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Introduction to Weak Methods in Theorem Proving 13.1 The General Problem
1Automated Reasoning
13
13.0 Introduction to Weak
Methods in Theorem
Proving
13.1 The General Problem
Solver and Difference
Tables
13.2 Resolution Theorem
Proving
13.3 PROLOG and
Automated Reasoning
13.4 Further Issues in
Automated Reasoning
13.5 Epilogue and
References
13.6 Exercises
2
Chapter Objective
? Learn about general-purpose theorem proving
in predicate calculus.
3
The problem
? Given:
a knowledge base (a set of sentences)
? Prove:
a sentence
Formally,
? Given:
a Knowledge Base (KB),
a sentence α
? Show whether:
KB |= α (does KB entail α ? Or
does α follow from KB ?)
4
The tool
? Modus ponens
KB: p → q
p
question: q
answer: yes
{p → q, p} |= {q}
? We can form arbitrarily long “chains” of
inference to prove a sentence
? We can use forward or backward reasoning
5Example
? If Mary goes to a party, Jane also does.
If Jane goes to a party, she cannot study.
If Jane cannot study, she fails.
Mary went to a party.
? Can we prove:
Jane will fail.
6
Example
? If Mary goes to a party, Jane also does.
M J
If Jane goes to a party, she cannot study.
J C
If Jane cannot study, she fails.
C F
Mary went to a party.
M
? Can we prove:
Jane will fail.
F
Does {M → J, J → C, C → F, M} entail {F}?
7
Example
1. M → J
2. J → C
3. C → F
4. M
Modus ponens on 1 and 4:
5. J
Modus ponens on 2 and 5:
6. C
Modus ponens on 3 and 6:
7. F proven! 8
Another tool
? Modus tollens
KB: p → q
?q
entails ? p.
? So, a theorem proving process involves
applying such rules until the desired sentence
is proven.
? We call this a “proof” because the rules we
use are sound (correct).
9Using modus ponens
? solves a lot of practical problems and is fairly
efficient in terms of “searching” for a proof.
? Unfortunately, fails to prove some sentences
which should be entailed by a KB (it is
incomplete)
10
Example
If Mary goes to the party, Jane also will.
M J
If Mary does not go to the party, Jane will.
?M J
{ M → J, ?M → J} should entail {J} because
either M is true, or ?M is true and either way J
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