### Table 1 Complexity results of abduction from default theories.

2002

"... In PAGE 5: ... 2 Abstracting from the particular kind of expla- nation, the main decision problems in the context of abductive reasoning are the following: consistency: Does a given nite default abduc- tion problem possess an explanation? relevance: Given some nite default abduction problem P = hH; M; W; i and a set H0 H, is H0 relevant for P? necessity: Given some nite default abduction problem P = hH; M; W; i and a set H0 H, is H0 necessary for P? Concerning the computational complexity of these tasks, as shown in [12], these problems are located between the second and fourth level of the polynomial hierarchy. Table1 gives the speci c results; each entry C represents completeness of the corresponding problem for the class C.1 As can be seen from these results, skeptical abduction is always one level harder than brave abduction.... ..."

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### Table 8 Protocol theory complexity overview Bounded # Roles Unbounded # Roles

1982

"... In PAGE 35: ...3. Protocol complexity matrix Table8 shows a summary of the complexity results for the main theorems pre- sented in this paper. The two main columns consider the case of whether the number of roles is bounded or unbounded.... In PAGE 36: ... Because the number is fixed, the nonces can be assumed to have been produced during initialization, and not within the roles themselves. The two rows of Table8 consider whether the term size k is fixed in all instances of the problem, or whether the term size is allowed to vary as a parameter of the problem. For each entry of the matrix in Table 8, we show the complexity result for that case, using P to indicate the problem is in polynomial time, NPC forNP-complete, DEXPC forDEXP-complete, and Undec .... In PAGE 36: ... The two rows of Table 8 consider whether the term size k is fixed in all instances of the problem, or whether the term size is allowed to vary as a parameter of the problem. For each entry of the matrix in Table8 , we show the complexity result for that case, using P to indicate the problem is in polynomial time, NPC forNP-complete, DEXPC forDEXP-complete, and Undec . for Undecidable.... In PAGE 36: ... Table 9 is a more detailed summary of the complexity results, where we show more detail about the results for the upper and lower bounds. The columns are the same as in Table8 , but now the two main rows consider whether the intruder is allowed to generate fresh values or not. These rows are further subdivided into the cases where the roles can perform disequality tests which would allow them to determine whether two fresh values are different from each other.... In PAGE 36: ... If disequality is not allowed, then this test is not performed. Table 9 shows the complexity results for these cases, using the same notation as for Table8 . The numeric references indicate the propositions about specific lower or upper bounds which we discuss in the following sections.... ..."

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### Table 1. Theories for networks, and their complexities.

"... In PAGE 15: ...1 160.0 Table1 presents six inductive theories that cover the examples. Theory T 1 is the most general theory n3e.... In PAGE 15: ... Finally, theory T 6 concisely expresses the reachability relation using a recursive rule. Table1 also presents the code lengths of the theories, according to the theory complexity measure presented in Section 2.4.... In PAGE 15: ...Thus the prior probability on theories corresponds at least to some extent to our subjective notion of theory simplicity. Note also that the n14 relation on theory complexity connects strongly with the n13 relation on models: for the theories of Table1 wehaveLn28T 1 n29n14Ln28T 4 n29n14Ln28T 6 n29 and Qn28T 1 n29 n13 Qn28T 4 n29 n13 Qn28T 6 n29. Table 2 presents the evaluation of each theory according to the MC and PC measures.... ..."

### Table 1 Summary of complexity results for some edge modification problems. Boldfaced results are obtained in this work, NPC indicates an NP-complete problem, P a polynomial problem, and ? an open problem.

"... In PAGE 2: ... In this work we prove new NP-completeness results for these problems in some classes of graphs, such as interval, circular-arc, permutation, circle, bridge, weakly chordal and clique-Helly graphs. Table1 summarizes the known complexities of edge mod- ification problems in different graph classes, including those obtained in this work (which are boldfaced). Some preliminary results of this work appear in [5].... In PAGE 27: ... Proof: Circular-arc, interval, chordal, perfect, comparability and permutation graphs verify the hypotheses of Theorem 24. The results of Table1 and The- orem 24 imply this corollary. a50 Bipartite edge modification problems can be defined in analogous way to edge modification problems.... In PAGE 29: ...chain deletion are NP-complete. Proof: Chain graphs verify the hypotheses of Theorem 28, hence the results of Table1 and Theorem 28 imply this corollary. a50 Note that the complexity of chain editing is still unknown (see Table 1).... In PAGE 29: ... Proof: Chain graphs verify the hypotheses of Theorem 28, hence the results of Table 1 and Theorem 28 imply this corollary. a50 Note that the complexity of chain editing is still unknown (see Table1 ). We only know that this problem is reducible in polynomial time to biclique-Helly chain bipartite editing.... ..."

