The VLSAT-2 Benchmark Suite

09/08/2021
by   Pierre Bouvier, et al.
0

This report presents VLSAT-2 (an acronym for "Very Large Boolean SATisfiability problems),the second part of a benchmark suite to be used in scientific experiments and softwarecompetitions addressing SAT-solving issues.VLSAT-2 contains 100 benchmarks (50 satisfiable and 50 unsatisfiable formulas)of increasing complexity, proposed in DIMACS CNF format undera permissive Creative Commons license.25 during the 2020 and 2021 editionsof the International SAT Competition.

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1 Benchmark Description

VLSAT-2111https://cadp.inria.fr/resources/vlsat/2.html is a collection of 100 SAT formulas. Many of these formulas are difficult to handle by current SAT solvers. One half of these formulas is satisfiable, while the other half is not.

Each formula is provided as a separate file, expressed in Conjunctive Normal Form and encoded in the DIMACS CNF format222http://www.satcompetition.org/2009/format-benchmarks2009.html. Each file is then compressed using bzip2 to save disk space and allow faster downloads. The 100 formulas require 5.2 gigabytes of disk space and 1.4 gigabytes when compressed using bzip2.

The VLSAT-2 benchmarks are licensed under the CC-BY Creative Commons Attribution 4.0 International License333License terms available from http://creativecommons.org/licenses/by/4.0.

25% of the VLSAT-2 benchmarks have been selected by the organizers of recent SAT Competitions: 7 satisfiable and 7 unsatisfiable formulas have been chosen for the SAT Competition 2020, and 5 satisfiable and 8 unsatisfiable formulas have been chosen for the SAT Competition 2021 [2].

2 Scientific Context

Interesting Boolean formulas can be generated as a by-product of our recent work [3] on the decomposition of Petri nets into networks of automata, a problem that has been around since the early 70s. Concretely, we developed a tool chain that takes as input a Petri net (which must be ordinary, safe, and hopefully not too large) and produces as output a network of automata that execute concurrently and synchronize using shared transitions. Precisely, this network is expressed as a Nested-Unit Petri Net (NUPN) [4], i.e., an extension of a Petri net, in which places are grouped into sets (called units) that denote sequential components. A NUPN provides a proper structuring of its underlying Petri net, and enables formal verification tools to be more efficient in terms of memory and CPU time. Hence, the NUPN concept has been implemented in many tools and adopted by software competitions, such as the Model Checking Contest444https://mcc.lip6.fr [8, 7] and the Rigorous Examination of Reactive Systems challenge555http://rers-challenge.org [5, 9, 6]. Each NUPN generated by our tool chain is flat, meaning that its units are not recursively nested in each other, and unit-safe, meaning that each unit has at most one execution token at a time.

Our tool chain works by reformulating concurrency constraints on Petri nets as logical problems, which can be later solved using third-party software, such as SAT solvers, SMT solvers, and tools for graph coloring and finding maximum cliques [3]. We applied our approach to a large collection of more than 12,000 Petri nets from multiple sources, many of which are related to industrial problems, such as communication protocols, distributed systems, and hardware circuits. We thus generated a huge collection of Boolean formulas, from which we carefully selected a subset of formulas matching the requirements of the SAT Competition.

3 Structure of Formulas

Each of our formulas was produced for a particular Petri net. A formula depends on three factors:

  • [itemsep=-0.4ex]

  • the set of the places of the Petri net;

  • a concurrency relation defined over , such that iff both places and may simultaneously have an execution token; and

  • a chosen number of units.

A formula expresses whether there exists a partition of into subsets () such that, for each , and for any two places and of , . A model of this formula is thus an allocation of places into units, i.e., a valid decomposition of the Petri net. This can also be seen as an instance of the graph coloring problem, in which colors are to be used for the graph with vertices defined by the places of and edges defined by the concurrency relation. A formula is only satisfiable if the value of is large enough (namely, greater than or equal to the chromatic number of the graph), so that at least one decomposition exists.

More precisely, each formula was generated as follows. For each place and each unit , we created a propositional variable that is true iff place belongs to unit . We then added constraints over these variables:

  • For each unit and each two places and such that and , where is a bijection from places names to the interval , we added the constraint to express that two concurrent places cannot be in the same unit.

  • For each place , we could have added the constraint to express that belongs to at least one unit, but this constraint was too loose and allowed similar solutions, just by permuting unit names. We thus replaced this constraint by a stricter one that breaks the symmetry between units: for each place , we added the refined constraint , where is a bijection from unit names to the interval .

Figure 1 illustrates, for a typical Petri net, how the resolution time evolves as the chosen number of units increases. If the value of is small (resp. large) enough, it is relatively easy to prove that the formula is satisfiable (resp. unsatisfiable) because there are not enough (resp. too many) colors. The difficulty comes when gets close to the chromatic number (which is equal to 13 on Fig. 1), as the number of combinations to be examined for proving unsatisfiability explodes, despite the introduction of symmetry-breaking constraints, which make unsatisfiable formulas much easier and satisfiable formulas slightly harder.

4 Selection of Benchmarks

Using the approach presented in Sections 2 and 3, we previously published a test suite, named VLSAT-1 [1], of 100 formulas. However, VLSAT-1 only contains satisfiable formulas, as it was designed for the Model Counting Competition, which seeks formulas accepting a large number of models. For the SAT Competition, we therefore undertook the production of a different collection containing both satisfiable and unsatisfiable formulas, depending on the number of units chosen for a given Petri net.

