First-Order Theorem Proving

159 Pages·2005·0.72 MB·English

First-Order Theorem Proving
Peter Baumgartner
NICTA, Logic and Computation Program, Canberra
[email protected]
Slides partially based on material by Uli Furbach, Harald Ganzinger, John Slaney,
Viorica Sofronie-Stockermans and Uwe Waldmann
First-OrderTheoremProving–PeterBaumgartner–p.1
Contents
Part I: What FOTP is about
Part II: First-Order Predicate Logic (from the viewpoint of ATP)
Part III: Proof Systems, including Resolution
Part IV: Tableaux and Model Generation
First-OrderTheoremProving–PeterBaumgartner–p.2
Part I – What First-Order Theorem Proving is About
Mission statement
A glimpse at First-Order Theorem Proving
First-OrderTheoremProving–PeterBaumgartner–p.3
Mission Statement
Theorem proving is about . . .
Logics: Propositional, First-Order, Higher-Order, Modal, Description, . . .
Calculi and proof procedures: Resolution, DPLL, Tableaux, . . .
Systems: Interactive, Automated
Applications: Knowledge Representation, Verification, . . .
First-OrderTheoremProving–PeterBaumgartner–p.4
Mission Statement
Theorem proving is about . . .
Logics: Propositional, First-Order, Higher-Order, Modal, Description, . . .
Calculi and proof procedures: Resolution, DPLL, Tableaux, . . .
Systems: Interactive, Automated
Applications: Knowledge Representation, Verification, . . .
Milestones
60s: Calculi: DPLL, Resolution, Model Elimination
70s: Logic Programming
80s: Logic Based Knowledge Representation
90s: Modern Theory and Implementations, “A Basis for Applications”
2000s: Ontological Engineering, Verification
First-OrderTheoremProving–PeterBaumgartner–p.4
Mission Statement
In this talk, theorem proving is about . . .
Logics: Propositional, First-Order, Higher-Order, Modal, Description, . . .
Calculi and proof procedures: Resolution, DPLL, Tableaux, . . .
Systems: Interactive, Automated
Applications: Knowledge Representation, Verification, . . .
Milestones
60s: Calculi: DPLL, Resolution, Model Elimination
70s: Logic Programming
80s: Logic Based Knowledge Representation
90s: Modern Theory and Implementations, “A Basis for Applications”
2000s: Ontological Engineering, Verification
First-OrderTheoremProving–PeterBaumgartner–p.5
Application: Compiler Validation
Problem: prove equivalence of source and target program
Example:
1: y := 1 1: y := 1
2: if z = x*x*x 2: R1 := x*x
3: then y := x*x + y 3: R2 := R1*x
4: endif 4: jmpNE(z,R2,6)
5: y := R1+1
To prove: (indexes refer to values at line numbers; index 0 = initial values)
y ≈ 1 ∧ z ≈ x ∗ x ∗ x ∧ y ≈ x ∗ x + y
1 0 0 0 0 3 0 0 1
′ ′ ′ ′ ′ ′
y ≈ 1 ∧ R1 ≈ x ∗ x ∧ R2 ≈ R1 ∗ x ∧ z ≈ R2 ∧ y ≈ R1 + 1
1 2 0 0 3 2 0 0 3 5 2
′ ′ ′ ′
∧ x ≈ x ∧ y ≈ y ∧ z ≈ z |= y ≈ y
0 0 0 0 0 0 3 5
First-OrderTheoremProving–PeterBaumgartner–p.6
A Glimpse at FOTP
A logical puzzle:
Someone who lives in Dreadbury Mansion killed Aunt Agatha.
Agatha, the butler, and Charles live in Dreadbury Mansion, and are
the only people who live therein. A killer always hates his victim,
and is never richer than his victim. Charles hates no one that Aunt
Agatha hates. Agatha hates everyone except the butler. The butler
hates everyone not richer than Aunt Agatha. The butler hates
everyone Aunt Agatha hates. No one hates everyone. Agatha is not
the butler.
First-OrderTheoremProving–PeterBaumgartner–p.7
A Glimpse at FOTP
A logical puzzle:
Someone who lives in Dreadbury Mansion killed Aunt Agatha.
Agatha, the butler, and Charles live in Dreadbury Mansion, and are
the only people who live therein. A killer always hates his victim,
and is never richer than his victim. Charles hates no one that Aunt
Agatha hates. Agatha hates everyone except the butler. The butler
hates everyone not richer than Aunt Agatha. The butler hates
everyone Aunt Agatha hates. No one hates everyone. Agatha is not
the butler.
Who killed Aunt Agatha?
First-OrderTheoremProving–PeterBaumgartner–p.7
A Glimpse at FOTP
Before solving the problem with a theorem prover we have to formalize it:
First-OrderTheoremProving–PeterBaumgartner–p.8