### Table 1. Search runtimes of PB formulas with and without SBPs (for generators only) using PBS. Size of original instances and SBPs is shown. Symmetry statistics including symmetry detection runtime, number of symmetries, and generators are also provided. All runtimes are reported in seconds. Alternative PB encoding

"... In PAGE 8: ... Symmetry-breaking predicates [3] are applied to generators of the symmetry group found by SAUCY. Table1 and Table 2 list symmetry detection runtimes, the number of symmetries, and symmetry generators. The size of the original formula and the SBP, in terms of the number of variables, clauses, and PB constraints, are shown too.... In PAGE 8: ... The tables also compare runtimes for solving original instances and instances augmented with SBPs. Table1 reports on the PB formulation and Table 2 reports on a CNF- only formulation derived by converting the PB constraints using the exponential transformation de- scribed in [1]. S/U indicates if the formula is satisfiable or unsatisfiable.... ..."

### Table 1: CHAFF runtime on original SAT instances is compared to the combined runtime of symmetry detection and CHAFF on instances with symmetry-breaking clauses (the right-most column). The full name of benchmark 2dlx ca mc is 2dlx ca mc ex bp f . The numbers of symmetry generators and max cycles used per generator are shown (10 or all). Pure search speed-up (that does not take symmetry detection into account) is also given. Results for opportunistic window-based symmetry finding are also given and in most cases discover all or a large fraction of all symmetries. Instance Satis- #varables Plain Time Symmetries Speed-up ratios:

2002

"... In PAGE 1: ... Another such family was constructed by Urquhart in terms of expander graphs and with considerable use of random- ization [27]. Indeed, state-of-the-art SAT solvers, such as CHAFF, take a very long time to solve those instances (see Table1 ), but the relevance of such pathological cases to Design Automation is ques- tionable. In particular, lower bounds for SAT are typically proven for unsatisfiable instances, and it remains to be seen whether any satisfiable instances can be difficult for state-of-the-art solvers.... In PAGE 5: ... SAT solvers without random restarts will also need to solve many pigeon-hole sub-instances before finding satisfying solutions. Em- pirical results for these satisfiable instances (fpga)in Table1 show that they are very difficult for CHAFF. We observed that these sat- isfiable instances become more difficult with the increase of the difference between the throughput of the small switchboxes and the capacities of the channels that lead to them.... In PAGE 5: ...1 (chnl and fpga) and 5.2 (grout), Table1 lists six standard pigeon- hole instances (hole), five families of artificially constructed ran- domized Urquhart benchmarks (Urq) [27] and seven recent bench- marks from the micro-processor verification domain [28]. CHAFF runtimes in Table 1 are averages of (up to) 20 indepen- dent starts.... In PAGE 5: ...2 (grout), Table 1 lists six standard pigeon- hole instances (hole), five families of artificially constructed ran- domized Urquhart benchmarks (Urq) [27] and seven recent bench- marks from the micro-processor verification domain [28]. CHAFF runtimes in Table1 are averages of (up to) 20 indepen- dent starts. Such experimental protocol is required because CHAFF... In PAGE 7: ... Permutation gener- ators are specified by cycles. For each SAT instance, Table1 lists NAUTY/GRAPE/GAP runtime in seconds excluding I/O. the total number of symmetries and the number of permutation generators.... In PAGE 7: ... After symmetry-breaking clauses were added to the original CNF instance, the resulting preprocessed instance was solved with CHAFF. Table1 lists average runtimes of 20 independent runs of CHAFF for each instance. There were no time-outs for any of pre-processed CNFs.... In PAGE 7: ... There were no time-outs for any of pre-processed CNFs. The last column in Table1 shows relative speed-up ratios due to the use of symmetry-breaking clauses. For a given CNF instance, the first number in that column is the ratio of (i) CHAFF runtime on original instance, and (ii) the total runtime of symmetry detection and CHAFF on preprocessed instances.... In PAGE 7: ... We observe good empirical performance with win- dows of size of 1000. Results in Table1 show that, when applied to micro-processor verification benchmarks [28], our window-based opportunistic symmetry finding method found all or a significant portion of all symmetries in a fraction of runtime spent by com- plete symmetry finding. If a randomized variable ordering is used, one could combine local permutation generators found for different orderings.... In PAGE 8: ... For any solver slower than CHAFF, one can expect our flow to provide more significant speed-ups. For example, when we performed experiments described in Table1 with GRASP [26] in- stead of CHAFF [8], our flow demonstrated speed-up for all micro- processor verification benchmarks we considered. The speed-up ranged from 1.... ..."

