### Table 1: Major modal axioms into a TSL is the following (p0 is the TSL concept corre- sponding to the modal formula p):

### 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 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. Formulas and their intended meaning

"... In PAGE 12: ... Additional aspects appearing on the stage in specific cases may be addressed by refining the system and adding new axioms. Table1 gives a number of modal formulas appearing in this paper, together with their intended meanings. The symbol a15 denotes a proposition, but all these formulas also appear with respect to an action a24.... ..."

### Table 1: Modal encoding of the eight base relations [2].

1998

"... In PAGE 5: ... Bennett encoded these constraints in modal logic by introducing an S4- operator I which he interpreted as an interior operator [2]. Table1 displays these constraints for the eight base relations. In order to combine these constraints to a single modal formula, Bennett introduced a strong S5-operator 2, where 2 apos; is written for every model constraint apos; and :2 for every entailment constraint [2].... ..."

Cited by 36

### Table 1. Formulas wemodelchecked. S is the set frt i

1997

"... In PAGE 11: ... Intuitively, formulas capture eventuality properties and formulas capture invariant properties. The modal mu-calculus formulas we model checked are listed in Table1 . Their meaning can be understood as follows: { DLF: Deadlock freedom.... ..."

Cited by 1

### Tableau systems are de nable for modal logics in di erent formats: either signed or unsigned tableaux, and either pre xed or unpre xed ones. In this work we refer to unpre xed signed tableaux [5]. Although the pre xed version is closer to the matrix method, the use of unpre xed tableaux is preferable in this setting. In fact it restricts the application of rules expanding formulae of possible force (when a rule is applied to one of such formulae, the others are \lost quot;), therefore allowing for a better analysis of the structure of a tableau proof. A signed formula is either TX or FX, where X is a formula. A compact formulation of the modal expansion rules can be obtained by de ning -formulae (formulae of necessary force) and -formula (formulae of possible force), with their 0 and 0, as follows.

### Table 3 Bennetts encoding of the eight base relations in modal logic [3].

1999

"... In PAGE 7: ... For this purpose the topological interior operator i is used. This operator must satisfy the following axiom schemata for arbitrary sets ; U [3]: i( ) ; (1) i(i( )) = i( ); (2) i(U) = U; (3) i( \ ) = i( ) \ i( ): (4) The model and entailment constraints can be encoded in modal logic, where regions correspond to propositional atoms, the interior operator i corresponds to a modal operator I (see Table3 ), and the universe U corresponds to the set of all worlds W [3]. The axiom schemata for i must also hold for the modal operator I, which results in the following axiom schemata [3] for arbitrary modal formulas ; : I ! ; (5) II $ I ; (6) I gt; $ gt; (for any tautology gt;); (7) I( ^ ) $ I ^ I : (8) the encoding based on closed regions [3].... ..."

Cited by 91