We selected 50 satisfiable and 50 unsatisfiable formulas, carefully chosen among a large collection of more than 132,000 formulas generated by our tool chain. We used five SAT solvers (namely, CaDiCal 1.3.0, Kissat 1.0.3, MathSAT 5.6.5, MiniSAT 2.2.0, and Z3 4.8.9) to reject all formulas that can be solved in less than one minute of CPU time by at least one of these solvers, and that can be solved within two hours by each of these solvers (experiments done on a Xeon E5-2630 v4 machine with 256 gigabytes of RAM). We also tried to provide formulas of increasing complexities, with a good compromise between the size of a formula and the time taken by the fastest solver to process this formula.

Figure 1: Resolution times for a typical NUPN
Figure 2: Dispersion of the VLSAT-2 benchmarks
variables clauses type difficulty
544 8738 UNSAT 3
600 11,440 UNSAT 3
684 9417 UNSAT 3
684 13,953 UNSAT 2
708 10,259 UNSAT 3
736 11,022 UNSAT 2
798 17,543 UNSAT 3
1000 22,250 SAT 4
1022 14,955 UNSAT 3
1125 15,795 UNSAT 3
1134 26,703 UNSAT* 4
1155 42,917 UNSAT* 4
1183 26,975 UNSAT 3
1209 29,889 UNSAT 3
1326 35,956 UNSAT 3
1352 42,432 UNSAT 3
1560 54,564 UNSAT 3
1586 57,751 UNSAT 3
2231 68,844 UNSAT 2
2340 70,758 UNSAT 2
2403 100,259 UNSAT 2
2548 119,204 UNSAT 3
3252 372,331 UNSAT 3
3456 192,912 SAT 2
3480 190,496 UNSAT 3
4424 545,056 UNSAT* 3
5152 824,642 UNSAT* 3
5525 327,765 UNSAT 3
5568 1,124,240 UNSAT 3
5600 1,042,700 UNSAT* 3
452 334,035 SAT 3
11,130 1,186,888 SAT* 0 (148 s)
11,280 4,223,777 UNSAT* 3
11,374 1,150,943 SAT* 1 (1802 s)
11,664 5,532,624 UNSAT 4
12,690 1,481,522 SAT 2
14,016 1,374,747 SAT 4
14,280 6,781,327 UNSAT* 3
14,424 7,585,190 UNSAT* 4
14,637 1,778,453 SAT 2
14,640 1,323,246 UNSAT 5
15,249 1,937,993 SAT 2
15,440 1,409,906 UNSAT 5
15,704 1,804,650 SAT 3
15,960 1,464,039 UNSAT 5
16,269 2,203,672 SAT 3
16,297 1,562,268 UNSAT 5
16,676 1,598,591 SAT 2
16,788 9,021,307 UNSAT 3
17,688 1,741,702 UNSAT 5
variables clauses type difficulty
18,250 2,995,915 SAT 2
19,565 3,665,001 SAT 2
20,424 2,157,568 UNSAT 5
21,114 2,240,429 UNSAT 5
21,190 2,597,791 SAT 3
21,546 2,615,351 SAT 3
21,573 2,289,124 SAT 3
22,032 3,022,731 SAT 2
23,961 2,714,844 UNSAT 5
24,450 2,770,239 SAT 1 (721 s)
26,104 3,131,630 UNSAT 5
26,606 3,191,844 SAT 2
26,988 3,251,923 UNSAT 5
27,507 3,314,450 SAT 3
28,930 6,497,511 SAT 4
29,040 3,874,024 SAT 2
29,205 3,712,921 UNSAT 5
29,456 6,615,638 SAT 2
29,736 3,780,419 SAT* 2
29,945 3,796,274 SAT 4
30,195 3,855,554 UNSAT 5
30,744 3,925,645 SAT 5
31,552 5,400,750 SAT 4
32,480 4,362,044 UNSAT 5
33,040 4,437,242 SAT 4
33,582 4,529,625 UNSAT 5
34,099 4,695,729 SAT 3
34,161 4,607,712 SAT 4
35,929 5,082,743 UNSAT 5
36,518 5,166,057 SAT 4
36,603 5,137,412 SAT 3
36,792 5,273,558 SAT 2
37,758 5,364,539 SAT* 4
39,552 5,878,762 UNSAT 5
40,170 5,970,608 SAT 5
40,896 6,104,639 UNSAT 5
41,535 6,200,014 SAT 4
41,875 6,420,498 SAT 4
45,150 7,165,285 SAT 4
46,440 7,369,989 SAT 2
47,817 1,337,056 SAT 2
57,038 10,572,502 SAT 5
58,380 10,775,711 SAT 4
59,204 10,973,962 SAT 4
67,996 13,708,722 SAT 5
68,760 13,862,744 SAT 5
69,524 14,016,766 SAT 5
70,288 14,170,788 SAT* 4
71,816 14,478,832 SAT 3
83,334 20,350,783 SAT 4
Table 1: List of VLSAT-2 formulas

The VLSAT-2 formulas are listed in Table 1. Those marked with a plus (resp. a star) in the table have been selected by the organizers of the SAT Competition 2020 (resp. 2021). The column “difficulty” contains a number from 0 (easy) to 5 (hard) indicating how many of the five aforementioned SAT solvers failed to solve the corresponding formula within two hours. For the easy values 0 and 1, the average number of seconds taken by the tools that managed to solve the formula is given.

Figure 2 shows the dispersion of the VLSAT-2 benchmarks for both satisfiable and unsatisfiable formulas. In general, and as confirmed by Fig. 1, satisfiable formulas need to be much larger (in the number of variables and clauses) than unsatisfiable ones to reach the same level of difficulty.

Acknowledgements

The experiments presented in this paper were carried out using the Grid’5000666https://www.grid5000.fr testbed, supported by a scientific interest group hosted by Inria and including Cnrs, Renater and several Universities as well as other organizations.

References

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