### Table 3. Succinct ow logic for the functional fragment.

1998

"... In PAGE 6: ... We express this as follows: (Rd F ; Rc F ; MF ; SF ; WF ) satis es R; M e : S1 ! S2 amp; W Here the proposed solution consists of the ve caches of Table 2 and the entities R; M; S1; S2 and W: R 2 d Env is the environment in which e is to be analysed, M 2 P(Mem) is the set of contexts in which e is to be analysed, S1 2 d Store is the store that is possible immediately before e, S2 2 d Store is the store that is possible immediately after e, and W 2 c Val is the value that e can evaluate to. Since the ve caches of Table 2 remain \constant quot; throughout the veri cation we shall dispense with listing them when de ning the \ quot; relation in Table3 . Note that the clauses are de ned compositionally and hence clearly are well-de ned.... In PAGE 6: ... Given the caches of Example 4 we may verify the following formula for the program of Example 1 [ ]; f g program : [ ] ! [ ] amp; f( ; (y; 3))g re ecting that the initial environment is empty, that the initial context is the empty call string, that the program does not manipulate the store (which hence is empty) and that the nal value is described by f( ; (y; 3))g. The veri cation will amount to a proof using the clauses of Table3 as rules and axioms; if successful, the proof and the caches constitute the analysis of the program. 2 The clause for variables merely demands that the store after x equals the store possible before x and that the value associated with x in the environment equals... In PAGE 8: ... Containments versus equalities. Since the speci cation in Table3 is concerned with verifying whether or not a proposed solution is acceptable it is sensible that the clause for function application employs a containment like takek(l; M) MF ( ) rather than an equality like takek(l; M) = MF ( ). The reason is that there might be other instances of the clause where the label of the application... In PAGE 9: ... In fact it would be incorrect to replace the containment by an equality: if M 6 = ;, k gt; 0 and li1 6 = li2 then it is impossible to obtain takek(li; M) = MF ( ) for all i. Although the clauses in Table3 contain no explicit equalities they do contain a lot of implicit equalities because the same ow variable is used more than once in the same clause. One can avoid this by introducing new variables and then linking them explicitly by containments as illustrated below.... In PAGE 10: ...F ( )dXl dc(M; W1; )e v Rc F( ) ^ takek(l; M) MF( ) for some R1; M1; S11; S12; W1; R2; M2; S21; S22; W2 Clearly there will be proposed solutions that are acceptable according to the modi ed speci cation but that are not acceptable according to Table3 . This motivates being explicit about what we mean by the best solution.... In PAGE 10: ... In other words, we can change containments to equalities if we \collect quot; all terms de ning the same entity. 3 Attribute Grammar Formulations The ow logic of Table3 can be transformed into an attribute grammar. The basic idea behind attribute grammars is as follows.... In PAGE 10: ... We shall now proceed in two stages. First we show that a minor transformation will turn the speci cation of Table3 into an extended attribute grammar with global attributes and side conditions. The second stage will then transform the extended attribute grammar into an attribute grammar using global attributes and de ning the attributes by containments (rather than equalities).... In PAGE 11: ... The global attributes can be used as constants in the construction of terms for the attributes and their values can be further constrained by explicit conditions associated with the syntactic rules. It is now easy to see that Table 4 can be obtained from Table3 and vice versa by simply changing the notation. Hence it should be clear that the two speci cations admit the same acceptable solutions and therefore that the best solution for one equals the best solution for the other.... In PAGE 15: ... In doing so we shall exploit the presence of labels on all subexpressions. We shall write the analysis of an expression tl as (Rd F ; Rc F ; MF ; SF ; WF ; RL; ML; WL; SL) satis es tl and (as in Table3 ) we shall be explicit about the analysis of subexpressions. Allowing minor changes in notation this results in the speci cation of Table 8.... ..."

