Propositional Clausal Logic - VUBai.vub.ac.be/~ydehauwe/decl_prog/2_Logic.pdf · Semantics The...

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Declarative Programming

Propositional Clausal Logic

Yann-Michaël De Hauwereydehauwe@vub.ac.be

Course contents• Introduction to declarative programming

• Clausal logic

• Logic programming in Prolog

• Search

• Natural language processing

• Reasoning using incomplete information

• Inductive logic programming

Clausal logicA logic system consists of:

1. Syntax: which “sentences” are legal

2. Semantics: what is the truth value of a sentence

3. Proof theory: how to derive new sentences (theorems) from assumed ones (axioms) by means of inference rules

A logic system should be

1. Sound: anything you can prove is true

2. Complete: anything true, can be proven

Prolog is neither!

propositional

relational

definite

propositions that are true or false

contains constants, variables and their relations

contains function symbols & compound terms

true prolog: clauses only have one true literal

Different clausal logic systems

ss married;bachelor :- man,adult

likes(peter,S) :- student_of(peter,S)

loves(X,person_loved_by(X)).

A :- B1, B2,..., Bn.

SyntaxConnectives:! :-! “if”! ;! “or”! ,! “and”

Example:

Someone is married or a bachelor if he is a man and an adult

Propositional clausal logic

clause : head [:- body]head : [proposition [; proposition]* ]body : [proposition [, proposition]* ]proposition: atomatom : single word starting with lowercase

married; bachelor :- man, adult

Logic programPropositions are denoted as atoms, f.i. married denotes that someone is married.

Combining atoms result in clauses: married;bachelor :- man,adultmeaning ‘somebody is married or a bachelor if he is a man and an adult’

Combining clauses, results in a program:

woman;man :- human.human :- man.human :- woman.

(human ⇒ (woman ∨ man))∧ (man ⇒ human)∧ (woman ⇒ human)

is equivalent to

(¬human ∨ woman ∨ man) ∧ (¬man ∨ human)∧ (¬woman ∨ human)

B ⇒ H ≡ ¬B ∨ H

ClausesIn general a clause

H1 ; ... ; Hn :" B1 , ... , Bm

is equivalent with

H1 ∨ ... ∨Hn ∨ ¬B1 ∨ ... ∨ ¬Bm

A clause can also be defined asL1 ∨ L2 ∨ ... ∨ Ln

where each Li is a literal, i.e. Li = Ai or Li = ¬Ai , with Ai a proposition.

Special clausesEmpty body stands for true

Empty head stands for false

Entire program in standard logic:

(true ⇒ man) ∧ (impossible ⇒ false)

man ∧ ¬ impossible

man :- . % usually written as man.

:- impossible

true ⇒ man

impossible ⇒ false

SemanticsThe Herbrand base βP is the set of all atoms occurring in P

A Herbrand interpretation i is a mapping of all atoms in βP to a truth value

An interpretation is a model for a clause if the clause is true under the interpretation

An interpretation is a model for a program if it is a model for each clause in the program.

βP = {woman, man, human}i = {(woman, true), (man, false), (human, true)}I = {woman, human}

Semantics: example

βP = {woman, man, human}

I1 = {woman} I2 = {man}I3 = {human} I4 = {woman, human}

I5 = {man, human} I6 = {woman, man}I7 = {woman, man, human} I8 = {}

Semantics: example

βP = {woman, man, human}

I1 = {woman} I2 = {man}I3 = {human} I4 = {woman, human}

I5 = {man, human} I6 = {woman, man}I7 = {woman, man, human} I8 = {}

Logical ConsequenceA clause C is a logical consequence of a program P, denoted

P ⊨ C

if every model of P is also a model of C.

models of P:

P ⊨ human

I1 is preferred since it does not assume anything that does not have to be true.

woman.woman;man :- human.human :- man.human :- woman.

I1 = {woman, human} I2 = {woman, man, human}

Best model = minimal model, obtained by taking the intersection of all models.

However...

M1 ∩ M2 ∩ M3 = {human} which is not a model of the first clause...

A definite clausal logic program has a unique minimal model

Minimal models

I1 = {woman, human} I2 = {woman, man, human}

woman;man :- human.human.

M1 = {woman, human} M2 = {man, human}M3 = {woman, man, human}

Proof theoryChecking every model of a program to compute the logical consequences is in general unfeasible.

Use resolution as an inference rule

married;bachelor :- man,adult.has_wife :- man,married.

If someone is a man and an adult, then he is a bachelor or married; but if he is married, he has a wife; therefore, if someone is a man and an adult, the he is a bachelor or he has a wife

¬man ∨ ¬adult ∨ married ∨ bachelor¬man ∨ ¬married ∨ has_wife

either married and then ¬man ∨ has_wifeor ¬married and then ¬man ∨ ¬adult ∨ bachelor

thus ¬man ∨ has_wife ∨ ¬man ∨ ¬adult ∨ bachelor

B ⇒ H ≡ ¬B ∨ H

Proof theory (2)

A proof of derivation of a clause C from a program P, is a sequence of clauses such that each clause is either in the program, or the resolvent of two previous clauses, and the last clause is C.

