### Table 3. A pre xed-based tableau calculus for MELL

1999

"... In PAGE 6: .... The one-branch tree apos; is a tableau for apos;. 2. If T is a tableau for apos; and T results from T by the application of a tableau expansion rule from Table3... In PAGE 8: ... Then T is closed by C, W, and . The following theorems state that the tableau calculus in Table3 is correct and complete. In order to follow the proof sketches prior knowledge of [17] is required.... In PAGE 12: ...ragment M?LL. We rst present the calculus and repeat some de nitions. A Tableau Calculus for M?LL. The tableau calculus for M?LL is similar to the calculus for MELL presented in Table3 . Since ? and ! can only occur, respec- tively, with positive and negative polarity we do not need the -rule anymore.... ..."

### Table 1: Tableau calculus for INT

1996

Cited by 4

### Table 1. The tableau rules for T pro (i; o; %; +) : apos; ^

1999

"... In PAGE 11: ...biguity, T pro, is given in Table1 . The rules may seem somewhat overwhelming, but most of them are familiar ones.... In PAGE 13: ... The threading of contextual information may seem a bit confusing, but it is hard to display the dynamics of the instantiation of the context variables on `static apos; paper. It may be helpful to read the tableau rules in Table1 as Prolog clauses, where the context variables of the parent of a rule unify with the context variables of the node the rule is applied to. 6 Results The tableau calculus T pro has a number of advantages over a resolution-based approach to pronoun resolution, as provided in [MdR98b].... ..."

Cited by 4

### Table 6. A pre xed-based tableau calculus for M?LL fml(F,Pol,P,F1,F2,F3,PrN,Ctr) is used to specify the rules of our pre x- based tableau calculus. It succeeds if there is a rule to expand the formula F. Pol is the polarity, P, F1, F2, F3, and PrN are the position p, formulas F1, F2, F3, and the new pre x character s0, respectively. Ctr is bound to c if a contraction (rule) is applied. According to our calculus we need 18 clauses to specify all rules.

1999

"... In PAGE 12: ... Let AF be the set of all predicate symbols in the formula F. A tableaux for a for- mula F is de ned as usual (see De nition 4) but with the tableau expansion rules from Table6 where Ap A0 is a predicate symbol and Xp X is a predicate variable. Let T be a tableau, : M!( M [ M) be a string substitution, and X:X!(AF [ A0) be a predicate substitution.... ..."

### Tableau with clauses expanded once: TB

2000

Cited by 4

### Table 1 presents the rules of a generic free variable semantic tableau calculus. Starting from the initial tableau for a given closed formula of L+ , such rules allow to prolongate tableau branches in the standard way, as described for instance in [9]. We also refer the reader to [9] for all related basic notions, such as those of closed branch, closed tableau, satis able tableau, etc. The -, -, and -rules are the standard ones, so they deserve no further explanation. Concerning the -rule, we will characterize its proviso in such a way as to enforce soundness and encompass the -rule variants present in literature that de ne Skolem terms in a syntactical way.

"... In PAGE 32: ...47 10.81 Table1 . Complexity of the Case Studies... In PAGE 33: ... The last two examples contain mutually recursive operators. Table1 illustrates the complexity of the examples. It contains the number of lemmas (constant for all heuristics), and, for our novel heuristics with mandatory and obligatory literals, the number of manual interactions (manually applied inference rules + manually chosen induction order), the number of automatically applied inference rules (including the later deleted ones), the number of deleted inference rules due to a failed relief test and the runtime in seconds measured by a CMU Common Lisp system on a machine with a 1330 MHz AMD processor and 512 MB RAM.... In PAGE 60: ... - The term f( !S ) in the -rule is computed by a given function S (T ;m;n), where T is the current tableau, m is the index of the branch to be expanded, and n is the position of the -formula to be instantiated. Table1 . Tableau rules for a generic calculus.... In PAGE 61: ... We indicate with sko = (P; F [ sko) the augmented signature and with L sko the language over sko. The Skolem term f( !S ) in the -rule in Table1 consists of a function symbol f 2 sko of arity n 0 and an n-tuple !S of terms in L+ sko, whose variables belong to Var+. In general, the constraints that f( !S ) must satisfy may depend on the current tableau T , on the branch which is about to be expanded, and on the -formula on that is about to be instantiated.... In PAGE 62: ... Then we put: S (T ; m; n) =Def f( !H ) : (1) Section 4 illustrates how to apply our generic -rule to show the correctness of some -rules in literature. But before doing that, we will show that the tableau calculus described in Table1 is sound, provided that its associated Skolem terms construction rule satis es the above conditions C1-C8. It will be enough to show that tableau satis ability is preserved by the ex- pansion rules in Table 1 and substitution applications.... In PAGE 62: ... But before doing that, we will show that the tableau calculus described in Table 1 is sound, provided that its associated Skolem terms construction rule satis es the above conditions C1-C8. It will be enough to show that tableau satis ability is preserved by the ex- pansion rules in Table1 and substitution applications. To this purpose, it is convenient to stratify the language L+ sko, and then show how we can expand a given structure for L+ to a canonical structure for L+ sko.... In PAGE 64: ...Soundness of the generic -rule We are now ready to show that the tableau calculus in Table1 is sound, provided that the Skolem terms construction rule is de ned as in (1) and conditions C1- C8 hold. This will plainly be entailed by the following theorem.... In PAGE 64: ...t. Let A be an assignment in Msko. By the inductive hypothesis there exists a branch on T such that (Msko; A) j= . Let T 0 be the tableau resulting from an application of one of the expansion rules in Table1 or from an application of a substitution to T . If T 0 = T , then it can be shown that Msko satis es T 0 (cf.... In PAGE 84: ... f:(memb(C, A)), :(memb(C, B)), memb(C, intersect(A, B))g. Table1 . Timing and clauses of OSHL, Otter, Vampire, E-SETHEO and DCTP on set of theorems [-1-left for various values of n.... ..."

### Tableau ATMS Derived Tableau

1993

Cited by 1

### TABLEAU(A VERSIONOFTHEDAVIS-PUTNAMALGORITHM) VERSUSUSINGTABLEAUALONE.(ALL EXPERIMENTSWERE ONA 100-MzSGI CHALLENGEWORKSTATION.) vat-s clauses bounds and tableau tableau only

1996

Cited by 113

### TABLEAU(A VERSIONOFTHEDAVIS-PUTNAMALGORITHM) VERSUSUSINGTABLEAUALONE.(ALL EXPERIMENTSWERE ONA 100-MzSGI CHALLENGEWORKSTATION.) vat-s clauses bounds and tableau tableau only

1996

Cited by 113