Cited by 28

### Table 4: CHAFF run time on original SAT instances is compared to the combined run time of symme- try detection and CHAFF on instances with symmetry-breaking clauses added. The rightmost column also shows pure search speed-up (that does not take symmetry detection into account). The full name of bench- mark 2dlx ca mc is 2dlx ca mc ex bp f . The numbers of symmetry generators and max cycles used per generator (10 or all) are shown. The benchmarks we generated for these experiments are available at http://gigascale.org/bookshelf/Slots/SATbench

2003

"... In PAGE 27: ... Crosses represent input connections mated to channels, and every dot indicates the absence of a link. Empirical results in Table4 are shown for six routing configurations (chnl) in which one tries to route (a) 11, 12 or 13 connections through 10 tracks, and (b) 12, 13 or 20 connections through 11 tracks. These instances are extremely difficult for the leading-edge SAT solver CHAFF [41] and also have many symmetries.... In PAGE 27: ... On average, at least several such instances must be solved before a good partition is found. Empirical results for these satisfiable instances (fpga) in Table4 show that they are difficult for CHAFF. We observe that these instances be- come more harder when the difference between the throughput of the small switchboxes and the capacities of the channels that lead to them is increased.... In PAGE 30: ...nd 1GB of RAM. All codes were compiled with g++ 2.95.4 -O3 and ran on Debian Linux. In addition to the instances described in Section 4 (chnl and fpga)and(grout), Table4 lists six standard pigeon-hole instances (hole), five families of artificially constructed randomized Urquhart benchmarks (Urq) [52] and seven recent benchmarks from the micro- processor verification domain [54]. CHAFF run times in Table 4 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of different runs may vary significantly.... In PAGE 30: ... In addition to the instances described in Section 4 (chnl and fpga)and(grout), Table 4 lists six standard pigeon-hole instances (hole), five families of artificially constructed randomized Urquhart benchmarks (Urq) [52] and seven recent benchmarks from the micro- processor verification domain [54]. CHAFF run times in Table4 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of different runs may vary significantly. All runs not completed in 1000 seconds were aborted and did not contribute to averages.... In PAGE 30: ...n Section 2. Those graphs are subsequently processed by the NAUTY program [38, 39]. For each run, the result is a list of permutation generators of the group of symmetries, specified by their cycles. For each SAT instance, Table4 lists NAUTY run time in seconds excluding I/O, the total number of symmetries and the number of permutation generators. Those symmetry detection implementations are deterministic and not affected by re-ordering of vertices in the input graph.... In PAGE 30: ... After new clauses were added, the preprocessed CNF instance was solved with CHAFF. Table4 lists CHAFF run times for each instance. Because CHAFF run time on a given instance fluctuates from run to run, we report averages of 20 independent runs for each instance.... In PAGE 30: ... Pre-processed CNFs never timed out in our experiments. The last column in Table4 shows the relative speed-up ratios due to the use of symmetry- breaking clauses. For a given CNF instance, the first number is the ratio of (i) the CHAFF run time on original instance, and (ii) the total run time of symmetry detection and CHAFF on pre- processed instances.... ..."

