Download hilbert systems - CSA

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