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First-Order Theorem Proving and Vampire Laura Kov acs (Chalmers - PowerPoint PPT Presentation

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First-Order Theorem Proving and Vampire Laura Kov acs (Chalmers University of Technology) Andrei Voronkov (The University of Manchester) Outline Introduction First-Order Logic and TPTP Inference Systems Saturation Algorithms Redundancy

  1. First-Order Logic and TPTP ◮ Language: variables, function and predicate (relation) symbols. A constant symbol is a special case of a function symbol. Variable names start with upper-case letters. ◮ Terms: variables, constants, and expressions f ( t 1 , . . . , t n ) , where f is a function symbol of arity n and t 1 , . . . , t n are terms. Terms denote domain (universe) elements (objects).

  2. First-Order Logic and TPTP ◮ Language: variables, function and predicate (relation) symbols. A constant symbol is a special case of a function symbol. Variable names start with upper-case letters. ◮ Terms: variables, constants, and expressions f ( t 1 , . . . , t n ) , where f is a function symbol of arity n and t 1 , . . . , t n are terms. Terms denote domain (universe) elements (objects). ◮ Atomic formula: expression p ( t 1 , . . . , t n ) , where p is a predicate symbol of arity n and t 1 , . . . , t n are terms.

  3. First-Order Logic and TPTP ◮ Language: variables, function and predicate (relation) symbols. A constant symbol is a special case of a function symbol. Variable names start with upper-case letters. ◮ Terms: variables, constants, and expressions f ( t 1 , . . . , t n ) , where f is a function symbol of arity n and t 1 , . . . , t n are terms. Terms denote domain (universe) elements (objects). ◮ Atomic formula: expression p ( t 1 , . . . , t n ) , where p is a predicate symbol of arity n and t 1 , . . . , t n are terms. Formulas denote properties of domain elements. ◮ All symbols are uninterpreted, apart from equality = .

  4. First-Order Logic and TPTP ◮ Language: variables, function and predicate (relation) symbols. A constant symbol is a special case of a function symbol. Variable names start with upper-case letters. ◮ Terms: variables, constants, and expressions f ( t 1 , . . . , t n ) , where f is a function symbol of arity n and t 1 , . . . , t n are terms. Terms denote domain (universe) elements (objects). ◮ Atomic formula: expression p ( t 1 , . . . , t n ) , where p is a predicate symbol of arity n and t 1 , . . . , t n are terms. Formulas denote properties of domain elements. ◮ All symbols are uninterpreted, apart from equality = . FOL TPTP ⊥ , ⊤ $false , $true ¬ F ˜F F 1 ∧ . . . ∧ F n F1 & ... & Fn F 1 ∨ . . . ∨ F n F1 | ... | Fn F 1 → F n F1 => Fn ( ∀ x 1 ) . . . ( ∀ x n ) F ! [X1,...,Xn] : F ( ∃ x 1 ) . . . ( ∃ x n ) F ? [X1,...,Xn] : F

  5. More on the TPTP Syntax %---- 1 * x = x fof(left identity,axiom,( ! [X] : mult(e,X) = X )). %---- i(x) * x = 1 fof(left inverse,axiom,( ! [X] : mult(inverse(X),X) = e )). %---- (x * y) * z = x * (y * z) fof(associativity,axiom,( ! [X,Y,Z] : mult(mult(X,Y),Z) = mult(X,mult(Y,Z)) )). %---- x * x = 1 fof(group of order 2,hypothesis, ! [X] : mult(X,X) = e ). %---- prove x * y = y * x fof(commutativity,conjecture, ! [X,Y] : mult(X,Y) = mult(Y,X) ).

