Fast Sampling of Perfectly Uniform Satisfying Assignments & Panos Theodoropoulos 2 1 University of California, Santa Cruz Santa Cruz, CA, USA {dimitris, zayd}@ucsc.edu 2 University of Athens Athens, Greece ptheodor@di.uoa.gr July 10, 2018 Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 1 / 25 Zayd Hammoudeh 1 ( � ) Dimitris Achlioptas 1 , 2
Introduction Setting Primary Application: Constrained Random Verifjcation (CRV) Basic Idea: Input test vector into hardware design and verify outputs Problem: Too many valid inputs to test all of them Standard Approach: Select valid test inputs u.a.r. Key Contribution We discuss approximate counting in the next session Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 2 / 25 From Boolean Formula F , select s ≥ 1 satisfying assignments u.i.r. SPUR : Perfectly uniform SAT sampler > 400 × faster than the state-of-the-art
Exact Model Counting Exact Model Counting Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 3 / 25
Exact Model Counting sharpSAT sharpSAT sharpSAT : State-of-the-art exact model counter [1] Developed by Marc Thurley First published at SAT-06 and iteratively improved since. DPLL-based Integrates both SAT techniques like CDCL, UP, two-watch literals, VS heuristics and counting specifjc optimizations e.g., component decomposition, subformula caching New Caching Paradigm : sharpSAT ’s primary difgerentiator Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 4 / 25
Exact Model Counting sharpSAT July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) If yes, sharpSAT uses the stored model count and backtracks 5 / 25 Structure : Set of key-value pairs Benefjt : Prevents re-analysis of duplicate residual formulas Caching : Analogue of CDCL for SAT residual formulas CDCL : Memoizes UNSAT assignments in new, implied clauses Caching Key : Boolean formula F ′ Value : Model count of F ′ sharpSAT places all SAT residual formulas, F ( σ ) , into the cache Before analyzing any F ( σ ) , sharpSAT checks if F ( σ ) is in the cache. Otherwise, sharpSAT analyzes F ( σ ) as normal
Exact Model Counting sharpSAT July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) . . . . . . . . . . . . The sharpSAT Tree 6 / 25 · · ·
Reservoir Sampling Reservoir Sampling Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 7 / 25
Reservoir Sampling Framing the Sampling Problem July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) Storage Capacity: s 3 “No looking back” on the stream 2 1 Additional Constraints : 8 / 25 Setting : From arbitrary, fjnite set P select s elements u.i.r. P 1 ∪ P 2 ∪ . . . ∪ P j ∪ . . . P m is an arbitrary partition of P ’s elements P is revealed as a stream of disjoint subsets (bundles) P 1 P 2 . . . P m |P| and m are unknown Solution : Reservoir Sampling Select s bundles P j with replacement where Pr[ Select P j ] ∝ | P j | From each selected bundle P j , select one member element p ∈ P j u.i.r.
Reservoir Sampling Let’s Build an Intuition about Reservoir Sampling Question : What is a reservoir? Answer : A multiset ( think memory ) of fjxed size s and m are unknown? Answer : Invariant – After analyzing each bundle P k , the statistical Question : Is reservoir sampling an iterative procedure? Answer : Yes. Identical iterative procedure for each stream bundle. An example will help clarify... Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 9 / 25 Question : How can reservoir sampling provide statistical guarantees if | P | guarantees are satisfjed w.r.t. ∪ k j = 1 P j (i.e., everything seen so far ).
