CS 245 W15 A. Lubiw, U. Waterloo Lecture 5 Recall

```CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
Example: analyzing and simplifying code
Recall
if (C1 or not C2) then
- definition of tautology, equivalence
if (not (C2 and C3)) then
- logical identities (e.g. De Morgan’s laws, commutativity)
P1
- application to circuits
else
if (C2 and not C3) then
P2
else
P3
else
P4
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
Proof Methods
Theorem. The proof method of applying logical identites is sound and
complete.
How do we construct valid proofs?
A proof should start from agreed truths, continue using agreed methods.
axioms
sound: If we can use logical identities to prove a formula α then α is a
tautology.
inference rules
Issues:
- soundness
- completeness
complete: If α is a tautology then we can prove it using logical identities.
- algorithms
- ease of checking
- matches human proofs
Note that this is a proof about proofs!
In what proof system do we do this proof?
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
Proof Methods
How do we construct valid proofs?
A proof should start from agreed truths, continue using agreed methods.
axioms
inference rules
Proof Methods for propositional logic
- apply logical identities (“transformational proofs”)
- axiomatic systems
- natural deduction
- semantic tableaux
Dear Reader: Enclosed is a check for ninety-eight cents.
Using your work, I have proven that this equals the amount
you requested.
- resolution refutation
- Davis Putnam
- DPLL (Davis Putnam Logeman Loveland)
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
Proof Methods — history
Gerhard Gentzen (1909 − 1945)
- introduced natural deduction
- more natural than e.g. Russell and
Whitehead, Principia Mathematica, an attempt to
do all mathematics from a set of axioms and
inference rules.
deduction, especially as it has been developed by Frege,
Russell, and Hilbert, is rather far removed from the forms of
deduction used in practice in mathematical proofs. Considerable
formal advantages are achieved in return.
In contrast, I intended first to set up a formal system which
comes as close as possible to actual reasoning.”
CS 245 W15
Lecture 5
Natural Deduction
Recall some proof methods from MATH 135
- direct proof
- case analysis
Natural deduction models these.
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
Notation
Rules of Natural Deduction
α⊢β
means there is a proof (using natural deduction) of β from premise α
α1, . . . ,αn ⊢ β
means there is a proof (using natural deduction) of β from premises
α1, . . . ,αn
⊢β
means we can prove β without any premises
The symbol ⊢ is called the “turnstile”
Note the difference between ⊨ and ⊢
soundness: if α ⊢ β then α ⊨ β
completeness: if α ⊨ β then α ⊢ β
CS 245 W15
Rules of Natural Deduction
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Rules of Natural Deduction
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
What I expect you to know/do:
- use logical identities
- use natural deduction
notation α ⊢ β
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