Phase Transitions of PP-Complete Satisfiability Problems D. Bailey, V. Dalmau, Ph.G. Kolaitis Computer Science Department UC Santa Cruz

Phase Transition Phenomena First observed in NP-complete problems: SAT, CSP, Number Partitioning, … First observed in NP-complete problems: SAT, CSP, Number Partitioning, … Recently also observed in problems that are complete for higher complexity classes Recently also observed in problems that are complete for higher complexity classes Observed in PSPACE-complete problems: Observed in PSPACE-complete problems:  QBF: Cadoli et al. ; Gent & Walsh  SSAT: Littman; Littman-Majercik-Pitassi

PP: Probabilistic NP PP: there is a polynomial time NTM such that an input is accepted iff at least half of the computations are accepting. Simon, 1975; Gill, 1977 PP: there is a polynomial time NTM such that an input is accepted iff at least half of the computations are accepting. Simon, 1975; Gill, 1977 Prototypical PP-complete problem: MAJSAT: given a CNF-formula, is it satisfied by at least half of the possible truth assignments? Prototypical PP-complete problem: MAJSAT: given a CNF-formula , is it satisfied by at least half of the possible truth assignments? PP-complete problem in AI: Orponen, 1990; Roth, 1996 PP-complete problem in AI: Orponen, 1990; Roth, 1996

PP vs. Other Complexity Classes PP contains both NP and coNP. PP contains both NP and coNP. PP is contained in PSPACE. PP is contained in PSPACE. P PP = P # P (Angluin, 1980). Hence, P PP = P # P (Angluin, 1980). Hence,  PP captures the complexity of counting.  PP is highly intractable, since P # P contains the polynomial hierarchy PH (Toda, 1990).

Phase Transitions in PP Problems Study PP-complete satisfiability problems under the fixed clauses-to-variables model. Study PP-complete satisfiability problems under the fixed clauses-to-variables model. First natural choice to study: MAJ 3SAT First natural choice to study: MAJ 3SAT

Phase Transitions in PP Problems Study PP-complete satisfiability problems under the fixed clauses-to-variables model. Study PP-complete satisfiability problems under the fixed clauses-to-variables model. First natural choice to study: MAJ 3SAT. However, First natural choice to study: MAJ 3SAT. However, MAJ 3SAT is not known to be PP-complete. MAJ 3SAT is not known to be PP-complete. MAJ 3SAT has no phase transition: for every r, almost all random 3CNF formulas are satisfied by less than half of all possible truth assignments. MAJ 3SAT has no phase transition: for every r, almost all random 3CNF formulas are satisfied by less than half of all possible truth assignments.

Square Root 3SAT Square Root 3SAT - #3SAT(  2 n/2 ): given a 3CNF-formula, is it satisfied by at least 2 n/2 truth assignments? Square Root 3SAT - #3SAT(  2 n/2 ): given a 3CNF-formula , is it satisfied by at least 2 n/2 truth assignments? Intuitively, #3SAT(  2 n/2 ) asks whether at least one of the first n/2 bits of the number of satisfying truth assignments is equal to 1. Intuitively, #3SAT(  2 n/2 ) asks whether at least one of the first n/2 bits of the number of satisfying truth assignments is equal to 1.

Square Root 3SAT Square Root 3SAT - #3SAT(  2 n/2 ): given a 3CNF formula, is it satisfied by at least 2 n/2 truth assignments? Square Root 3SAT - #3SAT(  2 n/2 ): given a 3CNF formula , is it satisfied by at least 2 n/2 truth assignments? Intuitively, #3SAT(  2 n/2 ) asks whether at least one of the first n/2 bits of the number of satisfying truth assignments is equal to 1. Intuitively, #3SAT(  2 n/2 ) asks whether at least one of the first n/2 bits of the number of satisfying truth assignments is equal to 1. Theorem: #3SAT(  2 n/2 ) is PP-complete. Theorem: #3SAT(  2 n/2 ) is PP-complete.

Phase Transition Definition F(n,r): space of all 3CNF-formulas  with n variables and rn clauses. F(n,r): space of all 3CNF-formulas  with n variables and rn clauses. X: random variable on F(n,r), such that X  = number of satisfying assignments of . X: random variable on F(n,r), such that X  = number of satisfying assignments of . S: an assertion about X. S: an assertion about X. X has phase transition at r* iff for all r X has phase transition at r* iff for all r  If r < r*, then Pr[ S is true ]  1;  If r > r*, then Pr[ S is true ]  0.

Phase Transition Conjecture F(n,r): space of all 3CNF-formulas  with n variables and rn clauses. F(n,r): space of all 3CNF-formulas  with n variables and rn clauses. X: random variable on F(n,r) such that X  = number of satisfying assignments of . X: random variable on F(n,r) such that X  = number of satisfying assignments of . Conjecture: There is a ratio r* such that Conjecture: There is a ratio r* such that  If r < r*, then Pr[ X  2 n/2 ]  1;  If r > r*, then Pr[ X  2 n/2 ]  0.

