PPT-Boolean Satisfiability

and SAT Solvers SAV March 18 th 2015 Boolean Satisfiability Goal find a model that satisfies a propositional formula The original NPcomplete problem As hard as any other problem in NP. Satisfiability. and Constraint Satisfaction Problems. by Carla P. Gomes, Bart Selman, . Nuno. . Crato. and . henry. . Kautz. Presented by . Yunho. Kim. Provable Software Lab, KAIST. Contents. Heavy-Tailed Phenomena in . Goal: . Show . how . propositional equivalences . are established . & introduce . the most . important such . equivalences.. Copyright © Peter . Cappello. 2. Equivalence. Name. p .  T . ≡ p; p . TopinpointthecomputationalcomplexityofCONN(S)andST-CONN(S),weneedtointroducecertainnewtypesofrelations.Denition3.LetRf0;1gkbealogicalrelation.1.Riscomponentwisebijunctiveifeveryconnectedcomponentoft M. AL- . Towaileb. 1. Boolean Functions. In Boolean algebra we work with the . set {0,1}. , . where:. 0 ≡ F . (False) & . 1 ≡ T . (True).. The 3 Operations used in Boolean Algebra are:. Complementation ( . Goal: . Show . how . propositional equivalences . are established . & introduce . the most . important such . equivalences.. Copyright © Peter . Cappello. 2. Equivalence. Name. p .  T . ≡ p; p . Satisfiability. Modulo Theories . Frontiers . of . Computational Reasoning . 2009 . –. MSR Cambridge. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. is a set . B. of values together with: . - two binary operations, commonly denoted by and ∙ , . - a unary operation, usually denoted by . ˉ or ~ or . ’. ,. - two elements usually called . Algebra. Huntington’s Postulates. Truth Tables. Graphic Symbols. Boolean Algebra Theorems. 1. Boolean . Algebra. 2. Boolean . Algebra. A fire sprinkler system should spray water if high heat is sensed and the system is set to . Karem A. Sakallah. EECS Department. University of Michigan. João Marques Silva. Informatics Department. Technical University of Lisbon. IST/INESC, CEL. SAT tutorial. 2. Context. SAT is the quintessential NP-complete problem. Karem A. Sakallah. EECS Department. University of Michigan. João Marques Silva. Informatics Department. Technical University of Lisbon. IST/INESC, CEL. SAT tutorial. 2. Context. SAT is the quintessential NP-complete problem. Algebra. Huntington’s Postulates. Truth Tables. Graphic Symbols. Boolean Algebra Theorems. 1. Boolean . Algebra. 2. Boolean . Algebra. A fire sprinkler system should spray water if high heat is sensed and the system is set to . Microchips (processors) . do exactly whatever instructions are fed into it, and that too without a single mistake.. Boolean . Logic was first introduced by George . Boole. The . basic . Boolean . operation can be further mapped into operations using bits and bytes. The most basic idea of Boolean Logic can be explained using logic gates. When the logic required becomes complex, these logic gates can be combined into more complex forms to get the required output. We have seen how we can represent information in binary, now we will explore. Why we use binary. How to compute using binary. How to implement binary operations using Boolean algebra (such as binary addition). Modulo Theories . Manchester 2009. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. Modulo Theories. PSpace. -complete. (QBF). Undecidable. (First-order logic).