{"type":"ScholarlyArticle","title":"Phase Transitions of PP-Complete Satisfiability Problems","authors":[{"name":"Delbert D. Bailey","slug":"delbert-d-bailey","given":"Delbert D.","family":"Bailey","affiliation":""},{"name":"Víctor Dalmau","slug":"victor-dalmau","given":"Víctor","family":"Dalmau","affiliation":""},{"name":"Phokion G. Kolaitis","slug":"phokion-g-kolaitis","given":"Phokion G.","family":"Kolaitis","affiliation":""}],"abstract":"The complexity class PP consists of all decision problems solvable by polynomial-time probabilistic Turing machines. It is well known that PP is a highly intractable complexity class and that PP-complete problems are in all likelihood harder than NP-complete problems. We investigate the existence of phase transitions for a family of PPcomplete Boolean satisfiability problems under the fixed clauses-to-variables ratio model. A typical member of this family is the decision problem #3S AT ( 2n=2 ): given a 3CNF-formula, is it satisfied by at least the square-root of the total number of possible truth assignments? We provide evidence to the effect that there is a critical ratio r3;2 at which the asymptotic probability of #3S AT ( 2n=2 ) undergoes a phase transition from 1 to 0. We obtain upper and lower bounds for r3;2 by showing that 0:9227 r3;2 2:595. We also carry out a set of experiments on random instances of #3S AT (2n=2 ) using a natural modification of the Davis-Putnam-Logemann-Loveland (DPLL) procedure. Our experimental results suggest that r3;2 2:5. Moreover, the average number of recursive calls of this modified DPLL procedure reaches a peak around 2:5 as well.","doi":null,"url":"https://ijcai.org/proceedings/2001/25","pdf":"https://ijcai.org/proceedings/2001/0025.pdf","markdown":"https://ijcai.org/proceedings/2001/25.md","bibtex":"https://ijcai.org/proceedings/2001/bibtex/25","volume":{"slug":"2001","title":"Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2001","url":"https://ijcai.org/proceedings/2001/","year":2001,"location":"Seattle, Washington, USA","dates":"","editors":"","isbn":"","publisher":"International Joint Conferences on Artificial Intelligence","doi_prefix":""},"section":null,"track":"","number":25,"pages":{"first":183,"last":191},"keywords":[],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}