Beyond NP: the QSAT phase transition

  • Authors:

  • Ian P. Gent;Toby Walsh

  • Affiliations:

  • -;-

  • Venue:
  • AAAI '99/IAAI '99 Proceedings of the sixteenth national conference on Artificial intelligence and the eleventh Innovative applications of artificial intelligence conference innovative applications of artificial intelligence
  • Year:
  • 1999

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Abstract

We show that phase transition behavior similar to that observed in NP-complete problems like random 3-SAT occurs further up the polynomial hierarchy in problems like random 2-QSAT. The differences between QSAT and SAT in phase transition behavior that Cadoli et al report are largely due to the presence of trivially unsatisfiable problems. Once they are removed, we see behavior more familiar from SAT and other NP-complete domains. There are, however, some differences. Problems with short clauses show a large gap between worst case behavior and median, and the easy-hard-easy pattern is restricted to higher percentiles of search cost. We compute the "constrainedness" of k-QSAT problems for any k, and use this to predict the location of phase transitions. We conjecture that these predictions are less accurate than in NP-complete problems because of the super-exponential size of the state space, and of the weakness of first moment methods in complexity classes above NP. Finally, we predict that similar phase transition behavior will occur in other PSPACEcomplete problems like planning and game playing.