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HILBERT SYSTEMS Mathematical Logic & Theorem Proving Asst. Prof. Deepak D'Souza CSA, IISc Raghavendra K R Msc (Engg), CSA, IISc. Agenda Definition Axioms Rules of Inference Deduction Theorem Soundness Completeness Definition A proof in a Hilbert system is a finite sequence Z1, Z2, ... Zn of formulas such that each term is either an axiom or follows from earlier terms by one of the rules of inference. A derivation in a Hilbert system from a set S of formulas is a finite sequence Z1, Z2, ..., Zn of formulas such that each term is either an axiom, or is a member of S, or follows from earlier terms by one of the rules of inference. X is a theorem of a Hilbert sytem if X is the last line of a proof. |- X X is a consequence of a set S if X is the last line of a derivation from S. S |- X Axioms 1. X > ( Y > X) 2. (X > (Y > Z)) > ((X > Y) > (X > Z)) 3. L > X 4. X > T 5. ~~X > X 6. X > (~X > Y) 7. a > a1 8. a > a2 9. (b1 > X) > ((b2 > X) > (b > X) Rule of Inference Modus Ponens: X, X > Y Y X>X (X > ((X > X) > X) { Axiom 1 } (X > ((X > X) > X) > ((X > (X > X)) > (X > X)) {Axiom 2 } ((X > (X > X)) > (X > X)) { MP } (X > (X > X)) { Axiom 1 } (X > X) (~X > X) > X Axiom 9, b = (~X > X) (~~X > X) > ((X > X) > ((~X > X) > X)). ~~X > X {Axiom} ((X > X) > ((~X > X) > X)) { MP } X>X (~X > X) > X) Deduction Theorem S U {X} |- Y <=> S |- (X > Y) With axioms 1 and 2 and Modus Ponens If: S |- (X > Y) Using MP, S U {X} |- Y. Only If: Derivation 1: Z1, Z2, ... Zn is a derivation of Y from S U {X}, Zn = Y Derivation 2: Prefix X >. X > Z1, X > Z2, .... X > Y If Zi is an axiom or a member of S, then insert Zi and Zi > (X > Zi) If Zi is the formula X, insert steps of derivation of X > X If Zi comes from MP, then there exists Zj and Zk with j, k < i and Zk = Zj > Zi Insert (X > (Zj > Zi)) > ((X > Zj) > (X > Zi)) and (X > Zj) > (X > Zi) The resulting Derivation 2 constitutes a derivation of X > Y from S. Soundness S |- X => S |= X There is a derivation of the form Z1, Z2, Z3, ... Zn of X Proof is by induction on the length of the derivation Every Zi is either an instance of axiom or a member of S or obtained by applying MP Suffices to show that the axioms define valid formulas and MP preserves validity. Completeness S |= X => S |- X Define C = { S : S does not |- X } To show that C is a consitency property. Not both A, ~A belong to S ~T or L does not belong to S ~~Z belongs to S => S U {Z} belongs to C a belongs to S => S U {a1, a2} belongs to C b belongs to S => S U {b1} or S U {b2} belongs to C Completeness ... contd. S |= X => S |- X Contrapositive Proof. Suppose X is not derivable of S. Then S is Consistent. S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. Thank You If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven? One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it. No one shall expel us from the Paradise that Cantor has created Hilbert (1862 – 1943), a German Mathematician References First Order Logic and Automated Theorem Proving – Melvin Fitting An Introduction to Logic – Madhavan Mukund