for modal predicate logics with and without the Barcan formula can be found in [32]. Just like the identity of individuals gives rise to many philosophical ques- tions in modal predicate logic, it also gives rise to many deep mathematical questions. As a result, various alternative semantic frameworks have been developed for modal predicate logic during the 1990s, including the Kripke bundles of Shehtman and Skvortsov [37] and the category-theoretic seman- tics proposed by Ghilardi [16] The notion of (axiomatic) completeness is another source of interesting mathematical questions in modal predicate logic. It turns out that the mini- mal predicate logical extension of many well-behaved and complete proposi- tional modal logics need not be complete. The main (negative) result in this area is that among the extensions of S4, propositional modal logics L whose minimal predicate logical extension is complete must have either L S5 or

### Table 1. The complexity of the satisfiability problem for modal logics

2005

"... In PAGE 2: ... We also show that the satisfiability problem of modal formulas with modal depth bounded by 1 in K4, KD4, and S4 is NP-complete; the satisfiability problem of sets of Horn modal clauses with modal depth bounded by 1 in K, K4, KD4, and S4 is PTIME-complete. In Table1 , we summarize the complexity of the basic modal logics under the mentioned restrictions. There, mdepth stands for modal depth ; PS-cp, NP-cp, and PT-cp respectively stand for PSPACE-complete, NP-complete, and PTIME-complete.... ..."

Cited by 3

### Table 2. Modal logics and frame restriction

"... In PAGE 5: ... Di erent modal logics are distinguished by their respective additional axiom schemata. Some of the most popular modal logics together with their axiom schemata are listed in Table2 . We refer to the properties of the accessibility relation of a modal logic L as the L-frame axioms or L-frame restrictions.... In PAGE 5: ... Given two normal modal logics L and L0, we say that L0 is a normal extension of L, and write L L0, if all L-frame restrictions are also L0-frame restrictions. Let L be one of the modal logics listed in Table2 . For a binary relation R, we use ExtL(R) to denote the least extension of R that satis es all L-frame axioms, excluding the axiom D.... ..."

### Table 2. Modal logics and frame restriction

"... In PAGE 5: ... Different modal logics are distin- guished by their respective additional axiom schemata. Some of the most popular modal logics together with their axiom schemata are listed in Table2 . We refer to the properties of the accessibility relation of a modal logic L as the L-frame axioms or L-frame restrictions.... In PAGE 5: ... By normal modal logics we call logics that are extensions of the logic K. Let L be one of the modal logics listed in Table2 . For a binary relation CA, we use BXDCD8C4B4CAB5 to denote the least extension of CA that satisfies all L-frame axioms, excluding the axiom D.... In PAGE 15: ... If AU is a ground formula then we write C5BN DB AF AU to denote that C5BN CEBN DB AF AU for some CE (note that what CE does not matter). Let C4 be the name of some propositional normal modal logic given in Table2 , e.... ..."

### Table 1: Encoding of the base relations in modal logic

1998

"... In PAGE 3: ... In order to distinguish between spatial variables and the corre- sponding propositional atoms we will write proposi- tional atoms as X; Y. Table1 displays the constraints for the eight base relations. In order to combine them to a single modal formula, Bennett introduced an S5- operator1 2, where 2 apos; is written for every model con- straint apos; and :2 for every entailment constraint (Bennett 1995).... ..."

Cited by 36

### Table 1 Corresponding notions for description logics and modal logics

"... In PAGE 3: ... Therefore we prefer using the description logic notions. Table1 compares the di erent notions used by the modal logic community with the corresponding notions used by the description logic community. The standard semantics of modal and description logics allows one to translate all T-Box and A-Box information into rst-order predicate logic (FOL).... ..."

### Table 1: A summary of results on logical reasoning tasks.

2003

Cited by 16

### Table 1. A Taxonomy for CMC Technologies Temporality Anonymity Modality Spatiality

"... In PAGE 8: ... TESTING THE EFFECTS OF DIFFERENT CMC TECHNOLOGIES In these studies, we focused on two common CMC technologies: the Palace and the ChatNet. As outlined in Table1 , the Palace and ChatNet, two popular CMC applications, can be contrasted across the attributes of Modality and Spatiality. The Palace is a multimedia two-dimensional chat program, while ChatNet is a text-only chat program.... ..."