Cited by 9

### Table 3: Expressivity and succinctness of XPath dialects.

### Table 1: M and N Matrices for Various Stationary Iterative Solvers Solver M Matrix N Matrix

"... In PAGE 9: ...Stationary Multiplicative Iterative Solvers At the core of the AMG smoothers S(Al; ul; bl) are stationary multiplicative solvers of the form u(k+1) = Mlu(k) + Nlb ; (11) where Ml and Nl are derived from Al via matrix splitting. Table1 illustrates the structure of the M and N matrices for the Jacobi, Gauss-Seidel, and SOR iterative methods. D is the diagonal of A and ?L and ?U are the upper and lower triangular parts of A, respectively [13].... In PAGE 22: ... As in the 2D example, the AMG runtimes include the time required to setup of the coarse levels and interpolation operators. Table1 0: 3D Pillar Analysis Runtimes - Single Workstation Mesh PCG Runtime AMG Runtime AMG Runtime (swap) (minutes) (minutes) (minutes) Uniform Elements 32 47 1,366 Non-uniform Elements 125 98 3,038 As with the 2D foundation example, the runtimes indicate that PCG is better suited for problems involving uniform meshes and AMG is better suited to meshes involving poorly scaled elements. In addition, some swap delays did occur during the generation of the coarse levels for the non-uniform mesh on the single workstation run of the AMG algorithm.... In PAGE 26: ...Table1 1: Numerical Results for 3D Pillar Analysis - Uniform Mesh Displacement Stress (PCG) Stress (AMG) (kN m2 ) (kN m2 ) 0.050 -25.... In PAGE 26: ...33 -59.96 Table1 2: Distributed Algorithms Speedups - No Swapping Uniform Mesh Algorithm Workstations 2 3 4 5 Distributed Levels 0.64 0.... In PAGE 27: ...Table1 3: 3D Pillar Analysis Runtimes - Distributed Algorithms/Uniform Mesh Algorithm Workstations 2 3 4 5 Runtime Runtime Runtime Runtime (minutes) (minutes) (minutes) (minutes) Distributed Levels 73 72 71 N/A Distributed Levels (parallel) 62 63 62 N/A Distributed A1 91 94 104 120 Table 14: 3D Pillar Analysis Runtimes - Distributed Algorithms/Non-uniform Mesh Algorithm Workstations 2 3 4 5 Runtime Runtime Runtime Runtime (minutes) (minutes) (minutes) (minutes) Distributed Levels 92 93 93 N/A Distributed Levels (parallel) 79 78 80 N/A Distributed A1 118 133 150 155 [5] H. Regler and U.... In PAGE 27: ...Algorithm Workstations 2 3 4 5 Runtime Runtime Runtime Runtime (minutes) (minutes) (minutes) (minutes) Distributed Levels 73 72 71 N/A Distributed Levels (parallel) 62 63 62 N/A Distributed A1 91 94 104 120 Table1 4: 3D Pillar Analysis Runtimes - Distributed Algorithms/Non-uniform Mesh Algorithm Workstations 2 3 4 5 Runtime Runtime Runtime Runtime (minutes) (minutes) (minutes) (minutes) Distributed Levels 92 93 93 N/A Distributed Levels (parallel) 79 78 80 N/A Distributed A1 118 133 150 155 [5] H. Regler and U.... In PAGE 27: ....J. Evans, editor, Sparsity and Its Applications, Cambridge, MA, 1984. Cambridge University Press. Table1 5: Distributed Algorithms Speedups - Swapping Uniform Mesh Algorithm Workstations 2 3 4 5 Distributed Levels 18.71 18.... ..."

### Table 1: A comparison of Cassatt with other SAT solvers on difficult benchmarks from the SAT 2002 competition [28]. Each solver was given 3600 CPU seconds per benchmark. The number of benchmarks completed within that time as well as the total CPU time for each suite of benchmarks is shown.

2005

"... In PAGE 32: ...ttp://www.satlive.org/SATCompetition/2002/onlinereport/index.html Appendix A: Cassatt on Other Benchmarks Table1 shows how Cassatt and a few well known SAT solvers fare on a subset of benchmarks taken from the SAT02 competition [28]. The benchmarks were run on machines with 2.... ..."

Cited by 3

### Table 3: Complexity of Explanations and Partial Explanations: Succinct Representation

"... In PAGE 21: ...olds. This includes, e.g., descriptions of a241 in terms of propositional formulas a86 over a2 such that the models of a86 describe the contexts in a241 . Table3 shows our complexity results for some of the problems in Sections 3 and 4 in the setting where contexts are succinctly represented. More precisely, recognizing explanations and partial explanations in the case of succinct context sets is complete for a60a11a29 a61 (resp.... ..."

