### Table 1: Rules generated for the queries to compare in the reduction from Theorem 4.1. Rules Range DIFF INCON

1997

"... In PAGE 7: ... Note that while negation is still used in DIFF and INCON, it only applies to constant predicates. The rules shown in Table1 relate to Theorem 4.1 as follows: rules (Aij) compute ^ D; (Kj) de nes predi- cate r00 j for relation Rj in D [ ^ D; (B0 i) represent the fact that Qi(D [ ^ D) 6 Vi; (D0 j) de nes predicate r0 j for relation Rj in U(D [ ^ D); (Fj) de nes predicate ^ r0 j for relation Rj in U( ^ D); (Hk) de nes predicate ^ v0 k for Qk(U( ^ D)), the new state of view Vk that derives from ^ D after the update; (Ik) expresses the fact that Qk(U(D [ ^ D)) 6 = Qk(U( ^ D)).... ..."

Cited by 36

### Table 1: Point reductions of Theorem 1 and Lemma 2 and 3 (POWER3 in- stances). Groups: Number of groups. Terminals: Number of terminals in Hanan grid. Theorem 1: Number of terminals after removing free points. Lemma 2 and 3: Number of terminals after additional point reductions.

"... In PAGE 21: ...Table1 shows the e ect of Theorem 1 and Lemma 2 and 3. A considerable fraction of the terminals can be removed, in particular as a result of applying Theorem 1.... ..."

### Table 4: Lemma 1 Program reductions commute with proof reductions. Proof. See Albrecht and Crossley [1].2 As a corollary we have: Theorem 2 Every sequence of proof-reductions and program-reductions terminates. Proof. By the lemma we can perform all the proof-reductions rst. This sequence of reductions terminates by theorem 1. Next, all program-reductions are length reducing hence any such sequence terminates.2

### Table 3: Parallel Reduction

1993

"... In PAGE 30: ... The subject reduction theorem is stated for the relation !1 of one-step reduction. This relation is de ned to be the restriction of the parallel reduction relation de ned in Table3 (which we now read for the simpli ed system) obtained by dropping the rule of re exivity, eliminating the premises on the rules, and duplicating the remaining rules so that reduction is performed in only one of the two possible subterms. It is easy to see that ! 1 and ! coincide.... ..."

Cited by 571

### Table 3: Parallel Reduction

1993

"... In PAGE 30: ... The subject reduction theorem is stated for the relation !1 of one-step reduction. This relation is defined to be the restriction of the parallel reduction relation defined in Table3 (which we now read for the simplified system) obtained by dropping the rule of reflexivity, eliminating the premises on the rules, and duplicating the remaining rules so that reduction is performed in only one of the two possible subterms. It is easy to see that ! 1 and ! coincide.... ..."

Cited by 571

### Table 3: Proof reductions The basic result for proof terms in natural deduction and in the arithmetic de ned below is that all reductions terminate leaving a term which is then said to be in normal form. This prop- erty is called strong normalization. (Weak normalization is when some sequence of reductions terminates.) For convenience we shall call these reductions proof reductions. Theorem 1 (Girard [5]) Every proof term strongly normalizes and the normal form is unique (up to renaming of variables). 2 We now specify a further reduction process. First we delete all the types as in [7], except that we quot;remember quot; the type of the original term, i.e. the outermost type which is the formula whose proof is represented by the original proof term. Program-reductions are as shown in table 4. The resulting terms are called simpli ed proof terms of type . A ()

### Table 3: The basic result for proof terms in natural deduction and in the arithmetic de ned below is that all reductions terminate leaving a term which is then said to be in normal form. This prop- erty is called strong normalization. (Weak normalization is when some sequence of reductions terminates.) For convenience we shall call these reductions proof reductions. Theorem 1 Every proof term strongly normalizes and the normal form is unique (up to re- naming of variables). We now specify a further reduction process. First we delete all the types. Program-reductions are as shown in table 4. However, we quot;remember quot; the type of the original term, i.e. the outermost type which is the formula whose proof is represented by the original proof term. The resulting terms are called simpli ed proof terms of type . A

### Table 1: Comparing reductions of CNF symmetry detection to graph automorphism. V is the number of variables in the original CNF instance, C is the number of clauses, C2 is the number of binary clauses, CA2 BP C A0 C2. The 2xEDGES reduction is not practical with NAUTY be- cause NAUTY does not support double edges in graphs. CNF instances for which the DAC02 reduction finds spurious symmetries are characterized in Theorem 2.3.2.

2003

"... In PAGE 18: ...16 of binary clauses, and our construction DAC02 leads to non-trivial run time savings in practice. Table1 summarizes the main properties of various reductions of CNF symmetry finding to graph automorphism. Additionally, we empirically compare MIN3C, DAC02, the reduction from [18] and a corrected version of that reduction.... ..."

Cited by 27

### Table 1: Comparing reductions of CNF symmetry extraction to graph automorphism. V is the number of variables in the original CNF instance, C is the number of clauses, C2 is the number of binary clauses, CA2 BP C A0C2. The 2xEDGES reduction is not practical with NAUTY because NAUTY does not support double edges in graphs. CNF instances for which the DAC02 reduction finds spurious symmetries are characterized in Theorem 2.3.2.

2003

"... In PAGE 18: ...16 of binary clauses, and our construction DAC02 leads to non-trivial run time savings in practice. Table1 summarizes the main properties of various reductions of CNF symmetry extraction to graph automorphism. Additionally, we empirically compare MIN3C, DAC02, the reduction from [18] and a corrected version of that reduction.... ..."

Cited by 27

### Table 2. Results for the compiler veri cation example It seems, that Otter apos;s, Protein apos;s and Setheo apos;s heuristics for using axioms are already strong enough, to avoid all axioms involving sorts (encoded as constants), which do not occur in the theorem. In a at speci cation structure, this set is already a good approximation to the set of relevant axioms. To see, how the heuristics of Otter and Setheo (Protein is very similar to Setheo, so we did not try it) would behave in general, we nally tried an example with the opposite characteristic: Only few sorts, but many operations. The example is from the KIV library of standard speci cations: There, a speci cation Graph is de ned. The full set of axioms contains over 500 axioms. The 40 theorems considered were theorems on the sets of nodes. Axiom reduction yields below 100 relevant axioms for all these theorems. Table 3 gives the results, which clearly show the positive e ect of axiom reduction.

1998

Cited by 17