Cited by 27

### Table 4: CHAFF run time on original SAT instances is compared to the combined run time of symme- try extraction and CHAFF on instances with symmetry-breaking clauses added. The rightmost column also shows pure search speed-up (that does not take symmetry extraction into account). The full name of bench- mark 2dlx ca mc is 2dlx ca mc ex bp f . The numbers of symmetry generators and max cycles used per generator (10 or all) are shown. The benchmarks we generated for these experiments are available at http://gigascale.org/bookshelf/Slots/SATbench

2003

"... In PAGE 27: ... Crosses represent input connections mated to channels, and every dot indicates the absence of a link. Empirical results in Table4 are shown for six routing configurations (chnl) in which one tries to route (a) 11, 12 or 13 connections through 10 tracks, and (b) 12, 13 or 20 connections through 11 tracks. These instances are extremely difficult for the leading-edge SAT solver CHAFF [41] and also have many symmetries.... In PAGE 27: ... On average, at least several such instances must be solved before a good partition is found. Empirical results for these satisfiable instances (fpga) in Table4 show that they are difficult for CHAFF. We observe that these instances be- come more harder when the difference between the throughput of the small switchboxes and the capacities of the channels that lead to them is increased.... In PAGE 30: ...nd 1GB of RAM. All codes were compiled with g++ 2.95.4 -O3 and ran on Debian Linux. In addition to the instances described in Section 4 (chnl and fpga)and(grout), Table4 lists six standard pigeon-hole instances (hole), five families of artificially constructed randomized Urquhart benchmarks (Urq) [52] and seven recent benchmarks from the micro- processor verification domain [54]. CHAFF run times in Table 4 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of different runs may vary significantly.... In PAGE 30: ... In addition to the instances described in Section 4 (chnl and fpga)and(grout), Table 4 lists six standard pigeon-hole instances (hole), five families of artificially constructed randomized Urquhart benchmarks (Urq) [52] and seven recent benchmarks from the micro- processor verification domain [54]. CHAFF run times in Table4 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of different runs may vary significantly. All runs not completed in 1000 seconds were aborted and did not contribute to averages.... In PAGE 30: ...n Section 2. Those graphs are subsequently processed by the NAUTY program [38, 39]. For each run, the result is a list of permutation generators of the group of symmetries, specified by their cycles. For each SAT instance, Table4 lists NAUTY run time in seconds excluding I/O, the total number of symmetries and the number of permutation generators. Those symmetry extraction implementations are deterministic and not affected by re-ordering of vertices in the input graph.... In PAGE 30: ... After new clauses were added, the preprocessed CNF instance was solved with CHAFF. Table4 lists CHAFF run times for each instance. Because CHAFF run time on a given instance fluctuates from run to run, we report averages of 20 independent runs for each instance.... In PAGE 30: ... Pre-processed CNFs never timed out in our experiments. The last column in Table4 shows the relative speed-up ratios due to the use of symmetry- breaking clauses. For a given CNF instance, the first number is the ratio of (i) the CHAFF run time on original instance, and (ii) the total run time of symmetry extraction and CHAFF on pre- processed instances.... ..."

Cited by 27

### Table 1. Generating set of the grid symmetries

2000

Cited by 7

### Table 4: Number of generated triangles and locates done by the lazy perturbation algorithm.

### Table 2. Search runtimes of CNF formulas with and without SBPs (for generators only) using PBS. Size of original instances and SBPs is shown. Symmetry statistics including symmetry detection runtime, number of symmetries, and generators are also provided. All runtimes are reported in seconds. The CNF-only formulation is derived by converting the PB constraints using the exponential transformation described in [1].

"... In PAGE 8: ... Symmetry-breaking predicates [3] are applied to generators of the symmetry group found by SAUCY. Table 1 and Table2 list symmetry detection runtimes, the number of symmetries, and symmetry generators. The size of the original formula and the SBP, in terms of the number of variables, clauses, and PB constraints, are shown too.... In PAGE 8: ... The tables also compare runtimes for solving original instances and instances augmented with SBPs. Table 1 reports on the PB formulation and Table2 reports on a CNF- only formulation derived by converting the PB constraints using the exponential transformation de- scribed in [1]. S/U indicates if the formula is satisfiable or unsatisfiable.... ..."