### Table 3 Complexity of Explanations and Partial Explanations: Succinct Representation

"... In PAGE 23: ...in terms of propositional formulas a153 over a11 such that the models of a153 describe the contexts in a18 . Table3 shows our complexity results for some of the problems in Sections 3 and 4 in the setting where contexts are succinctly represented. More precisely, recog- nizing explanations and partial explanations in the case of succinct context sets is complete for a67 a34 a68 (resp.... ..."

### Table 2: Results of the best general purpose Sat Solvers on various problem suites. Bracketed numbers indicate number of failures for that family. Ranking is with respect to all 23 solvers on Sat-Ex. The best times are in bold and times have been standardized to the Sat-Ex machine times.

2002

Cited by 11

### Table 2: Results of the best general purpose Sat Solvers on various problem suites. Bracketed numbers indicate number of failures for that family. Ranking is with respect to all 23 solvers on Sat-Ex. The best times are in bold and times have been standardized to the Sat-Ex machine times.

2002

Cited by 11

### Table 1: Complexity of problems speci#0Ced succinctly. The list is not

in Complexity of Hierarchically and One-dimensional Periodically-specified Problems I: Hardness Results

"... In PAGE 29: ... Our idea then, is to lift the known reduction from 3SAT to problem #05 when the instance is speci#0Ced non-hierarchically, and thus obtain a suitable reduction from L-3SAT to the problem L-#05. This approach helps one to prove easily many more PSPACE-hardness results for hierarchically speci#0Ced instances #28see Table1 #29. We also point out that most of the reductions are quasi-linear size and quasi-linear time reductions and thus provide tightlower bounds on the deterministic time complexityofthe problem #28under standard complexity theoretic assumptions#29.... In PAGE 33: ...heorem 6.2. The following problems are PSPACE-hard for 1-FPN-or L-speci#0Ced planar graphs of O#28log N #29 bandwidth: 3-COLORING, INDEPEN- DENT SET#28IS#29, DOMINATING SET, VERTEX COVER, PARTITION INTO TRIANGLES and HAMILTONIAN PATH#28HP#29. Table1 contains a sample of the results wehave obtained for succinctly speci#0Ced problems. As another example of the applicability of our results we consider the complexity of the monotone circuit value problem for L- and 1-FPN#28BC#29-speci#0Ced inputs.... In PAGE 33: ... We also outlined how our techniques could be used to prove the PSPACE- hardness of several other L-, 1-FPN-, 1-FPN#28BC#29- or 1-PN-speci#0Ced prob- lems. Table1 contains example of the results for L-, 1-FPN-, 1-FPN#28BC#29- or 1-PN-speci#0Ced problems that can be obtained by the techniques presented here. In the following, we explain the various entries in Table 1 and also... In PAGE 33: ... Table 1 contains example of the results for L-, 1-FPN-, 1-FPN#28BC#29- or 1-PN-speci#0Ced problems that can be obtained by the techniques presented here. In the following, we explain the various entries in Table1 and also... In PAGE 34: ...33 1. An entry 1 and 4 in Table1 denotes that the corresponding problem is PSPACE-hard, even when restricted to L- or 1-PN-speci#0Ccations of O#28log N #29 bandwidth bounded instances. These results have pre- viously not appeared in the literature; 4 implies that the result is proved formally in #5BMH+96#5D.... In PAGE 34: ... 2. An entry 2 in Table1 denotes that the problem L-#05 or the problem 1-PN-#05 was shown to be PSPACE-hard in #5BLW92#5Dor#5BOr82a#5D. This can also be shown by the ideas and techniques of this paper, even when restricted to L and 1-PN-speci#0Ccations of O#28log N #29 bandwidth bounded instances.... In PAGE 34: ... 3. An entry 3 in Table1 denotes that the corresponding problem is polynomial time solvable. This is shown in #5BMH+96#5D.... ..."

### Table 5 provides a succinct summary of the various indicators of some

1998