"... In PAGE 8: ... If a proof of Conjecture 8 were found it might be possible to find a closed formula for the number of perfect Skolem sets of any given order. Table4 contains the number of perfect Skolem sets of orders 1 to 20. Table 4 Number of perfect Skolem sets.... ..."

"... In PAGE 12: ... As a non-completely trivial example, in Figure 5 we sketch the proof of axiom 32p 23p @ 0, characteristic of the logic of reflexive, weakly directed frames, in C : = QK[fre ; wconng. Rules re = Str(T) and wconn = Str(3) are visible in Table4 . This proof is possible, as we expect, since the property of weak connectedness is strictly stronger than that of weak directedness.... In PAGE 13: ...5 Discussion The methodology outlined earlier on allows us to build sequent calculi for any FO-axiomatisable QML (with or without equality). As an extended example, Table4 shows the rules obtained by application of the strengthening procedure to sentences in Table 1. We have given them mnemonic names, such as re = Str(T), and so on.... In PAGE 13: ... We have given them mnemonic names, such as re = Str(T), and so on. As usual, labels in the rules of Table4 are really placeholders. Besides adding to the elegance of the presentation, the properties of modularity and uniformity are useful for the implementation of these logics.... ..."

"... In PAGE 13: ... We have given them mnemonic names, such as re = Str(T), and so on. As usual, labels in the rules of Table4 are really placeholders.... ..."

"... 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.... ..."

"... In PAGE 7: ...We can observe that since in the structural problem constant names are local within clauses (so called Skolem constants), it does not matter whether we work with skolemized clauses, or variabilized clauses so that links between literals are preserved in the variabilization. Given P, we build the new instance space showed in Table1 . Let us more closely con- sider how example E is reformulated.... ..."

"... In PAGE 14: ... We here consider sKizzo [2,3,1], a skolemization-based, hybrid QBF solver (incorporating tree-reconstruction) that can be configured to exercise the following strategies1: symbolic BDD-based incomplete reasoning (abbreviated in S ), symbolic resolution-based solving ( R ), branching reasoning with backjumping and learning ( B ), SAT-based solution of ground sub-problems ( G ), and q-resolution reasoning ( Q ). The preliminary results in Table3 concern different reasoning styles, and show that advantages are expected to cover a broad family of QBF solvers. 1 Different strategies can be combined together to obtain solving personalities .... ..."

"... In PAGE 10: ... add to Ft the Skolem function(s) introduced at the previous step; 3. build a 2LK-derivation of ; S ! (see Table5 in Subsection 4.1) in which every sequent labelling a leaf contains only constraints, or ; when using rule l8 , avoid copying the main formula into the premise; 4.... In PAGE 20: ... Subcase (I) Let the set of subformulae of [ contain no 2-formulae and let FrmAxS(QL) be empty. By structural induction on the shape of the subformulae of [[ ]] and [[ ]], it is clear that every node N 2 is labelled by a 2LK-rule displayed in Table5 , except r8 and l8 . But, each of these rules is the translation of a single CQL-rule (recall the proof of implication 1-2); therefore, by Definition 15, every node in is a single, compact trail.... ..."