### Table 1: CHAFF runtime on original SAT instances is compared to the combined runtime of symmetry detection and CHAFF on instances with symmetry-breaking clauses (the right-most column). The full name of benchmark BECSD0DC CRCP D1CR is BECSD0DC CRCP D1CR CTDC CQD4 CU . The numbers of symmetry generators and max cycles used per generator are shown (10 or all). Pure search speed-up (that does not take symmetry detection into account) is also given. Results for opportunistic window-based symmetry-finding are also given and in most cases discover all or a large fraction of all symmetries. All benchmarks that we generated for these experiments are available at CWD8D8D4BMBBBBDBDBDBBACTCTCRD7BAD9D1CXCRCWBACTCSD9BBDICUCPD0D3D9D0BBCQCTD2CRCWD1CPD6CZD7BACWD8D1D0 Instance Satis- #variables Plain Time Symmetries Speed-up ratios:

2002

"... In PAGE 1: ... Another such family was con- structed by Urquhart in terms of expander graphs and with considerable use of randomization [24]. Indeed, state-of-the-art SAT solvers, such as CHAFF, take a long time to solve those instances (see Table1 ), but the relevance of such pathological cases to Design Automation is ques- tionable. While lower bounds for SAT are often proven for unsatisfiable instances, it remains to be seen whether satisfiable instances can be dif- ficult for the best solvers.... In PAGE 3: ... The one on the left entails routing N B7k connections through N tracks and yields unsatisfiable in- stances that for k BP 1 resemble the well-known pigeon-hole benchmarks. Empirical results in Table1 are shown for six routing configurations (CRCWD2D0) in which one tries to route (a) 11, 12 or 13 connections through 10 tracks, and (b) 12, 13 or 20 connections through 11 tracks. These in- stances are extremely difficult for the leading-edge SAT solver CHAFF [6] and also have many symmetries.... In PAGE 4: ... SAT solvers without random restarts will also need to solve many pigeon-hole sub-instances before finding satisfying solutions. Empirical results for these satisfiable instances (CUD4CVCP) in Table1 show that they are very diffi- cult for CHAFF. We observe that these instances become more difficult when the difference between the throughput of the small switchboxes and the capacities of the channels that lead to them is increased.... In PAGE 4: ... Then we applied the SAT encoding above. Table1 shows empirical results for the five most difficult instances (CVD6D3D9D8). 5.... In PAGE 4: ... In addition to the instances described in Section 4 (CRCWD2D0 and CUD4CVCP)and(CVD6D3D9D8), Ta- ble 1 lists six standard pigeon-hole instances (CWD3D0CT), five families of ar- tificially constructed randomized Urquhart benchmarks (CDD6D5) [24] and seven recent benchmarks from the micro-processor verification domain [25]. CHAFF runtimes in Table1 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of dif- ferent runs may vary significantly. All runs that did not complete in 1000 seconds were aborted and did not contribute to averages.... In PAGE 4: ... Permutation generators are specified by cycles. For each SAT instance, Table1 lists NAUTY runtime in seconds exclud- ing I/O. the total number of symmetries and the number of permutation generators.... In PAGE 4: ... After new clauses were added, the preprocessed CNF instance was solved with CHAFF. Table1 lists average runtimes of 20 independent runs of CHAFF for each instance. Pre-processed CNFs never timed out in our experiments.... In PAGE 4: ... Pre-processed CNFs never timed out in our experiments. The last column in Table1... In PAGE 6: ... We observe good empir- ical performance with windows of size 1000. Results in Table1 show that our window-based technique found all or a significant portion of all symmetries for the micro-processor verification benchmarks [25] in a fraction of the runtime spent by complete symmetry-finding. If a ran- domized variable ordering is used, one could combine local permutation generators found for different orderings.... In PAGE 6: ... Our proposed flow does not require source code modifications and should work with all complete SAT solvers. In experiments described in Table1 but performed with GRASP [23] instead of CHAFF [6], our flow demonstrated speed-ups 1.5-5 times even for the micro-processor verification benchmarks.... ..."