If a proof exists, it is denoted as P ⊦ C16

square :- rectangle, equal_sides.rectangle :- parallelogram, right_angles.

square :- rectangle,equal_sides. rectangle :- parallelogram,right_angles.

square :- equal_sides, parallelogram, right_angles.

has_wife :- man,married. married;bachelor :- man,adult.

has_wife;bachelor :- man,adult.

Resolution in traditional notation

E1 ∨ E2 ¬E2 ∨ E3

E1 ∨ E3

A ¬A ∨ B

AA ⇒ B

Modus Ponens

¬A ∨ B¬B

A ⇒ B

Modus Tollens

Meta-theory: soundnessResolution is sound if anything you can prove is true

IF P ⊦ C THEN P ⊧ C

Every model of the input clauses is also a model of the resolvent!

Every model of

is also a model of

Proven by validating the options for the literal resolved upon

has_wife;bachelor :- man,adult.

Meta-theory: completenessResolution is incomplete!!!

Take for instance the tautology

It is true under any interpretation, so P ⊧ a:-a, since every model of P will be a model of a tautology. Even if a does not occur in P!

This can not be established with resolution.

This is not what resolution is used for...

→ ‘is C a logical consequence of P?’, rather than ‘What are the logical consequences of P?’

a :- a

Meta-theory: proof by refutationIf P set of clauses is inconsistent, it is always possible to derive ��by resolution.

Suppose we want to know if P ⊧ a

→ An interpretation I of P is a model of a iff I is not a model of :-a.

→ Check if P ∪ {:-a} are inconsistent

inconsistent set has no model → every model is a model of another clause, in particular �

IF P ⊧ a THEN P ∪ {:-a} ⊦ �

Meta-theory: exampleProof by refutation:

⊧ P ⊧ C

⊦ �

P ∪ ¬ C

happy :- has_friends.friendly :- happy. friendly :- has_friends

happy :- has_friends.friendly :- happy.has_friends.:- friendly

:- friendly friendly :- happy

:- happy happy :- has_friends

:- has_friends has_friends

Meta-theory: refutation complete

P ⊧ C ⇔ each model of P is also a model of C

⇔ no model of P is a model of ¬C

⇔ P∪¬C has no model

P∪¬C is inconsistent

entailment reformulated

it can be shown that:

if Q is inconsistent then Q ⊦ �if P ⊧ C then P∪¬C ⊦ �

refutation-complete

it derives the empty clause from any inconsistent set of

clauses

C = L1∨L2∨...∨Ln

¬C = ¬L1∧¬L2...∧¬Ln

= {¬L1,¬L2...,¬Ln} = set of clauses itself

empty clause false :- true for which no model exists

Meta-theory: completenessResolution is refutation complete

IF P ⊧ C THEN P ∪ ¬C ⊢ �

where Q ⊢ � means Q is inconsistent

i.e. C is a logical consequence of P if and only if one can derive, using resolution, the empty clause (�) from P ∪ ¬C

Relational Clausal Logic

Relational clausal logic

propositional

relational

definite

A :- B1, B2,..., Bn.

Syntax

- same connectives as propositional clausal logic

- introduces constants and variables

For any S, if S is a student of peter, the peter likes S

constant : single word starting with lowercasevariable : single word starting with uppercaseterm : constant | variablepredicate : single word starting with lowercase atom : predicate[(term[,term]*)]clause : head [:- body]head : [atom [; atom]* ]body : atom [, atom]*

likes(peter,S) :- student_of(S,peter).student_of(maria,peter).

SemanticsThe Herbrand universe of a program P is the set of all ground terms occurring in it

The Herbrand base βP of P is the set of all ground atoms that can be constructed using predicates in P and arguments in the Herbrand universe of P

A Herbrand interpretation I of P is a subset of βP, consisting of ground atoms that are true

βP = { likes(peter,peter), likes(peter,maria), likes(maria,peter), likes(maria,maria), student_of(peter,peter), student_of(peter,maria), student_of(maria,peter), student_of(maria,maria) }I = {likes(peter,maria), student_of(maria,peter)}

Substitutions & grounding

if σ={S/maria}, then Cσ is

ground instances :likes(peter,peter):-student_of(peter,peter).likes(peter,maria):-student_of(maria,peter).student_of(maria,peter).

A substitution is a mapping σ : Var → Term.For a clause C, the result of σ on C, denoted Cσ is obtained by replacing all occurrences of X ∈ Var in C by σ(X). Cσ is an instance of C.

likes(peter,maria) :- student_of(maria,peter).

A ground instance of a clause C is the result of some substitution such that Cσ contains but ground atoms.

ModelsAn interpretation I is a model of a clause C iff it is a model of every ground instance of C. An interpretation is a model of a program P iff it is a model of each clause C ∈ P.

