Foundations of Artiﬁcial Intelligence 30. Propositional Logic: Reasoning and Resolution Malte Helmert

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Foundations of Artificial Intelligence

30. Propositional Logic: Reasoning and Resolution

Malte Helmert

University of Basel

April 26, 2021

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Reasoning Resolution Summary

Propositional Logic: Overview

Chapter overview: propositional logic 29. Basics

30. Reasoning and Resolution 31. DPLL Algorithm

32. Local Search and Outlook

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Reasoning

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Reasoning: Intuition

Generally, formulas only represent an incomplete description of the world.

In many cases, we want to know

if a formula logically follows from (a set of) other formulas.

What does this mean?

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Reasoning: Intuition

example: ϕ= (P ∨Q)∧(R∨ ¬P)∧S S holds in every model of ϕ.

What about P,Q andR?

consider all models of ϕ:

P Q R S F T F T F T T T T F T T T T T T

Observation

In all models ofϕ, the formulaQ∨R holds as well.

We say: “Q∨R logically follows fromϕ.”

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Reasoning: Formally

Definition (logical consequence)

Let Φ be a set of formulas. A formulaψlogically follows from Φ (in symbols: Φ|=ψ) if all models of Φ are also models of ψ.

German: logische Konsequenz, folgt logisch In other words: for each interpretationI, ifI |=ϕfor all ϕ∈Φ, then alsoI |=ψ.

Question

How can we automatically compute if Φ|=ψ?

One possibility: Build a truth table. (How?)

Are there “better” possibilities that (potentially) avoid generating the whole truth table?

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Reasoning: Deduction Theorem

Proposition (deduction theorem)

LetΦbe a finite set of formulas and let ψbe a formula. Then

Φ|=ψ iff (^

ϕ∈Φ

ϕ)→ψ is a tautology.

German: Deduktionssatz Proof.

iff

Φ|=ψ

iff for each interpretationI: ifI |=ϕ for all ϕ∈Φ, then I |=ψ iff for each interpretationI: ifI |=V

ϕ∈Φϕ, then I |=ψ iff for each interpretationI: I 6|=V

ϕ∈Φϕor I |=ψ iff for each interpretationI: I |= (V

ϕ∈Φϕ)→ψ iff (V

ϕ∈Φϕ)→ψ is tautology

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Reasoning: Deduction Theorem

Proposition (deduction theorem)

LetΦbe a finite set of formulas and let ψbe a formula. Then

Φ|=ψ iff (^

ϕ∈Φ

ϕ)→ψ is a tautology.

German: Deduktionssatz Proof.

iff

Φ|=ψ

ifffor each interpretation I: ifI |=ϕ for all ϕ∈Φ, then I |=ψ ifffor each interpretation I: ifI |=V

ϕ∈Φϕ, then I |=ψ ifffor each interpretation I: I 6|=V

ϕ∈Φϕor I |=ψ ifffor each interpretation I: I |= (V

ϕ∈Φϕ)→ψ iff(V

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Reasoning

Consequence of Deduction Theorem

Reasoning can be reduced to testing validity.

Algorithm

Question: Does Φ|=ψhold?

1 test if (V

2 if yes, then Φ|=ψ, otherwise Φ6|=ψ

In the following: Can we test for validity “efficiently”, i.e., without computing the whole truth table?

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Resolution

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Sets of Clauses

for the rest of this chapter:

prerequisite: formulas in conjunctive normal form clause represented as aset C of literals

formula represented as a set ∆ of clauses

Example

Letϕ= (P∨Q)∧ ¬P.

ϕin conjunctive normal form

ϕconsists of clauses (P ∨Q) and¬P

representation ofϕas set of sets of literals: {{P,Q},{¬P}}

Distinguish(empty clause) vs.∅ (empty set of clauses).

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Resolution: Idea

Observation

Testing for validity can be reduced to testing unsatisfiability.

formulaϕvalid iff¬ϕ unsatisfiable

Resolution: Idea

method to test formula ϕfor unsatisfiability

idea: derive new formulas fromϕthat logically follow fromϕ if empty clausecan be derived ϕunsatisfiable

German: Resolution

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The Resolution Rule

C₁∪ {`},C₂∪ {`}¯ C₁∪C₂

“From C₁∪ {`} andC₂∪ {`}, we can conclude¯ C₁∪C₂.”

C1∪C2 is resolventof parent clausesC1∪ {`} andC2∪ {`}.¯ The literals `and ¯`are called resolution literals,

the corresponding proposition is called resolution variable.

resolvent follows logically from parent clauses (Why?) German: Resolutionsregel, Resolvent, Elternklauseln, Resolutionsliterale, Resolutionsvariable

Example

resolvent of {A,B,¬C} and{A,D,C}?

resolvents of {¬A,B,¬C} and{A,D,C}?

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Resolution: Derivations

Definition (derivation)

Notation: R(∆)= ∆∪ {C |C is resolvent of two clauses in ∆}

A clauseD can be derivedfrom ∆ (in symbols∆`D) if there is a sequence of clausesC1, . . . ,Cn=D such that for all i ∈ {1, . . . ,n}

we haveCi ∈R(∆∪ {C₁, . . . ,Ci−1}).

German: Ableitung, abgeleitet Lemma (soundness of resolution) If∆`D, then ∆|=D.

Does the converse direction hold as well (completeness)?

German: Korrektheit, Vollst¨andigkeit

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Resolution: Completeness?

The converse of the lemma does not hold in general.

example:

{{A,B},{¬B,C}} |={A,B,C}, but {{A,B},{¬B,C}} 6` {A,B,C}

but: converse holds for special case of empty clause (no proof) Theorem (refutation-completeness of resolution)

∆is unsatisfiable iff∆`

German: Widerlegungsvollst¨andigkeit consequences:

Resolution is a complete proof method for testing unsatisfiability.

Resolution can be used for general reasoning by reducing to a test for unsatisfiability.

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Example

Let Φ ={P∨Q,¬P}. Does Φ|=Q hold?

Solution

test if ((P ∨Q)∧ ¬P)→Q is tautology

equivalently: test if ((P ∨Q)∧ ¬P)∧ ¬Q is unsatisfiable resulting set of clauses: Φ⁰: {{P,Q},{¬P},{¬Q}}

resolving {P,Q}with {¬P} yields{Q}

resolving {Q} with {¬Q} yields observation: empty clause can be derived, hence Φ⁰ unsatisfiable

consequently Φ|=Q

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Resolution: Discussion

Resolution is a complete proof method to test formulas for unsatisfiability.

In the worst case, resolution proofs can take exponential time.

In practice, a strategywhich determines the next resolution step is needed.

In the following chapter, we discuss theDPLL algorithm, which is a combination of backtracking and resolution.

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Summary

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Summary

Reasoning: the formulaψ follows from the set of formulas Φ if all models of Φ are also models of ψ.

Reasoning can be reduced to testing validity (with the deduction theorem).

Testing validity can be reduced to testing unsatisfiability.

Resolution is arefutation-completeproof method applicable to formulas in conjunctive normal form.

can be used to test if a set of clauses is unsatisfiable