### Table 6: Complexity of Equivalences with Bounded Relativization in Terms of Completeness Results.

in SEMANTICAL CHARACTERIZATIONS AND COMPUTATIONAL ASPECTS OF EQUIVALENCES IN STABLE LOGIC PROGRAMMING

2005

"... In PAGE 43: ...). Hence, the respective problem classes apply to programs P , Q, only if card(Atm(P [ Q) n A) k. Apparently, this class of problems contains strong and uniform equivalence in its unrelativized versions (k = 0). The complexity results are summarized in Table6 . In particular, we get that in the case of RSE all entries (except Horn/Horn) reduce to coNP-completeness.... ..."

### Table 5: Complexity of Relativized Equivalences in Terms of Completeness Results.

in SEMANTICAL CHARACTERIZATIONS AND COMPUTATIONAL ASPECTS OF EQUIVALENCES IN STABLE LOGIC PROGRAMMING

2005

"... In PAGE 37: ... Like in the previous section, we also consider different classes of programs. Our results are summarized in Table5 for both RSE and RUE at a glance by just highlighting where the complexity differs. Note that the only differences between RSE and RUE stem from the entries P 2 =coNP in the column for head-cycle free programs.... In PAGE 38: ...s special cases, i.e., if (V n A) = ;. We deal with bounded relativization explicitly in the subsequent section. Towards deriving the results from Table5 , we first consider model checking problems. Formally, for a set of atoms A, the problem of A-SE-model checking (resp.... ..."

### 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.... ..."

Cited by 2

### Table 4: Variables 3, Total Degree 3, Matrix Bound 2, Projection Bound 100.

2000

"... In PAGE 6: ... Note that one could use an alternative notion of a random polynomial based upon restricting the total degree rather than the degree in each term, and also by choosing distinct random monomials rather than just random monomials. Implementations reveal, see Table4 , that our algorithm is also effective under this distribution. 5 Implementation details The algorithm was implemented in C by the second author and programs run on a 550 MHz PC with 128 MB of RAM.... In PAGE 6: ... The parameters Total Degree and D. Terms are used in Table4 where we consider the slightly different notion of a random polynomial mentioned at the end of Section 4. In the following sections we make brief comments on the effectiveness of our algorithm referring to the tables in Appendix A.... In PAGE 7: ... However, up to this limit, absolute irreducibility of polynomials with a few hundred terms could be shown within minutes. Table4 gives information on polynomials of total degree bounded by 3. Here the mono- mials are distinct random monomials and thus column 1 contains the exact number of mono- mials in each polynomial.... ..."

Cited by 5

### Table 4: Variables 3, Total Degree 3, Matrix Bound 2, Projection Bound 100.

"... In PAGE 6: ... Note that one could use an alternative notion of a random polynomial based upon restricting the total degree rather than the degree in each term, and also by choosing distinct random monomials rather than just random monomials. Implementations reveal, see Table4 , that our algorithm is also e ective under this distribution. 5 Implementation details The algorithm was implemented in C by the second author and programs run on a 550 MHz PC with 128 MB of RAM.... In PAGE 6: ... The parameters \Total Degree quot; and \D. Terms quot; are used in Table4 where we consider the slightly di erent notion of a random polynomial mentioned at the end of Section 4. In the following sections we make brief comments on the e ectiveness of our algorithm referring to the tables in Appendix A.... In PAGE 7: ... However, up to this limit, absolute irreducibility of polynomials with a few hundred terms could be shown within minutes. Table4 gives information on polynomials of total degree bounded by 3. Here the mono- mials are distinct random monomials and thus column 1 contains the exact number of mono- mials in each polynomial.... ..."

### Table 1: Generic width g(n;; d) of polynomials of degree d in n variables.

1996

"... In PAGE 7: ... To be accurate, it is necessary to study each case separately, and this has been done mostly during the nineteenth century in the frame of invariant theory. Table1 summarizes known values of g(n;; d). Again, these values correspond to the generic case, and there are smaller and larger reachable values (see example in section 4).... ..."

Cited by 29

### Table D.1: Degree of polynomials

### Table 1: Price of anarchy of pure equilibria for linear latencies (left) and polynomial latencies of degree p (right).

in The

"... In PAGE 2: ... We also study both symmetric and asymmetric games. Our results (both lower and upper bounds) are summarized in the left part of Table1 . For the case of asymmetric games, the values hold also for network congestion games.... ..."

### Table 1 : Equivalent relativizations

2006

"... In PAGE 36: ... We have also established a few new cases of validity of the Conjecture. The main results are summarized in Table1 and 2... In PAGE 36: ...Table 1 : Equivalent relativizations Undirected graph classes Directed graph classes Uniformly k-sparse Uniformly k-sparse Line graphs Directed line graphs Quasi-series-parallel partial orders Finite interval graphs Finite partial orders of dimension 2 Interval graphs (for MS-OI) Partial orders of dimension 2 (for MS-OI) Table 2 : Proved relativizations. Table1 shows the equivalent relativizations. One could add the extensions of these classes by vertex and edge labellings.... ..."

Cited by 9