### Table 3: Mean and modal agreement scores

"... In PAGE 7: ... By participant An overall mean agreement score was computed for each participant using the above formula for the target and baseline algo- rithms in each domain. Results by item Table3 shows the mean and modal agreement scores obtained for both target and base- line in each domain type. At a glance, the target algo- rithm performed better than the baseline on the spatial domains, with a modal score of 1, indicating perfect agreement on 60% of the objects.... ..."

### Table 3: Mean and modal agreement scores

"... In PAGE 7: ... By participant An overall mean agreement score was computed for each participant using the above formula for the target and baseline algo- rithms in each domain. Results by item Table3 shows the mean and modal agreement scores obtained for both target and base- line in each domain type. At a glance, the target algo- rithm performed better than the baseline on the spatial domains, with a modal score of 1, indicating perfect agreement on 60% of the objects.... ..."

### 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 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: Frame properties

1999

"... In PAGE 16: ...4. Let be a modal formula, let be a possibly empty set of frame properties from Table1 , and let N be the set of clauses obtained by applying Cls r to . Then, 1.... In PAGE 17: ...5. Let be a modal formula, let f igi be a family of possibly empty sets of frame properties from Table1 , and let = Si i. If N = Cls r ( ), then 1.... ..."

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### Table 1: Frame properties

1999

"... In PAGE 16: ...4. Let be a modal formula, let be a possibly empty set of frame properties from Table1 , and let N be the set of clauses obtained by applying Cls r to . Then, 1.... In PAGE 17: ...5. Let be a modal formula, let f igi be a family of possibly empty sets of frame properties from Table1 , and let = Si i. If N = Cls r ( ), then 1.... ..."

Cited by 20

### Table 4: Language of QCTL

2006

"... In PAGE 11: ...depicted in Table4 and are obtained by enriching the quantum formulas with CTL modalities. The intuitive semantics of the temporal modalities is similar to those in classical CTL.... ..."

Cited by 3

### Table 4 kSAT, kSAT+USB, and kSAT+USH on TANCS 2000 modalized easy/medium problems we consider only the formulas on which the system did not fail, and write \{ quot; when the system fails on all the samples. As it can be seen, caching produces dramatic improvements. Many tests that are not solved within the timelimit by kSAT, are solved when using a caching mechanism.Indeed, the sets in Tables 1 and 2 are just a few. Consider also Tables 3 and 4, showing kSAT, kSAT+USB and kSAT+USH results on the easy/medium and modalized easy/medium formulas respectively. These Tables make clear the qualitative di erences between kSATon one side, and kSAT+USB and kSAT+USH on the other. kSAT solves only 51 (9%) out of the 544 samples on which it has been tested. kSAT+USB and kSAT+USH solve 414 (76%) and 390 (71%)

### Table 5: Sequent Calculus Rules for S4 our rules only require that at least one other formula should be appropriately modalised.McCarthy and Hayes [12, p. 472] do propose such a reading of modal operators in terms of rewrites, but their modalities are normal Kripkean ones in a classical theory, and they are unable to make a very precise application of the modal logic. Consider now the relation which holds between A and B if we have A ` B. This looks very like the rewrite relation. Firstly, it is transitive, since if we have A ` B and B ` C, then we also have A ` C by A ` B

"... In PAGE 10: ...2 The Global Theory: Linear Modalities Consider the rules for modal operators given in Table 4. Notice that they are not the same as the normal rules for an S4 modality { that is, the rules in Table5 . The side conditions on the left -rule and the right @-rule are di erent; whereas the normal S4 rules require all of the other formulae in these rules to be modalised, 4The corresponding proof search procedure was, in fact, also described by Wolfgang Bibel some time before Girard apos;s paper on linear logic; see [1].... ..."

### Table 6: Semantics of the modal -calculus. f[tt] g( )

1999

"... In PAGE 17: ... Intuitively, f[ ]g( ) denotes the set of all processes that satisfy under the environment . Formally, the semantic mapping f[ ] g : (F E) ?! 2P, where E stands for the set of all environments, is inductively de ned over the structure of formulas, as shown in Table6 . If is a closed formula, its semantics is independent of the environment.... ..."

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