Cited by 28

### Table 1: CHAFF runtime on original SAT instances is compared to the combined runtime of symmetry detection and CHAFF on instances with symmetry-breaking clauses (the right-most column). The full name of benchmark BECSD0DC CRCP D1CR is BECSD0DC CRCP D1CR CTDC CQD4 CU . The numbers of symmetry generators and max cycles used per generator are shown (10 or all). Pure search speed-up (that does not take symmetry detection into account) is also given. Results for opportunistic window-based symmetry-finding are also given and in most cases discover all or a large fraction of all symmetries. All benchmarks that we generated for these experiments are available at CWD8D8D4BMBBBBDBDBDBBACTCTCRD7BAD9D1CXCRCWBACTCSD9BBDICUCPD0D3D9D0BBCQCTD2CRCWD1CPD6CZD7BACWD8D1D0 Instance Satis- #variables Plain Time Symmetries Speed-up ratios:

2002

"... In PAGE 1: ... Another such family was con- structed by Urquhart in terms of expander graphs and with considerable use of randomization [24]. Indeed, state-of-the-art SAT solvers, such as CHAFF, take a long time to solve those instances (see Table1 ), but the relevance of such pathological cases to Design Automation is ques- tionable. While lower bounds for SAT are often proven for unsatisfiable instances, it remains to be seen whether satisfiable instances can be dif- ficult for the best solvers.... In PAGE 3: ... The one on the left entails routing N B7k connections through N tracks and yields unsatisfiable in- stances that for k BP 1 resemble the well-known pigeon-hole benchmarks. Empirical results in Table1 are shown for six routing configurations (CRCWD2D0) in which one tries to route (a) 11, 12 or 13 connections through 10 tracks, and (b) 12, 13 or 20 connections through 11 tracks. These in- stances are extremely difficult for the leading-edge SAT solver CHAFF [6] and also have many symmetries.... In PAGE 4: ... SAT solvers without random restarts will also need to solve many pigeon-hole sub-instances before finding satisfying solutions. Empirical results for these satisfiable instances (CUD4CVCP) in Table1 show that they are very diffi- cult for CHAFF. We observe that these instances become more difficult when the difference between the throughput of the small switchboxes and the capacities of the channels that lead to them is increased.... In PAGE 4: ... Then we applied the SAT encoding above. Table1 shows empirical results for the five most difficult instances (CVD6D3D9D8). 5.... In PAGE 4: ... In addition to the instances described in Section 4 (CRCWD2D0 and CUD4CVCP) and (CVD6D3D9D8), Ta- ble 1 lists six standard pigeon-hole instances (CWD3D0CT), five families of ar- tificially constructed randomized Urquhart benchmarks (CDD6D5) [24] and seven recent benchmarks from the micro-processor verification domain [25]. CHAFF runtimes in Table1 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of dif- ferent runs may vary significantly. All runs that did not complete in 1000 seconds were aborted and did not contribute to averages.... In PAGE 4: ... Permutation generators are specified by cycles. For each SAT instance, Table1 lists NAUTY runtime in seconds exclud- ing I/O. the total number of symmetries and the number of permutation generators.... In PAGE 4: ... After new clauses were added, the preprocessed CNF instance was solved with CHAFF. Table1 lists average runtimes of 20 independent runs of CHAFF for each instance. Pre-processed CNFs never timed out in our experiments.... In PAGE 4: ... Pre-processed CNFs never timed out in our experiments. The last column in Table1 shows relative speed-up ratios due to the... In PAGE 6: ... We observe good empir- ical performance with windows of size 1000. Results in Table1 show that our window-based technique found all or a significant portion of all symmetries for the micro-processor verification benchmarks [25] in a fraction of the runtime spent by complete symmetry-finding. If a ran- domized variable ordering is used, one could combine local permutation generators found for different orderings.... In PAGE 6: ... Our proposed flow does not require source code modifications and should work with all complete SAT solvers. In experiments described in Table1 but performed with GRASP [23] instead of CHAFF [6], our flow demonstrated speed-ups 1.5-5 times even for the micro-processor verification benchmarks.... ..."