Reservoir Sampling Reservoir July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) k successes s times Z Bent Coin Total Weight Next Total Weight Partition : Reservoir Already Processed s 1 P 1 P 2 P 3 P 4 P 5 2 4 3 5 10 / 25 · · · Pr[ Success ] = | P 5 | 6 · · · Z = � 4 j = 1 | P j | Z = � 5 i = 1 | P i |
Bringing Together Model Counting & Reservoir Sampling Bringing Together Model Counting & Reservoir Sampling Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 11 / 25
Bringing Together Model Counting & Reservoir Sampling Overview SPUR – Satisfying Perfectly Uniform Random assignment sampler Reservoir sampling performed on the stream of leaves encountered by sharpSAT during its normal execution Probability a particular leaf selected is proportional to its model count with sharpSAT . Low overhead – If sharpSAT can count the models, SPUR can sample them Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 12 / 25 SPUR := sharpSAT + Reservoir Sampling Rule of Thumb : Generating 1,000 samples takes ∼ 10 × as long as counting
Bringing Together Model Counting & Reservoir Sampling Implementation C++ Based and Open Source Repository : https://github.com/ZaydH/spur implementation sharpSAT ’s caching interferes with reservoir sampling Necessitates change to caching paradigm Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 13 / 25 sharpSAT + Reservoir Sampling : Straightforward idea, complex
Experimental Results Experimental Results Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 14 / 25
Experimental Results Overview State-of-the-Art SAT Sampler : UniGen2 [2] Developed by Chakraborty, Fremont, Meel, Seshia, Vardi Probabilistic “ almost-independent ,” “ almost uniform ” Dataset : 373 formulas from diverse domains Exclusively all the formulas in the UniGen2 and sharpSAT [1] papers # Clauses : 43 to 1.7M We compare SPUR and UniGen2 in two sets of experiments: Uniformity of selected samples Execution time to generate 1K samples Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 15 / 25 # Variables : 17 to > 300K
Experimental Results 400 July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) Figure 1: Uniformity comparison between the ideal distribution, SPUR and UniGen2 UniGen2 SPUR Ideal Occurrences 500 300 Sampler Uniformity 200 100 0 280 260 240 220 200 Uniformity – Both About the Same in Practice 16 / 25
Experimental Results 1hr July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) SPUR Time Out UniGen2 Time Out SPUR UniGen2 7hr 1min Execution Time Comparison 10sec 1sec 10hr 1hr 1min 10sec 1sec 17 / 25 0 . 1sec 0 . 1sec
Experimental Results Execution Time Comparison Execution Time Results Summary SPUR is faster than state-of-the-art on 371/373 benchmarks sharpSAT takes to count Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 18 / 25 Average speed-up : > 400 × Takeaway : 7 hours → 1 minute > 10 × faster on 369/373 benchmarks > 100 × faster on > 2 / 3 benchmarks On > 70 % of benchmarks, SPUR generated 1K samples within 10 × the time SPUR is 3 × more likely than UniGen2 to successfully generate 1K samples for formulas of > 10 K variables
Inner Workings of SPUR Inner Workings of SPUR Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 19 / 25
Inner Workings of SPUR Inter-leaf dependencies undermines sample independence July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) In only this case, caching satisfying assignments is safe dependencies leaves in the sharpSAT tree Turning sharpSAT into a Sampler Sharing cached satisfying assignments induces dependencies between cache hit Key Observation #1 : “ Model counts are reusable but samples are not. ” Problem : This simply doesn’t work... Enables trivially building complete assignments even at cache hit leaves assignments A First Idea : Enhance sharpSAT to cache model counts and satisfying 20 / 25 Key Observation #2 : If only sampling s = 1 model, there are no
Inner Workings of SPUR The sharpSAT Tree Meets Reservoir Sampling Procedure : Perform reservoir sampling on each leaf P j k : Number of samples to be selected from P j Since the reservoir only has partial assignments, SPUR must convert them to complete assignments eventually. Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 21 / 25 P : Set of satisfying assignments for Boolean formula F P j ∈ P : A leaf in the sharpSAT tree If k > 0 , store k and P j ’s partial assignment in the reservoir
Inner Workings of SPUR Top-Level SPUR Algorithm July 10, 2018 Fast SAT Sampling Achlioptas et. al. (UCSC & Univ. Athens) Preserves sample independence Recursion terminates faster Advantages : 22 / 25 Input : Boolean formula F and sample count s Output : S = { < s i , σ i > } s i : Sample count where � i s i = s σ i : Sample partial variable assignment For each < s i , σ i > ∈ S : a. F ( σ i ) is empty Set remaining bits in σ i u.a.r. b. F ( σ i ) is non -empty s i > 1 : Invoke SPUR recursively with formula F ( σ i ) and s i as normal s i = 1 : Invoke SPUR recursively with sample caching using key observation #2
Conclusions Conclusions Achlioptas et. al. (UCSC & Univ. Athens) Fast SAT Sampling July 10, 2018 23 / 25
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