### Table 3. Experimental results. Complexity of the learned theory.

"... In PAGE 6: ...Some statistics concerning the learned theories are reported in Table3 . It is noteworthy that the average number of examples covered by a rule is between 6 and 8 for all three concepts.... ..."

Cited by 1

### Table 1. Theories for networks, and their complexities. i

1994

"... In PAGE 15: ...1 160.0 Table1 presents six inductive theories that cover the examples. Theory T1 is the most general theory gt;.... In PAGE 15: ... Finally, theory T6 concisely expresses the reachability relation using a recursive rule. Table1 also presents the code lengths of the theories, according to the theory complexity measure presented in Section 2.4.... In PAGE 15: ... Thus the prior probability on theories corresponds at least to some extent to our subjective notion of theory simplicity. Note also that the relation on theory complexity connects strongly with the relation on models: for the theories of Table1 we have L(T1) L(T4) L(T6) and Q(T1) Q(T4) Q(T6). Table 2 presents the evaluation of each theory according to the MC and PC measures.... ..."

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### Table 1. Theories for networks, and their complexities. i

1994

"... In PAGE 15: ...1 160.0 Table1 presents six inductive theories that cover the examples. Theory T1 is the most general theory gt;.... In PAGE 15: ... Finally, theory T6 concisely expresses the reachability relation using a recursive rule. Table1 also presents the code lengths of the theories, according to the theory complexity measure presented in Section 2.4.... In PAGE 15: ... Thus the prior probability on theories corresponds at least to some extent to our subjective notion of theory simplicity. Note also that the relation on theory complexity connects strongly with the relation on models: for the theories of Table1 we have L(T1) L(T4) L(T6) and Q(T1) Q(T4) Q(T6). Table 2 presents the evaluation of each theory according to the MC and PC measures.... ..."

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### Table 6.2: Input-output complexity of computing interconnection queries. D is always assumed to conform to S. If S is a tree, the complexity is always polynomial in D and the output.

2004

Cited by 2

### Table 3: Complexity of ISAT on all restricted subdomains of A3. We conjecture that ISAT is NP-complete on 5 = f \; \ ; g. A proof to this conjecture will resolve the remaining four cases. Note that the large number of polynomial restrictions can be used to speed up the solution of a non-polynomial one, by reducing the required enumeration size: For a given instance, one can easily nd which polynomial restriction is \closest quot; to it, in the sense that a minimum number of relation sets in the instance fall outside the restriction. By xing the possible relations in those (hopefully few) relation sets, the resulting problem can be solved polynomially. We have used the framework and the terminology of Allen apos;s interval calculus for tem- poral reasoning, but the results apply in any context where the consistency of assertions about relations of intervals must be veri ed. The tools we have used were mainly from graph theory and combinatorics. We have hoped to demonstrate that the interconnection

"... In PAGE 31: ... Polynomiality was shown in some cases by pointing out the equivalence to well known polynomial graph theoretic problems, and in others by techniques developed here. Table3 summarizes the current complexity status on all restricted domains in A3. In the table, represents both and , and \ represents both \ and \ , (Note that although there are seven non-empty relation sets in A3, we need to consider only the power set of the ve relations ; \; \; ; \ , since if a relation appears in the input, its converse also appears implicitly.... ..."

### Table 1. Complexity of Equational Matching Problems Theory Decision Counting Theory Decision Counting

"... In PAGE 18: ... Using the theory of #P-completeness, we identi ed the complexity of #E-Matching problems for several equational theories E. Table1 summarizes our ndings and compares the complexity of counting problems in equational matching with the complexity of the corresponding decision problems. Although in most cases the NP-completeness of the decision problem is accompanied by the #P-completeness of the associated counting problem, it should be emphasized that in general there is no relation between the complexities of these two problems.... ..."