  6. More on the TPTP Syntax ◮ Comments; %---- 1 * x = x fof(left identity,axiom,( ! [X] : mult(e,X) = X )). %---- i(x) * x = 1 fof(left inverse,axiom,( ! [X] : mult(inverse(X),X) = e )). %---- (x * y) * z = x * (y * z) fof(associativity,axiom,( ! [X,Y,Z] : mult(mult(X,Y),Z) = mult(X,mult(Y,Z)) )). %---- x * x = 1 fof(group of order 2,hypothesis, ! [X] : mult(X,X) = e ). %---- prove x * y = y * x fof(commutativity,conjecture, ! [X,Y] : mult(X,Y) = mult(Y,X) ).

  7. More on the TPTP Syntax ◮ Comments; ◮ Input formula names; %---- 1 * x = x fof(left identity,axiom,( ! [X] : mult(e,X) = X )). %---- i(x) * x = 1 fof(left inverse,axiom,( ! [X] : mult(inverse(X),X) = e )). %---- (x * y) * z = x * (y * z) fof(associativity,axiom,( ! [X,Y,Z] : mult(mult(X,Y),Z) = mult(X,mult(Y,Z)) )). %---- x * x = 1 fof(group of order 2,hypothesis, ! [X] : mult(X,X) = e ). %---- prove x * y = y * x fof(commutativity,conjecture, ! [X,Y] : mult(X,Y) = mult(Y,X) ).

  8. More on the TPTP Syntax ◮ Comments; ◮ Input formula names; ◮ Input formula roles (very important); %---- 1 * x = x fof(left identity,axiom,( ! [X] : mult(e,X) = X )). %---- i(x) * x = 1 fof(left inverse,axiom,( ! [X] : mult(inverse(X),X) = e )). %---- (x * y) * z = x * (y * z) fof(associativity,axiom,( ! [X,Y,Z] : mult(mult(X,Y),Z) = mult(X,mult(Y,Z)) )). %---- x * x = 1 fof(group of order 2,hypothesis, ! [X] : mult(X,X) = e ). %---- prove x * y = y * x fof(commutativity,conjecture, ! [X,Y] : mult(X,Y) = mult(Y,X) ).

  9. More on the TPTP Syntax ◮ Comments; ◮ Input formula names; ◮ Input formula roles (very important); ◮ Equality %---- 1 * x = x fof(left identity,axiom,( ! [X] : mult(e,X) = X )). %---- i(x) * x = 1 fof(left inverse,axiom,( ! [X] : mult(inverse(X),X) = e )). %---- (x * y) * z = x * (y * z) fof(associativity,axiom,( ! [X,Y,Z] : mult(mult(X,Y),Z) = mult(X,mult(Y,Z)) )). %---- x * x = 1 fof(group of order 2,hypothesis, ! [X] : mult(X,X) = e ). %---- prove x * y = y * x fof(commutativity,conjecture, ! [X,Y] : mult(X,Y) = mult(Y,X) ).

  10. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input]

  11. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas;

  12. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus

  13. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus

  14. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus

  15. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus

  16. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus ◮ Proof by refutation, generating and simplifying inferences, unused formulas . . .

  17. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus ◮ Proof by refutation, generating and simplifying inferences, unused formulas . . .