Cited by 28

### Table 1: CHAFF runtime on original SAT instances is compared to the combined runtime of symmetry detection and CHAFF on instances with symmetry-breaking clauses (the right-most column). The full name of benchmark BECSD0DC CRCP D1CR is BECSD0DC CRCP D1CR CTDC CQD4 CU . The numbers of symmetry generators and max cycles used per generator are shown (10 or all). Pure search speed-up (that does not take symmetry detection into account) is also given. Results for opportunistic window-based symmetry-finding are also given and in most cases discover all or a large fraction of all symmetries. All benchmarks that we generated for these experiments are available at CWD8D8D4BMBBBBDBDBDBBACTCTCRD7BAD9D1CXCRCWBACTCSD9BBDICUCPD0D3D9D0BBCQCTD2CRCWD1CPD6CZD7BACWD8D1D0 Instance Satis- #variables Plain Time Symmetries Speed-up ratios:

2002

"... In PAGE 1: ... Another such family was con- structed by Urquhart in terms of expander graphs and with considerable use of randomization [24]. Indeed, state-of-the-art SAT solvers, such as CHAFF, take a long time to solve those instances (see Table1 ), but the relevance of such pathological cases to Design Automation is ques- tionable. While lower bounds for SAT are often proven for unsatisfiable instances, it remains to be seen whether satisfiable instances can be dif- ficult for the best solvers.... In PAGE 3: ... The one on the left entails routing N B7k connections through N tracks and yields unsatisfiable in- stances that for k BP 1 resemble the well-known pigeon-hole benchmarks. Empirical results in Table1 are shown for six routing configurations (CRCWD2D0) in which one tries to route (a) 11, 12 or 13 connections through 10 tracks, and (b) 12, 13 or 20 connections through 11 tracks. These in- stances are extremely difficult for the leading-edge SAT solver CHAFF [6] and also have many symmetries.... In PAGE 4: ... SAT solvers without random restarts will also need to solve many pigeon-hole sub-instances before finding satisfying solutions. Empirical results for these satisfiable instances (CUD4CVCP) in Table1 show that they are very diffi- cult for CHAFF. We observe that these instances become more difficult when the difference between the throughput of the small switchboxes and the capacities of the channels that lead to them is increased.... In PAGE 4: ... Then we applied the SAT encoding above. Table1 shows empirical results for the five most difficult instances (CVD6D3D9D8). 5.... In PAGE 4: ... In addition to the instances described in Section 4 (CRCWD2D0 and CUD4CVCP)and(CVD6D3D9D8), Ta- ble 1 lists six standard pigeon-hole instances (CWD3D0CT), five families of ar- tificially constructed randomized Urquhart benchmarks (CDD6D5) [24] and seven recent benchmarks from the micro-processor verification domain [25]. CHAFF runtimes in Table1 are averages of (up to) 20 independent starts because CHAFF uses randomization internally and results of dif- ferent runs may vary significantly. All runs that did not complete in 1000 seconds were aborted and did not contribute to averages.... In PAGE 4: ... Permutation generators are specified by cycles. For each SAT instance, Table1 lists NAUTY runtime in seconds exclud- ing I/O. the total number of symmetries and the number of permutation generators.... In PAGE 4: ... After new clauses were added, the preprocessed CNF instance was solved with CHAFF. Table1 lists average runtimes of 20 independent runs of CHAFF for each instance. Pre-processed CNFs never timed out in our experiments.... In PAGE 4: ... Pre-processed CNFs never timed out in our experiments. The last column in Table1... In PAGE 6: ... We observe good empir- ical performance with windows of size 1000. Results in Table1 show that our window-based technique found all or a significant portion of all symmetries for the micro-processor verification benchmarks [25] in a fraction of the runtime spent by complete symmetry-finding. If a ran- domized variable ordering is used, one could combine local permutation generators found for different orderings.... In PAGE 6: ... Our proposed flow does not require source code modifications and should work with all complete SAT solvers. In experiments described in Table1 but performed with GRASP [23] instead of CHAFF [6], our flow demonstrated speed-ups 1.5-5 times even for the micro-processor verification benchmarks.... ..."

Cited by 28