CS1502 Formal Methods in Computer Science Lecture Notes 10 Resolution and Horn Sentences.

CS1502 Formal Methods in CS1502 Formal Methods in Computer Science Computer Science

Lecture Notes 10

Resolutionand Horn Sentences

Resolution Theorem ProvingResolution Theorem Proving

Method for searching for proofs automaticallyMethod for searching for proofs automaticallySentences are translated into CNF, and then to Sentences are translated into CNF, and then to sets of clausessets of clausesAt each step, a new clause is derived from two At each step, a new clause is derived from two clauses you already haveclauses you already haveProof steps all use the same ruleProof steps all use the same rule

If reach If reach (_|_), sentences were not satisfiable (_|_), sentences were not satisfiable

Start: premises + negation of goalStart: premises + negation of goal

End: if reach End: if reach , premises |= goal, premises |= goal

Conversion to Clausal FormConversion to Clausal Form

UseUse P P Q Q equivequiv ~P v Q ~P v Q to removeto remove UseUse P P Q Q equivequiv ((~P v Q) ^ (~Q v P)) ((~P v Q) ^ (~Q v P)) to removeto remove

New 1st Step! We hadn’t done conditionals yetNew 1st Step! We hadn’t done conditionals yet

Use Use DeMorgan’s lawsDeMorgan’s laws and and ~Elim~Elim to move to move ~~ as far inward as far inward as possible as possible (gives NNF)(gives NNF)

Use the Use the distributive lawsdistributive laws until the sentence is in until the sentence is in CNFCNF

Clausal FormClausal Form

Given a sentence S written in CNFGiven a sentence S written in CNF S = ( ) S = ( ) ( ) ( ) . . . . . . ( ) ( )

Convert each ( ) into a clause - a set Convert each ( ) into a clause - a set consisting of each of the literals in ( ).consisting of each of the literals in ( ).

Example: Example: S=S= (A) (A) ( (B B C C D) D) ( (AA D) D)Clauses are: Clauses are: {A}, {{A}, { B, C, D}, { B, C, D}, { A, D} A, D}


Method for searching for proofs automaticallyMethod for searching for proofs automatically

Sentences are translated into CNF, and then to sets of Sentences are translated into CNF, and then to sets of clausesclauses

At each step, a new clause is derived from two clauses At each step, a new clause is derived from two clauses you already haveyou already have

Proof steps all use the same ruleProof steps all use the same rule

If reach If reach (_|_), sentences were not (_|_), sentences were not satisfiablesatisfiableStart: premises + negation of goalStart: premises + negation of goal


Satisfying a Set of ClausesSatisfying a Set of Clauses

Assigning truth-values to the atomic Assigning truth-values to the atomic sentences so the CNF sentence the set sentences so the CNF sentence the set corresponds to is true.corresponds to is true.

{{A, {{A, C, D}, {C},{C, D}, {C},{A},{A, D}}A},{A, D}} is satisfied by is satisfied by

A = FalseA = False C = True C = True D = True D = True

Empty ClauseEmpty Clause

{} denoted by {} denoted by . �. �The empty clause is not satisfiableThe empty clause is not satisfiableWe want We want {{A}, {{{A}, { A}} A}} to lead to to lead to {}{} ( ( ��))The basic resolution step involves “canceling” The basic resolution step involves “canceling” pos and neg matching literals from two clausespos and neg matching literals from two clauses

So, we derive So, we derive {}{} ( ( ��) from ) from {{A}, {{{A}, { A}} A}}

Example: not satisfiableExample: not satisfiable

P P Q Q PP

QQ

{P, Q} {P}

{Q}

{P, Q} {P} {Q}



Sentences are translated into CNF, and then to Sentences are translated into CNF, and then to sets of clausessets of clauses

At each step, a new clause is derived from two At each step, a new clause is derived from two clauses you already haveclauses you already have

Proof steps all use the same ruleProof steps all use the same ruleIf reach If reach (_|_), sentences were not satisfiable (_|_), sentences were not satisfiable



Resolution StepResolution Step

Res. Step with Larger ClausesRes. Step with Larger Clauses

ResolventResolvent

Clause R is a resolvent of clauses CClause R is a resolvent of clauses C11 and and

CC22 if there is a literal in C if there is a literal in C11 whose negation whose negation

is in Cis in C22 and R consists of all the remaining and R consists of all the remaining

literals in either clause.literals in either clause.