  18. Proof by Vampire (Slightliy Modified) Refutation found. Thanks to Tanya! 203. $false [subsumption resolution 202,14] 202. sP1(mult(sK,sK0)) [backward demodulation 188,15] 188. mult(X8,X9) = mult(X9,X8) [superposition 22,87] 87. mult(X2,mult(X1,X2)) = X1 [forward demodulation 71,27] 71. mult(inverse(X1),e) = mult(X2,mult(X1,X2)) [superposition 23,20] 27. mult(inverse(X2),e) = X2 [superposition 22,10] 23. mult(inverse(X4),mult(X4,X5)) = X5 [forward demodulation 18,9] 22. mult(X0,mult(X0,X1)) = X1 [forward demodulation 16,9] 20. e = mult(X0,mult(X1,mult(X0,X1))) [superposition 11,12] 18. mult(e,X5) = mult(inverse(X4),mult(X4,X5)) [superposition 11,10] 16. mult(e,X1) = mult(X0,mult(X0,X1)) [superposition 11,12] 15. sP1(mult(sK0,sK)) [inequality splitting 13,14] 14. ˜sP1(mult(sK,sK0)) [inequality splitting name introduction] 13. mult(sK,sK0) != mult(sK0,sK) [cnf transformation 8] 12. e = mult(X0,X0) (0:5) [cnf transformation 4] 11. mult(mult(X0,X1),X2)=mult(X0,mult(X1,X2))[cnf transformation 3] 10. e = mult(inverse(X0),X0) [cnf transformation 2] 9. mult(e,X0) = X0 [cnf transformation 1] 8. mult(sK,sK0) != mult(sK0,sK) [skolemisation 7] 7. ? [X0,X1] : mult(X0,X1) != mult(X1,X0) [ennf transformation 6] 6. ˜! [X0,X1] : mult(X0,X1) = mult(X1,X0) [negated conjecture 5] 5. ! [X0,X1] : mult(X0,X1) = mult(X1,X0) [input] 4. ! [X0] : e = mult(X0,X0)[input] 3. ! [X0,X1,X2] : mult(mult(X0,X1),X2) = mult(X0,mult(X1,X2)) [input] 2. ! [X0] : e = mult(inverse(X0),X0) [input] 1. ! [X0] : mult(e,X0) = X0 [input] ◮ Each inference derives a formula from zero or more other formulas; ◮ Input, preprocessing, new symbols introduction, superposition calculus ◮ Proof by refutation, generating and simplifying inferences, unused formulas . . .

  19. Statistics Version: Vampire 3 (revision 2038) Termination reason: Refutation Active clauses: 14 Passive clauses: 28 Generated clauses: 124 Final active clauses: 8 Final passive clauses: 6 Input formulas: 5 Initial clauses: 6 Splitted inequalities: 1 Fw subsumption resolutions: 1 Fw demodulations: 32 Bw demodulations: 12 Forward subsumptions: 53 Backward subsumptions: 1 Fw demodulations to eq. taut.: 6 Bw demodulations to eq. taut.: 1 Forward superposition: 41 Backward superposition: 28 Self superposition: 4 Memory used [KB]: 255 Time elapsed: 0.005 s

  20. Vampire ◮ Completely automatic: once you started a proof attempt, it can only be interrupted by terminating the process.

  21. Vampire ◮ Completely automatic: once you started a proof attempt, it can only be interrupted by terminating the process. ◮ Champion of the CASC world-cup in first-order theorem proving: won CASC 28 times.

  22. Main applications ◮ Software and hardware verification; ◮ Static analysis of programs; ◮ Query answering in first-order knowledge bases (ontologies); ◮ Theorem proving in mathematics, especially in algebra;

  23. Main applications ◮ Software and hardware verification; ◮ Static analysis of programs; ◮ Query answering in first-order knowledge bases (ontologies); ◮ Theorem proving in mathematics, especially in algebra; ◮ Verification of cryptographic protocols; ◮ Retrieval of software components; ◮ Reasoning in non-classical logics; ◮ Program synthesis;

  24. Main applications ◮ Software and hardware verification; ◮ Static analysis of programs; ◮ Query answering in first-order knowledge bases (ontologies); ◮ Theorem proving in mathematics, especially in algebra; ◮ Verification of cryptographic protocols; ◮ Retrieval of software components; ◮ Reasoning in non-classical logics; ◮ Program synthesis; ◮ Writing papers and giving talks at various conferences and schools . . .

  25. What an Automatic Theorem Prover is Expected to Do Input: ◮ a set of axioms (first order formulas) or clauses; ◮ a conjecture (first-order formula or set of clauses). Output: ◮ proof (hopefully).

  26. Proof by Refutation Given a problem with axioms and assumptions F 1 , . . . , F n and conjecture G , 1. negate the conjecture; 2. establish unsatisfiability of the set of formulas F 1 , . . . , F n , ¬ G .

