### 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. Benchmarks comparing the TWB and LWB for modal logic K

"... In PAGE 15: ...hz), 1GB RAM, 1GB swap space; Software: Debian GNU/Linux OS, OCaml 3.09.2. Table2 show, for each class, how many formulae of each set could be solved. For each class, we generated all formulae with complexity up to 21 with a timeout of 100 seconds.... ..."

Cited by 1

### Table 6.2: An overview of complexity results for multi-modal hybrid logics

### Table 1. Data and combined complexity for fixpoint logics

2005

"... In PAGE 2: ... It turns out that in most cases the complexity of the model-checking problem over hierarchically defined input structures becomes EXP. Our results are collected in Table1 in Section 2. Proofs that are omitted due to space restrictions can be found in the full version [7].... In PAGE 5: ... As already explained in the introduction, we will be interested in combined complexity (both, the formula and the structure belong to the input) and data complexity (the formula is fixed and only the structure belongs to the input), where the structure is represented via an SLP. Table1 collects the known results as well as our new results concerning the (data and combined) complexity of the model-checking problem for the logics LFP, MLFP, and the modal -calculus. Only for the data complexity of MLFP and the modal -calculus on structures defined by c-bounded SLPs (for some fixed c 2 N) we do not obtain matching lower and upper bounds.... In PAGE 11: ... Together with Thm. 4 we get the EXP completeness results in Table1 . We start with the data complexity of LFP: Theorem 6.... ..."

Cited by 1

### Table 1. Data and combined complexity for fixpoint logics

2005

"... In PAGE 2: ... It turns out that in most cases the complexity of the model-checking problem over hierarchically defined input structures becomes EXP. Our results are collected in Table1 in Section 2. Proofs that are omitted due to space restrictions can be found in the full version [7].... In PAGE 5: ... As already explained in the introduction, we will be interested in combined complexity (both, the formula and the structure belong to the input) and data complexity (the formula is fixed and only the structure belongs to the input), where the structure is represented via an SLP. Table1 collects the known results as well as our new results concerning the (data and combined) complexity of the model-checking problem for the logics LFP, MLFP, and the modal -calculus. Only for the data complexity of MLFP and the modal -calculus on structures defined by c-bounded SLPs (for some fixed c 2 N) we do not obtain matching lower and upper bounds.... In PAGE 11: ... Together with Thm. 4 we get the EXP completeness results in Table1 . We start with the data complexity of LFP: Theorem 6.... ..."

Cited by 1

### 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).... ..."

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### 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).... ..."