Evidence for the Phase Transition Analytical results that yield upper and lower bounds for r*. Analytical results that yield upper and lower bounds for r*. Experimental results suggesting that r*  2.5 Experimental results suggesting that r*  2.5

Upper and Lower Bounds for r* Theorem:  r*  Theorem:  r*  Hint of Proof: Hint of Proof:  Upper Bound: Markov’s inequality.  Lower Bound: Covering partial assignments.

Upper Bound for r* From Markov’s inequality, Pr[ X  2 n/2 ]  E(X)/ 2 n/2 From Markov’s inequality, Pr[ X  2 n/2 ]  E(X)/ 2 n/2 E(X) = 2 n (7/8) rn E(X) = 2 n (7/8) rn Pr[ X  2 n/2 ]  2 n/2 (7/8) rn Pr[ X  2 n/2 ]  2 n/2 (7/8) rn If, 2 1/2 (7/8) r  1, then Pr [ X  2 n/2 ]  0 as n  If, 2 1/2 (7/8) r  1, then Pr [ X  2 n/2 ]  0 as n  So if, r  2.595, then Pr [ X  2 n/2 ]  0 as n  So if, r  2.595, then Pr [ X  2 n/2 ]  0 as n 

Lower Bound General Approach Show that a randomly selected formula  (x 1,…,x n ) has at least 2 n/2 satisfying assignments by finding a partial assignment with 2 n/2 variables that covers . Show that a randomly selected formula  (x 1,…,x n ) has at least 2 n/2 satisfying assignments by finding a partial assignment with  2 n/2 variables that covers . Covers means: the partial assignment makes the formula true, regardless of the truth value of the un-assigned variables. Covers means: the partial assignment makes the formula true, regardless of the truth value of the un-assigned variables.

Finding Covering Partial Assignments To show r* , derive from the existence of at least one satisfying assignment from To show r* , derive from the existence of at least one satisfying assignment from  a 2CNF theorem or from  a 3SAT lower bound theorem. To show r*  use Achlioptas’ technique to analyze a simple randomized algorithm, called the Extended Unit Clause. To show r*  use Achlioptas’ technique to analyze a simple randomized algorithm, called the Extended Unit Clause.

1/2 Lower Bound for r* Random n variable 3CNF  with m clauses, such that m/n  1/2. Random n variable 3CNF  with m clauses, such that m/n  1/2.  has at least one satisfying assignment,  by lower bound theorem for 3SAT.  has at least one satisfying assignment,  by lower bound theorem for 3SAT. Build partial assignment  with  n/2 variables that covers  for each of the  n/2 clauses in , choose some literal in the clause that is true under  and set  to also make that literal true. Build partial assignment  with  n/2 variables that covers  for each of the  n/2 clauses in , choose some literal in the clause that is true under  and set  to also make that literal true.

Experimental Results Implemented a threshold version of Birnbaum & Lozinskii’s CDP algorithm. Implemented a threshold version of Birnbaum & Lozinskii’s CDP algorithm. Experiments on random 3CNF-formulas with n = 10, 20, 30, 40, and 50 variables. Experiments on random 3CNF-formulas with n = 10, 20, 30, 40, and 50 variables. Probability curves cross at r  2.5 Probability curves cross at r  2.5 The average number of recursive calls peaks near 2.5 The average number of recursive calls peaks near 2.5

Birnbaum & Lozinskii’s CDP recursive function CDP(  n) recursive function CDP(  n) if  is empty, return 2 n if  is empty, return 2 n if  contains an empty clause, return 0 if  contains an empty clause, return 0 if  contains unit clause {t}, return CDP(  ,n-1), where if  contains unit clause {t}, return CDP(  ,n-1), where    contains all clauses in  that do not contain t ;  the literal  t is removed if present.

CDP (cont.) otherwise choose any variable x in  return CDP(  ,n-1) + CDP(  ,n-1), where otherwise choose any variable x in  return CDP(  ,n-1) + CDP(  ,n-1), where    contains all clauses in  that do not contain x, with the literal  x removed if present.    contains all clauses in  that do not contain  x, with the literal x removed if present.

Threshold CDP Accumulate partial counts in recursive calls of CDP. Accumulate partial counts in recursive calls of CDP. Return yes when accumulated count equals or exceeds threshold. Return yes when accumulated count equals or exceeds threshold. Return no otherwise. Return no otherwise. Can also use upper bound tracking to terminate and return no earlier. Can also use upper bound tracking to terminate and return no earlier.

Concluding Remarks Evidence for a phase transition in a natural PP-complete satisfiability problem. Evidence for a phase transition in a natural PP-complete satisfiability problem. Analytical upper bound obtained via Markov’s inequality is quite close to the value of the critical ratio suggested by the experiments. Analytical upper bound obtained via Markov’s inequality is quite close to the value of the critical ratio suggested by the experiments. Next steps: Next steps:  Obtain tighter upper and lower bounds.  Characterize the rate of growth of the average number of recursive calls as r and n vary.