  27. Proof by Refutation Given a problem with axioms and assumptions F 1 , . . . , F n and conjecture G , 1. negate the conjecture; 2. establish unsatisfiability of the set of formulas F 1 , . . . , F n , ¬ G . Thus, we reduce the theorem proving problem to the problem of checking unsatisfiability.

  28. Proof by Refutation Given a problem with axioms and assumptions F 1 , . . . , F n and conjecture G , 1. negate the conjecture; 2. establish unsatisfiability of the set of formulas F 1 , . . . , F n , ¬ G . Thus, we reduce the theorem proving problem to the problem of checking unsatisfiability. In this formulation the negation of the conjecture ¬ G is treated like any other formula. In fact, Vampire (and other provers) internally treat conjectures differently, to make proof search more goal-oriented.

  29. General Scheme (simplified) ◮ Read a problem; ◮ Determine proof-search options to be used for this problem; ◮ Preprocess the problem; ◮ Convert it into CNF; ◮ Run a saturation algorithm on it, try to derive ⊥ . ◮ If ⊥ is derived, report the result, maybe including a refutation.

  30. General Scheme (simplified) ◮ Read a problem; ◮ Determine proof-search options to be used for this problem; ◮ Preprocess the problem; ◮ Convert it into CNF; ◮ Run a saturation algorithm on it, try to derive ⊥ . ◮ If ⊥ is derived, report the result, maybe including a refutation. Trying to derive ⊥ using a saturation algorithm is the hardest part, which in practice may not terminate or run out of memory.

  31. Outline Introduction First-Order Logic and TPTP Inference Systems Saturation Algorithms Redundancy Elimination Equality

  32. Inference System ◮ inference has the form F 1 . . . F n , G where n ≥ 0 and F 1 , . . . , F n , G are formulas. ◮ The formula G is called the conclusion of the inference; ◮ The formulas F 1 , . . . , F n are called its premises. ◮ An inference rule R is a set of inferences. ◮ Every inference I ∈ R is called an instance of R . ◮ An Inference system I is a set of inference rules. ◮ Axiom: inference rule with no premises.

  33. Inference System: Example Represent the natural number n by the string | . . . | ε . ���� n times The following inference system contains 6 inference rules for deriving equalities between expressions containing natural numbers, addition + and multiplication · . x = y | x = | y ( | ) ε = ε ( ε ) x + y = z | x + y = | z (+ 2 ) ε + x = x (+ 1 ) x · y = u y + u = z ( · 2 ) ε · x = ε ( · 1 ) | x · y = z

  34. Derivation, Proof ◮ Derivation in an inference system I : a tree built from inferences in I . ◮ If the root of this derivation is E , then we say it is a derivation of E . ◮ Proof of E : a finite derivation whose leaves are axioms. ◮ Derivation of E from E 1 , . . . , E m : a finite derivation of E whose every leaf is either an axiom or one of the expressions E 1 , . . . , E m .

  35. Examples For example, || ε + | ε = ||| ε ||| ε + | ε = |||| ε (+ 2 ) is an inference that is an instance (special case) of the inference rule x + y = z | x + y = | z (+ 2 )

  36. Examples For example, || ε + | ε = ||| ε ||| ε + | ε = |||| ε (+ 2 ) is an inference that is an instance (special case) of the inference rule x + y = z | x + y = | z (+ 2 ) It has one premise || ε + | ε = ||| ε and the conclusion ||| ε + | ε = |||| ε .

