### Table 3. Five hard examples: SAT solving increases termination proving power

2006

"... In PAGE 13: ... However, for the SAT-based analyses, the overall runtimes are still extremely fast in comparison to the non-SAT-based configurations. Table3 highlights 5 examples which could not be solved by any tool in the termination competition 2005, whereas the SAT-based configuration proves ter- mination for all 5 in a total of 4.3 seconds.... ..."

Cited by 8

### Table 2. The 25 hardest tests for poSAT

2006

"... In PAGE 9: ... The current implementation does not decompose partial order constraints to their SCC-components (Lemma 3). The experimental results indicate that the implementation would not benefit from that: (a) Most of the tests are very fast without this decomposition; and (b) It is typical for hard cases of LPO termina- tion (see Table2 ) to have a large strongly connected component including the majority of the symbols. For experimentation we have taken all 751 term rewrite systems from the Termination Problem Data Base [18] which do not specify a theory or a strat- egy .... In PAGE 10: ... In contrast, poSAT demonstrates similar performance for both LPO and quasi-LPO. Table2 presents a detailed analysis for the 25 most challenging examples for poSAT chosen by maximum total time for strict- and quasi- LPO analysis. The two parts of the table present the respective results for strict- and quasi- LPO termination analyses.... In PAGE 10: ... The columns labeled poSAT and TTT indicate run times (in seconds) for the poSAT and TTT solvers. All of the tests in Table2 are not strict- nor quasi-LPO terminating. This is not surprising for the 25 hardest tests, as proving unsatisfiability is typi- cally harder than finding a solution for a satisfiable formula.... In PAGE 11: ... The differ- ence is due to the fact that in the case of poSAT the generation of a partial order formula never introduces trivial sub-formula ( true or false ), these are evaluated on-the-fly. Another observation based on the results of Table2 is that the partial or- der constraints derived from the tests typically have domain graphs with large strongly-connected components. Almost every test in the table has a core com- ponent including the majority of the symbols.... In PAGE 11: ... Therefore, it is unlikely that the performance of poSAT for the presented tests can be improved by using the SCC-based decomposition of the formula. As Table2 shows, the maximum CNF instance solved in our tests includes 12827 propositional variables and 18205 CNF clauses. This is well below the ca- pacity limits of MiniSat, which is reported to handle benchmarks with hundreds... In PAGE 12: ... However, in view of Lemma 3 we may assume that we are testing satisfiability for partial order constraints which have strongly connected domain graphs. Moreover, as indicated by our experimental evaluation (see Table2 ), the domain graphs for some of the more challenging examples have... ..."

Cited by 14

### Table 6: Total time for the longest path branching method to nd and prove optimality for a set of benchmark problems, using either 1 or 16 processors. The initial known solution is either optimal or 2% above optimum. A * indicates \out of memory quot; when solving the problem.

1995

"... In PAGE 18: ... Table 3 gives the number of nodes in the search tree in order to solve the problems to optimality and Table 4 gives the corresponding running times as well as the relative speed-up of the algorithm. In Table 5 and Table6 we show the results of using the longest path branching strategy by Brucker et al., which generates a larger set of new active nodes at each branching.... ..."

Cited by 7

### Table 5: Number of nodes bounded if we use the longest path branching method in order to nd and prove optimality for a set of benchmark prob- lems, using either 1 or 16 processors. The initial known solution is either optimal or 2% above optimum. A * indicates that the processor ran out of memory when solving the problem.

1995

"... In PAGE 18: ... Table 3 gives the number of nodes in the search tree in order to solve the problems to optimality and Table 4 gives the corresponding running times as well as the relative speed-up of the algorithm. In Table5 and Table 6 we show the results of using the longest path branching strategy by Brucker et al., which generates a larger set of new active nodes at each branching.... In PAGE 18: ..., which generates a larger set of new active nodes at each branching. Table5 and 6 are respectively the number of nodes searched and the running times when proving optimality. It is quite obvious from the tables that the problems LA fall into two categories | easy problems that require no branching in order to prove op- timality and di cult problems where a lot of nodes must be bounded before optimality is proved.... ..."

