1. Property Directed Reachability (PDR) Using Cubes of Non-state Variables With Property Directed Reachability Using Cubes of Non-state Variables With Property Directed Reachability PDR is a symbolic model checking algorithm for verifying safety properties. Ternary Valued Simulation With Gate Variables Shifting Time Frames A new SAT-Based algorithm for symbolic model checking has been gaining popularity. This algorithm, referred to as “Incremental Construction of Inductive Clauses for Indubitable Correctness” (IC3) or “Property Directed Reachability” (PDR), uses information learned from SAT instances of isolated time frames to either prove that an invariant exists, or provide a counter example. The information learned between each time frame is recorded in the form of cubes of the state variables. In this work, we study the effect of extending PDR to use cubes of intermediate variables representing the logic gates in the transition relation. We demonstrate that we can improve the runtime for satisfiable benchmarks by up to 3.2X, with an average speedup of 1.23X. Our approach also provides a speedup of up to 3.84X for unsatisfiable benchmarks. Ph.D. Candidate, University of Minnesota Associate Professor, University of Minnesota John Backes Marc Riedel Abstract SAT Results The algorithm solves SAT instances representing discrete time frames in isolation. Variables, Notation and Terms:Trace Properties BenchmarkTime States (s)Frames StatesInv. StatesTime Gates (s)Frames GatesInv. GatesTime Ratio 6s246.671360146.35136010.99 6s343984.897722841053.828910570.26 6s619.1918170921.592117961.13 bj08amba2g3f31.12104414.37104812.83 bjrb07amba10andenv2081.17112042024.4992460.97 bjrb07amba3andenv10.56910310.078730.95 bjrb07amba4andenv54.8777831.67660.58 bjrb07amba5andenv93.45813079.581090.85 bjrb07amba6andenv277.888160438.8181921.58 bjrb07amba7andenv176.7911148160.1291590.91 bjrb07amba9andenv974.8392531013.88141851.04 bob0569.0618407101.86215051.47 bobcohdoptdcd484.7317141751.611311440.61 bobsmi2c48.03501121193.3318711834.03 cmudme1--- 7342.07907148--- cmudme24255.069761972917.469947460.69 eijkbs151248.1117630331.361613120.65 eijks38228.025736834.35693801.23 eijks42064.0746416167.975011611.06 eijks444407.7562429237.49613940.58 eijks526142.326450980.75625000.57 intel00623.961154325.11136151.05 intel007243.7101580193.47815540.79 intel0262262.395265042183.984962390.97 intel054239.291910483052111291.27 intel05569.981730970.03192111 intel056146.262383591.82227200.63 intel057144.4322536134.2204940.93 intel05962.881854366.52205361.06 intel0622095.272841012077.672649090.99 nusmvguidancep523.981829226.35182921.1 nusmvguidancep777.492162878.24216281.01 nusmvguidancep819.92215520.17221551.01 nusmvguidancep921.252015421.25201541 nusmvreactorp21093.5817244372394.327389 2.19 nusmvreactorp61416.9716343901444.9716343901.02 pdtpmscoherence68.8216149144.51212170.65 pdtpmsheap18.722565311.47235030.61 pdtpmsretherrtf278.9751251048.65439250.17 pdtpmsvsar25.641126312.9112600.5 pdtswvibs8x8p118.22086713.45218130.74 pdtswvqis10x6p150.057319235.91662080.72 pdtswvqis8x8p1108.9356285187.01623391.72 pdtswvroz10x6p112.84587312.9956731.01 pdtswvroz10x6p2117.0388136105.51761660.9 pdtswvroz8x8p113.54506012.4350640.92 pdtswvroz8x8p262.527118390.46601351.45 pdtswvsam6x8p324.844028424.78393111 pdtswvtma6x4p2186.35521537221.316017581.19 pdtswvtma6x4p31006.195861501754.366478371.74 pdtswvtma6x6p1297.39501191184.714811870.62 pdtswvtma6x6p21573.626959441851.386970541.18 pdtswvtms10x8p1107.8116173597.071515210.9 pdtswvtms12x8p170.52161531201.813716252.86 pdtswvtms14x8p173.416136281.382313531.11 pdtvisbakery030.29324232.8332421.08 pdtvisbakery117.68214732.1731461.82 pdtvisbakery227.42324325.0127470.91 pdtvisgoodbakery024.07274426.7928401.11 pdtvisgoodbakery119.6425463826551.93 pdtvisgoodbakery215.28254326.1227471.71 pdtvisns3p0015.51119923.34111111.5 pdtvisns3p0112.55126812.3511820.98 pdtvisns3p0236.211412518.314990.51 pdtvisns3p037.1496215.2212872.13 pdtvisns3p0420.971310016.6411970.79 pdtvisns3p0531.821412613.4412820.42 pdtvisns3p0629.231112420.9911900.72 pdtvisns3p0719.04169110.2812840.54 pdtvistimeout04649.413516496--- pdtvisrethersqo449.123845431.97384500.65 pdtvisvending0154.216117658.741911021.08 Geometric Average--- 0.98 BenchmarkTime States(s)Frames StatesTime Gates (s)Frames GatesTime Ratio abp4p2ff12.34176.57150.53 abp4ptimoneg22.461819.99170.89 bc57sensorsp0353.5959248.85410.7 bc57sensorsp0neg339.2562353.39551.04 bc57sensorsp1217.0159550.17732.53 bc57sensorsp1neg595.5763428.22470.72 bc57sensorsp2468.5369274.03630.59 bc57sensorsp2neg460.6479586.85711.27 bc57sensorsp3731.4267227.82580.31 intel017--- 4878.43232--- intel0462274.65682191.27700.96 intel0452101.11701810.71700.86 intel0471371.72622643.31691.93 irstdme468.673121.67260.32 irstdme519.46266.46260.33 irstdme631.842917.94280.56 nusmvtcasp5n99.662478.58240.79 nusmvtcastp577.462267.15230.87 prodcellp0neg77.7160105.54781.36 prodcellp1155.6272141.58720.9 prodcellp1neg96.8764145.41631.5 prodcellp2181.0860141.76810.78 prodcellp2neg143.6582114.42620.8 prodcellp3117.758102.05560.87 prodcellp4146.5466143.28750.98 prodcellp4neg162.7380526.8623.23 Geometric Average--- 0.82 Generally better results for satisfiable benchmarks Some unsatisfiable benchmarks proved faster Blocking Phase: Propagation Phase: Why Use Cubes of Gate Variables? x 0,x 1,x 2,x 3 g 0,g 1 x4x4 0000011 0011011 0110101 1100011 Three cubes in terms of x 0,x 1,x 2,x 3 can by blocked by one cube in terms of g 0,g 1 ! UNSAT Results Experiment Original Transition RelationNew Transition Relation Gates g 0,g 1,g 2,g 3 have only state variables in their cone of influence (COI)