  37. Examples For example, || ε + | ε = ||| ε ||| ε + | ε = |||| ε (+ 2 ) is an inference that is an instance (special case) of the inference rule x + y = z | x + y = | z (+ 2 ) It has one premise || ε + | ε = ||| ε and the conclusion ||| ε + | ε = |||| ε . The axiom ε + ||| ε = ||| ε (+ 1 ) is an instance of the rule ε + x = x (+ 1 )

  38. Proof in this Inference System Proof of || ε · || ε = |||| ε (that is, 2 · 2 = 4). ε + ε = ε (+ 1 ) | ε + ε = | ε (+ 2 ) ε + || ε = || ε (+ 1 ) ε · || ε = ε ( · 1 ) || ε + ε = || ε (+ 2 ) | ε + || ε = ||| ε (+ 2 ) ( · 2 ) || ε + || ε = |||| ε (+ 2 ) | ε · || ε = || ε ( · 2 ) . || ε · || ε = |||| ε

  39. Derivation in this Inference System Derivation of || ε · || ε = ||||| ε from ε + || ε = ||| ε (that is, 2 + 2 = 5 from 0 + 2 = 3). ε + ε = ε (+ 1 ) | ε + ε = | ε (+ 2 ) ε + || ε = ||| ε ε · || ε = ε ( · 1 ) || ε + ε = || ε (+ 2 ) | ε + || ε = |||| ε (+ 2 ) ( · 2 ) || ε + || ε = ||||| ε (+ 2 ) | ε · || ε = || ε ( · 2 ) . || ε · || ε = |||| ε

  40. Arbitrary First-Order Formulas ◮ A first-order signature (vocabulary): function symbols (including constants), predicate symbols. Equality is part of the language. ◮ A set of variables. ◮ Terms are built using variables and function symbols. For example, f ( x ) + g ( x ) . ◮ Atoms, or atomic formulas are obtained by applying a predicate symbol to a sequence of terms. For example, p ( a , x ) or f ( x ) + g ( x ) ≥ 2. ◮ Formulas: built from atoms using logical connectives ¬ , ∧ , ∨ , → , ↔ and quantifiers ∀ , ∃ . For example, ( ∀ x ) x = 0 ∨ ( ∃ y ) y > x .

  41. Clauses ◮ Literal: either an atom A or its negation ¬ A . ◮ Clause: a disjunction L 1 ∨ . . . ∨ L n of literals, where n ≥ 0.

  42. Clauses ◮ Literal: either an atom A or its negation ¬ A . ◮ Clause: a disjunction L 1 ∨ . . . ∨ L n of literals, where n ≥ 0. ◮ Empty clause, denoted by � : clause with 0 literals, that is, when n = 0.

  43. Clauses ◮ Literal: either an atom A or its negation ¬ A . ◮ Clause: a disjunction L 1 ∨ . . . ∨ L n of literals, where n ≥ 0. ◮ Empty clause, denoted by � : clause with 0 literals, that is, when n = 0. ◮ A formula in Clausal Normal Form (CNF): a conjunction of clauses.

  44. Clauses ◮ Literal: either an atom A or its negation ¬ A . ◮ Clause: a disjunction L 1 ∨ . . . ∨ L n of literals, where n ≥ 0. ◮ Empty clause, denoted by � : clause with 0 literals, that is, when n = 0. ◮ A formula in Clausal Normal Form (CNF): a conjunction of clauses. ◮ A clause is ground if it contains no variables. ◮ If a clause contains variables, we assume that it implicitly universally quantified. That is, we treat p ( x ) ∨ q ( x ) as ∀ x ( p ( x ) ∨ q ( x )) .

  45. Binary Resolution Inference System The binary resolution inference system, denoted by BR is an inference system on propositional clauses (or ground clauses). It consists of two inference rules: ◮ Binary resolution, denoted by BR: p ∨ C 1 ¬ p ∨ C 2 ( BR ) . C 1 ∨ C 2 ◮ Factoring, denoted by Fact: L ∨ L ∨ C ( Fact ) . L ∨ C

  46. Soundness ◮ An inference is sound if the conclusion of this inference is a logical consequence of its premises. ◮ An inference system is sound if every inference rule in this system is sound.