Cited by 7

### Table 1. Benchmark problems solved by MiniSat+. GAC SAT: results from MiniSat+ with all CNF clauses; SAT: results from MiniSat+ with all CNF clauses but clause (4); Mono: results from a CP solver using the monolithic propagator; Decomp: results from a CP solver using the decomposition; |A|: number of activities; #: problem number; m: number of employees; sol: number of worked hours (boldfonted if best solution found amongst the different methods); time (s): CPU time in seconds to find and prove the optimality of a solution. Times are omitted when the search is suspended by a lack of memory; bt: number of backtracks (boldfonted if least back- tracks amongst methods that prove optimality); opt: solution was proved optimal. ILog solver did not prove any problems optimal within one hour of computation.

"... In PAGE 12: ... ILog solver was halted after one hour of computation as it never proved the optimality of a solution. Table1 presents the results for 17 satisfiable instances of the benchmark involving one or two activities. The CP model performed very well at finding a good solution.... ..."

### Table 1. Benchmark problems solved by MiniSat+. GAC SAT: results from MiniSat+ with all CNF clauses; SAT: results from MiniSat+ with all CNF clauses but clause (4); Mono: results from a CP solver using the monolithic propagator; Decomp: results from a CP solver using the decomposition; |A|: number of activities; #: problem number; m: number of employees; sol: number of worked hours (boldfonted if best solution found amongst the different methods); time (s): CPU time in seconds. Times are omitted when the search is suspended by a lack of memory; bt: number of backtracks (boldfonted if least backtracks amongst methods that prove optimality); opt: solution was proved optimal. ILog solver did not prove any problems optimal within one hour of computation.

"... In PAGE 12: ... 10.4.8 and ILog Solver on a AMD Dual Core Opteron 2.2 GHz with 4 Gb of RAM. The reader should be careful when comparing the times as the clock speeds of the computers are slightly different. Table1 presents the results for 17 satisfiable instances of the benchmark involving one or two activities. The CP model performed very well at finding a good solution.... ..."

### Table 3. The top-20 most similar paths to X solves Y .

2001

"... In PAGE 4: ... (c) Compute the similarity between p and the candidates that passed the filter using equation (2) and output the paths in descending order of their similarity to p. Table3 lists the Top-50 most similar paths to X solves Y generated by DIRT. Most of the paths can be considered as paraphrases of the original expression.... ..."

Cited by 41

### Table 3: Number of nodes bounded if we use the simple branching method, in order to nd and prove optimality for a set of benchmark problems, using either 1 or 16 processors. Reported with an initial solution that is either optimal or 2% above optimum.

1995

"... In PAGE 18: ... 5.1 Results of distributing nodes In Table3 and Table 4 we show the results of using the branching strategy by Carlier and Pinson, generating two new active nodes at each branching. Table 3 gives the number of nodes in the search tree in order to solve the problems to optimality and Table 4 gives the corresponding running times as well as the relative speed-up of the algorithm.... In PAGE 18: ...1 Results of distributing nodes In Table 3 and Table 4 we show the results of using the branching strategy by Carlier and Pinson, generating two new active nodes at each branching. Table3 gives the number of nodes in the search tree in order to solve the problems to optimality and Table 4 gives the corresponding running times as well as the relative speed-up of the algorithm. In Table 5 and Table 6 we show the results of using the longest path branching strategy by Brucker et al.... ..."

Cited by 7

### Table 4.1: Test set: Longest shortest path over all variables, avarage longest shortest path per variable (of the vig) and solve times in seconds using look-ahead solver march dl and conflict driven solver MiniSat.

in or

2006

### Table 1. Con gurable SAT solver run-time in cycles In order to evaluate the performance of this design, we built a C++ simula- tor and used DIMACS SAT problems as input [4]. Table 1 shows the number of hardware cycles needed to solve the problems. The rst column of data shows

1998

Cited by 7