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This file is licensed under the Creative Commons Attribution-NonCommercial 3.0 (CC BY-NC 3.0)
Dr. Harald Sack
Hasso Plattner Institute for IT Systems Engineering
University of Potsdam
Spring 2013
Semantic Web Technologies
Lecture 4: Knowledge Representations I07: Resolution
Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
2
Lecture 4: Knowledge Representations I
Open HPI - Course: Semantic Web Technologies
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
3
07 Resolution (PL)Open HPI - Course: Semantic Web Technologies - Lecture 4: Knowledge Representations I
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
430
A Calculator Machine for Logic
■Recall:
■A formula F is a logical consequence of a theory (knowledge base) T,
i.e., T ⊨ F, iff
all models of T are also models of F
■Problem:
■We have to consider all possible interpretations.
■How do we do this in practice?
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
530
A Calculator Machine for Logic
■Problem:
■We have to consider all possible interpretations.
■How do we do this in practice?
■Therefore, logical consequence is implemented via syntactical methods (= Calculus).
■For the logical calculus, there must be proven:
■Correctness: every syntactic entailment is also a semantic entailment, if T ⊢ F then T ⊨ F
■Completeness: all semantic entailments are also syntactic entailments, if T ⊨ F then T ⊢ F
Gottfried Wilhelm Leibniz (1646-1716)
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
630
Resolution
{F1,…,Fn} with F0 as logical Consequence
{F1,…,Fn} ⊨ F0
F1 ∧… ∧ Fn → F0 is a tautology
¬(F1 ∧… ∧ Fn → F0) is a contradiction
G1 ∧ …∧ Gk is a contradiction
□ The resolution procedure allows the entailment of a contradictionfrom G1 ∧ …∧ Gk.
Theory
equi
vale
nt a
sser
tions
John Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", Communications of the ACM, 5:23–41, 1965.
John Alan Robinson
□ Resolution is based on simple deduction rules and is a special form of enumeration
□ Instead of proofing that a formula is a tautology, it deducts a logical contradiction from its negation
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
730
Resolution (Propositional Logic)
■ If (p1∨…∨pk∨p∨¬q1∨…∨¬ql)∧(r1∨…∨rm∨¬p∨¬s1∨…∨¬sn) is true, then:
■ One of both p, ¬p has to be wrong.
■ Therefore: One of the other Literals must be true, i.e.
p1∨…∨pk∨¬q1∨…∨¬ql∨r1∨…∨rm∨¬s1∨…∨¬sn
must be true.
■ Therefore: If p1∨…∨pk∨¬q1∨…∨¬ql∨r1∨…∨rm∨¬s1∨…∨¬sn is a
contradiction, then(p1∨…∨pk∨p∨¬q1∨…∨¬ql)∧(r1∨…∨rm∨¬p∨¬s1∨…∨¬sn)is also a contradiction.
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
830 (p1∨…∨pk∨p∨¬q1∨…∨¬ql) (r1∨…∨rm∨¬p∨¬s1∨…∨¬sn)
p1∨…∨pk∨¬q1∨…∨¬ql∨r1∨…∨rm∨¬s1∨…∨¬sn
■ two clauses K1 and K2 are transformed into a new one K3
■ End of the resolution procedure:
■ If clauses are resolved that consist only of an atom and the negated atom, then a new „empty clause“ ⊥ can be resolved.
K2
K3
{K1,K 2} ⊨ K3Resolution step
K1
∧
Resolution (Propositional Logic)
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
930 • How to deduce a contradiction from a set M of clauses:
1.Select two clauses from M and create a new clause K via a
resolution step.
2. If K =⊥ , then a contradiction has been found.
3. If K ≠⊥ , K is added to the set M, continue with step 1.
Resolution (Propositional Logic)
• The Resolution Calculus (for propositional logic)is correct and complete
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1030 • How to proof that a formula ψ is a logical consequence of a Knowledge Base (Theory) Φ of clauses, i.e. Φ ⊨ ψ ?
1.Compute the negation of Φ → ψ, which is a contradiction:¬(Φ → ψ) ≣ Φ ∧ ¬ ψ
2.Determine the clausal form of Φ ∧ ¬ ψ
3.Select two clauses from Φ ∧ ¬ ψ and create a new clause K via a resolution step.
4. If K =⊥ , then a contradiction has been found.It holds that Φ ⊨ ψ. q.e.d.
5. If K ≠⊥ , K is added to the set Φ ∧¬ ψ, continue with step 3.
Resolution (Propositional Logic)
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1130 • If it rains, Jane brings her umbrella r ! u
• If Jane has an umbrella, she doesn't get wet u ! ¬w
• If it doesn't rain, Jane doesn't get wet ¬r ! ¬w
• Now, prove that Jane doesn't get wet ¬w
Example
Φ
knowledgebase
formula
ψ
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1230 • r ! u = ¬r ∨ u = {¬r,u}• u ! ¬w = ¬u ∨ ¬w = {¬u,¬w}• ¬r ! ¬w = r ∨ ¬w = {r,¬w}
Example
Φ
knowledgebase
formula
ψ
1. Transform knowledge base into clausal form
2. Add negated formula to knowledge base
• ¬¬w = w
1.{¬r,u}
2.{¬u,¬w}
3.{r,¬w}
4. {w}Φ∧¬ψ
new knowledge
base
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1330
Example3. Start Resolution
1.{¬r,u}
2.{¬u,¬w}
3.{r,¬w}
4.{w}
(3,4) {r,¬w}
{w}
5. {r}
Φ∧¬ψ
new knowledge
base
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1430
Example3. Continue Resolution
1.{¬r,u}
2.{¬u,¬w}
3.{r,¬w}
4.{w}
5.{r}
(2,4) {¬u,¬w}
{w}
6. {¬u}
Φ∧¬ψ
new knowledge
base
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1530
Example
1.{¬r,u}
2.{¬u,¬w}
3.{r,¬w}
4.{w}
5.{r}
6.{¬u}
(1,5) {¬r,u}
{r}
7. {u}
Φ∧¬ψ
new knowledge
base
3. Continue Resolution
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
1630
Example
1.{¬r,u}
2.{¬u,¬w}
3.{r,¬w}
4.{w}
5.{r}
6.{¬u}
7.{u}
(6,7) {¬u}
{u}
{⊥}
Φ∧¬ψ
new knowledge
base
4. We have found a contradiction in Φ∧¬ψ, therefore it holds that Φ ⊨ ψ , q.e.d.
3. Continue Resolution
Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
17
08 Resolution (FOL)Open HPI - Course: Semantic Web Technologies - Lecture 4: Knowledge Representations I