  47. Soundness ◮ An inference is sound if the conclusion of this inference is a logical consequence of its premises. ◮ An inference system is sound if every inference rule in this system is sound. BR is sound. Consequence of soundness: let S be a set of clauses. If � can be derived from S in BR , then S is unsatisfiable.

  48. Example Consider the following set of clauses {¬ p ∨ ¬ q , ¬ p ∨ q , p ∨ ¬ q , p ∨ q } . The following derivation derives the empty clause from this set: p ∨ q p ∨ ¬ q ¬ p ∨ q ¬ p ∨ ¬ q ( BR ) ( BR ) p ∨ p ¬ p ∨ ¬ p ( Fact ) ( Fact ) p ¬ p ( BR ) � Hence, this set of clauses is unsatisfiable.

  49. Can this be used for checking (un)satisfiability 1. What happens when the empty clause cannot be derived from S ? 2. How can one search for possible derivations of the empty clause?

  50. Can this be used for checking (un)satisfiability 1. Completeness. Let S be an unsatisfiable set of clauses. Then there exists a derivation of � from S in BR .

  51. Can this be used for checking (un)satisfiability 1. Completeness. Let S be an unsatisfiable set of clauses. Then there exists a derivation of � from S in BR . 2. We have to formalize search for derivations. However, before doing this we will introduce a slightly more refined inference system.

  52. Selection Function A literal selection function selects literals in a clause. ◮ If C is non-empty, then at least one literal is selected in C .

  53. Selection Function A literal selection function selects literals in a clause. ◮ If C is non-empty, then at least one literal is selected in C . We denote selected literals by underlining them, e.g., p ∨ ¬ q

  54. Selection Function A literal selection function selects literals in a clause. ◮ If C is non-empty, then at least one literal is selected in C . We denote selected literals by underlining them, e.g., p ∨ ¬ q Note: selection function does not have to be a function. It can be any oracle that selects literals.

  55. Binary Resolution with Selection We introduce a family of inference systems, parametrised by a literal selection function σ . The binary resolution inference system, denoted by BR σ , consists of two inference rules: ◮ Binary resolution, denoted by BR p ∨ C 1 ¬ p ∨ C 2 ( BR ) . C 1 ∨ C 2

  56. Binary Resolution with Selection We introduce a family of inference systems, parametrised by a literal selection function σ . The binary resolution inference system, denoted by BR σ , consists of two inference rules: ◮ Binary resolution, denoted by BR p ∨ C 1 ¬ p ∨ C 2 ( BR ) . C 1 ∨ C 2 ◮ Positive factoring, denoted by Fact: p ∨ p ∨ C ( Fact ) . p ∨ C

  57. Completeness? Binary resolution with selection may be incomplete, even when factoring is unrestricted (also applied to negative literals).

  58. Completeness? Binary resolution with selection may be incomplete, even when factoring is unrestricted (also applied to negative literals). Consider this set of clauses: ( 1 ) ¬ q ∨ r ( 2 ) ¬ p ∨ q ( 3 ) ¬ r ∨ ¬ q ( 4 ) ¬ q ∨ ¬ p ( 5 ) ¬ p ∨ ¬ r ( 6 ) ¬ r ∨ p ( 7 ) r ∨ q ∨ p

  59. Completeness? Binary resolution with selection may be incomplete, even when factoring is unrestricted (also applied to negative literals). It is unsatisfiable: Consider this set of clauses: ( 8 ) q ∨ p ( 6 , 7 ) ( 1 ) ¬ q ∨ r ( 9 ) q ( 2 , 8 ) ( 2 ) ¬ p ∨ q ( 10 ) r ( 1 , 9 ) ( 3 ) ¬ r ∨ ¬ q ( 11 ) ¬ q ( 3 , 10 ) ( 4 ) ¬ q ∨ ¬ p ( 12 ) � ( 9 , 11 ) ( 5 ) ¬ p ∨ ¬ r ( 6 ) ¬ r ∨ p Note the linear representation of ( 7 ) r ∨ q ∨ p derivations (used by Vampire and many other provers). However, any inference with selection applied to this set of clauses give either a clause in this set, or a clause containing a clause in this set.