Example:Example: {A,{A, C C , , D} D} andand {B, {B, CC} } havehave

resolventresolvent { {A, B, A, B, DD}}


Method for searching for proofs automaticallyMethod for searching for proofs automaticallySentences are translated into CNF, and then to Sentences are translated into CNF, and then to sets of clausessets of clausesAt each step, a new clause is derived from two At each step, a new clause is derived from two clauses you already haveclauses you already haveProof steps all use the same ruleProof steps all use the same rule

If reach If reach (_|_), sentences were not (_|_), sentences were not satisfiablesatisfiableStart: premises + negation of goalStart: premises + negation of goalEnd: if reach End: if reach , premises |= goal, premises |= goal

Resolution TheoremResolution Theorem

For any set of clauses that are For any set of clauses that are not not satisfiablesatisfiable, it is possible to arrive at the , it is possible to arrive at the empty clause by using successive empty clause by using successive resolutions.resolutions.



Sentences are translated into CNF, and then to sets of Sentences are translated into CNF, and then to sets of clausesclauses

At each step, a new clause is derived from two clauses At each step, a new clause is derived from two clauses you already haveyou already have

Proof steps all use the same ruleProof steps all use the same rule

If reach If reach (_|_), sentences were not satisfiable (_|_), sentences were not satisfiable



Using Resolution to Determine Validity of Using Resolution to Determine Validity of ArgumentsArguments

QQ is a logical consequence of is a logical consequence of P1, P2, …, P1, P2, …, Pn Pn iff iff P1 ^ P2 ^ … ^ PnP1 ^ P2 ^ … ^ Pn ^ ~Q^ ~Q is not is not satisfiablesatisfiableFor sentences in clausal form: For sentences in clausal form: Q Q is a logical consequence of a set of is a logical consequence of a set of clauses clauses SS iff iff SS { {Q} Q} is not satisfiable.is not satisfiable.

So, convert premises + negation of goal to So, convert premises + negation of goal to clausal form, and do resolution steps, trying to clausal form, and do resolution steps, trying to reach {} (reach {} () �) �

Is this Argument Valid?Is this Argument Valid?

B v C~C v ~DA v D~B v ~DA

ExampleExample

Show Show A A (B (B C) C) ( (C C D) D) (A (A D) D) ( (B B D)D) is not is not satisfiable.satisfiable.

Clauses: Clauses: {{ A}, {B, C}, { A}, {B, C}, {C, C, D}, {A, D}, {D}, {A, D}, {B, B, D}.D}.

{ {AA} {} {AA,D} {B,,D} {B,CC} {} {CC, , D}D}

{ {DD} {} {BB, , D} {D} {BB, , D}D}

{ {DD}}

��

ExampleExample

Modus PonesModus PonesPPQ Q PPQ or {Q or {P,Q}P,Q}

P P {P}{P}

Q negate to get Q negate to get {{Q}Q}

Apply resolution: Apply resolution: {{PP,Q} {,Q} {PP}}

{ {QQ} {} {QQ}}

What is a Horn sentence?What is a Horn sentence?

A A positive literalpositive literal is any literal that is not is any literal that is not preceded with a preceded with a . . For example, Cube(b) For example, Cube(b) and P are positive literals.and P are positive literals.

A sentence S is a A sentence S is a Horn sentenceHorn sentence if and if and only if it is only if it is in CNFin CNF and every conjunct has and every conjunct has at most one positive literal.at most one positive literal.

ExamplesExamples

(A (A B B C) C) ( (A A B)B)

(A (A B B C) C) (D) (D)

(A (A B) B) ( (C C D)D)

Horn sentence

Not Horn sentence

(A C) (B C) (A D) (B D)

Horn sentence

Alternate Form of Horn Alternate Form of Horn ConjunctConjunct

AA11 AA22...... AAnn B B

(A1 A2 ... An) B

(A1 A2 ... An) B Conditional Formof

Horn sentence

B :- A1, A2, …, An. In Prolog:

Rule

B if A1, A2, …, An

Special CasesSpecial Cases

No positive literalNo positive literalAA11 AA22...... AAnn

No negative literalsNo negative literalsBB

(A1 A2 ... An) False

True BIn Prolog, this

is a fact!

In Prolog, thisis a query!

Why are Horn sentences Why are Horn sentences important?important?

Very efficient algorithms exist for determining if a Very efficient algorithms exist for determining if a set of Horn sentences is satisfiableset of Horn sentences is satisfiable

CNF to PROLOGCNF to PROLOG

A A B B C is C is (A(AC) C) B is (A B is (A C) C)B B

B :- A, C.B :- A, C.A.A.C.C.

Query: :- B.Query: :- B.Answer: Yes.Answer: Yes.

Example Prolog ProgramExample Prolog Program

grandfather(X,Y) :-grandfather(X,Y) :- father(X,Z), father(X,Z), father(Z,Y). father(Z,Y).

grandfather(X,Y) :-grandfather(X,Y) :- father(X,Z), father(X,Z), mother(Z,Y). mother(Z,Y).

mother(ann,bill).mother(ann,bill).father(carl,ed).father(carl,ed).father(nick,ann).father(nick,ann).father(ed,sam).father(ed,sam).

CS1502 Formal Methods in Computer Science Lecture Notes 10 Resolution and Horn Sentences.

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