Satisfiability Checking Decidability and Decision Procedures –Some Historical Notes– Prof. Dr. Erika Ábrahám RWTH Aachen University Informatik 2 LuFG Theory of Hybrid Systems WS 19/20 Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 1 / 26 Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 branch and bound, Omega test Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 Peano arithmetic (N with + and ∗) undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 19/20 2 / 26 undecidable - But actually what does it mean “decidable” or “undecidable”? FO theories and their decidability Some first-order theories: Logic decidability algorithm Propositional logic decidable SAT-solving Equational logic decidable SAT-encoding Equational logic with uninterpr. functions decidable SAT-encoding Linear real algebra (R with +) decidable Simplex Real algebra (R with + and ∗) decidable CAD virtual substitution Presburger arithmetic (N with +) decidable branch and bound, Omega test Peano arithmetic (N with + and ∗) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University)