  60. Literal Orderings Take any well-founded ordering ≻ on atoms, that is, an ordering such that there is no infinite decreasing chain of atoms: A 0 ≻ A 1 ≻ A 2 ≻ · · · In the sequel ≻ will always denote a well-founded ordering.

  61. Literal Orderings Take any well-founded ordering ≻ on atoms, that is, an ordering such that there is no infinite decreasing chain of atoms: A 0 ≻ A 1 ≻ A 2 ≻ · · · In the sequel ≻ will always denote a well-founded ordering. Extend it to an ordering on literals by: ◮ If p ≻ q , then p ≻ ¬ q and ¬ p ≻ q ; ◮ ¬ p ≻ p .

  62. Literal Orderings Take any well-founded ordering ≻ on atoms, that is, an ordering such that there is no infinite decreasing chain of atoms: A 0 ≻ A 1 ≻ A 2 ≻ · · · In the sequel ≻ will always denote a well-founded ordering. Extend it to an ordering on literals by: ◮ If p ≻ q , then p ≻ ¬ q and ¬ p ≻ q ; ◮ ¬ p ≻ p . Exercise: prove that the induced ordering on literals is well-founded too.

  63. Orderings and Well-Behaved Selections Fix an ordering ≻ . A literal selection function is well-behaved if ◮ If all selected literals are positive, then all maximal (w.r.t. ≻ ) literals in C are selected. In other words, either a negative literal is selected, or all maximal literals must be selected.

  64. Orderings and Well-Behaved Selections Fix an ordering ≻ . A literal selection function is well-behaved if ◮ If all selected literals are positive, then all maximal (w.r.t. ≻ ) literals in C are selected. In other words, either a negative literal is selected, or all maximal literals must be selected. To be well-behaved, we sometimes must select more than one different literal in a clause. Example: p ∨ p or p ( x ) ∨ p ( y ) .

  65. Completeness of Binary Resolution with Selection Binary resolution with selection is complete for every well-behaved selection function.

  66. Completeness of Binary Resolution with Selection Binary resolution with selection is complete for every well-behaved selection function. Consider our previous example: A well-behave selection function must satisfy: ( 1 ) ¬ q ∨ r 1. r ≻ q , because of ( 1 ) ( 2 ) ¬ p ∨ q 2. q ≻ p , because of ( 2 ) ( 3 ) ¬ r ∨ ¬ q ( 4 ) ¬ q ∨ ¬ p 3. p ≻ r , because of ( 6 ) ( 5 ) ¬ p ∨ ¬ r There is no ordering that satisfies ( 6 ) ¬ r ∨ p these conditions. ( 7 ) r ∨ q ∨ p

  67. End of Lecture 1 Slides for lecture 1 ended here . . .

  68. Outline Introduction First-Order Logic and TPTP Inference Systems Saturation Algorithms Redundancy Elimination Equality

  69. How to Establish Unsatisfiability? Completess is formulated in terms of derivability of the empty clause � from a set S 0 of clauses in an inference system I . However, this formulations gives no hint on how to search for such a derivation.

  70. How to Establish Unsatisfiability? Completess is formulated in terms of derivability of the empty clause � from a set S 0 of clauses in an inference system I . However, this formulations gives no hint on how to search for such a derivation. Idea: ◮ Take a set of clauses S (the search space), initially S = S 0 . Repeatedly apply inferences in I to clauses in S and add their conclusions to S , unless these conclusions are already in S . ◮ If, at any stage, we obtain � , we terminate and report unsatisfiability of S 0 .

  71. How to Establish Satisfiability? When can we report satisfiability?

  72. How to Establish Satisfiability? When can we report satisfiability? When we build a set S such that any inference applied to clauses in S is already a member of S . Any such set of clauses is called saturated (with respect to I ).

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