I = {likes(peter,maria), student_of(maria, peter)}

is a model of all ground instances of clauses in P

likes(peter,peter):-student_of(peter,peter).likes(peter,maria):-student_of(maria,peter).student_of(maria,peter).

Proof theory: ResolutionNaive version: propositional resolution with all ground instances of clauses in P.

Herbrand universe = {peter, maria}

⇒ already 8 different grounding substitutions

⇒ exponential in the number of constants and variables

Relational clausal logic contains variables, allowing atoms to be made equal

likes(peter,S) :- student_of(S,peter).student_of(maria,T) :- follows(maria,C),teaches(T,C).

UnificationA substitution σ is a unifier of two atoms a1 and a2 ! a1σ = a2σ. If such a σ exists, a1 and a2 are called unifiable.

A substitution σ1 is more general than σ2 if σ2 = σ1θ for some substitution θ.

A unifier θ of a1 and a2 is a most general unifier (mgu) of a1 and a2 ! it is more general than any other unifier of a1 and a2.

If two atoms are unifiable then they their mgu is unique up to renaming.

Unification: examples

p(X, b) and p(a, Y) are unifiablewith most general unifier {X/a,Y/b}

q(a) and q(b) are not unifiable

q(X) and q(Y) are unifiable:

{X/Y} (or{Y/X}) is the most general unifier

{X/a, Y/a} is a less general unifier

Unification: example

student_of(maria,T) and student_of(S,peter) can be made equal:{S→maria,T→peter} resulting in:

Now consider:

{S→maria, X→maria, T→peter} OR {X→S, T→peter}

more general

likes(peter,S) :- student_of(S,peter).student_of(maria,T) :- follows(maria,C),teaches(T,C).

likes(peter,maria) :- student_of(maria,peter).student_of(maria,peter) :- follows(maria,C),teaches(peter,C).

likes(peter,S) :- student_of(S,peter).student_of(X,T) :- follows(X,C),teaches(T,C).

Proof theory: mguApply resolution on many clause-instances at once

Clausal logic

proof theory using mgu

“Do resolution on many clause-instances at once.”

C1 = L11 ∨ . . . L1

C2 = L21 ∨ . . . L2

L1i θ = ¬L2

j θ for some 1 ≤ i ≤ n1, 1 ≤ j ≤ n2

where θ = mgu(L1i , L2

j ), then

L11θ∨ . . .∨L1

i−1θ∨L1i+1θ∨ . . .∨L1

n1θ∨L2

1θ∨ . . .∨L2j−1θ∨L2

j+1θ∨ . . .∨L2n2

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Clausal logic

C1 = L11 ∨ . . . L1

C2 = L21 ∨ . . . L2

L1i θ = ¬L2

j θ for some 1 ≤ i ≤ n1, 1 ≤ j ≤ n2

j ), then

L11θ∨ . . .∨L1

i−1θ∨L1i+1θ∨ . . .∨L1

n1θ∨L2

1θ∨ . . .∨L2j−1θ∨L2

j+1θ∨ . . .∨L2n2

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Clausal logic

C1 = L11 ∨ . . . L1

C2 = L21 ∨ . . . L2

L1i θ = ¬L2

j θ for some 1 ≤ i ≤ n1, 1 ≤ j ≤ n2

j ), then

L11θ∨ . . .∨L1

i−1θ∨L1i+1θ∨ . . .∨L1

n1θ∨L2

1θ∨ . . .∨L2j−1θ∨L2

j+1θ∨ . . .∨L2n2

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Clausal logic

C1 = L11 ∨ . . . L1

C2 = L21 ∨ . . . L2

L1i θ = ¬L2

j θ for some 1 ≤ i ≤ n1, 1 ≤ j ≤ n2

j ), then

L11θ∨ . . .∨L1

i−1θ∨L1i+1θ∨ . . .∨L1

n1θ∨L2

1θ∨ . . .∨L2j−1θ∨L2

j+1θ∨ . . .∨L2n2

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Proof theory: example

:-likes(peter,N){N→maria} ∪ P ⊢ � Hence P ⊧ likes(peter,maria)

likes(peter,S) :- student_of(S,peter).student_of(S,T) :- follows(S,C),teaches(T,C).teaches(peter,decprog).follows(maria,decprog).

:- likes(peter,N) likes(peter,S) :- student_of(S,peter).

:- student_of(N,peter) student_of(S,T) :- follows(S,C),teaches(T,C).

:- follows(N,C),teaches(peter,C) follows(maria,decprog)

:- teaches(peter,decprog) teaches(peter,decprog).

{S→N}

{S→N, T→peter}

{N→maria, C→decprog}

Meta-theory: sound & completeRelational clausal logic is sound

IF P ⊢ C THEN P ⊧ C

Relational clausal logic is refutation complete

IF P ∪ {C} inconsistent THEN P ∪ {C} ⊢ �

different formulation because::- p(X).≡∀X·¬p(X)

while ¬(p(X).)≡¬(∀X·p(X))≡∃X·¬p(X)

Meta-theory: decidability

P ⊧ C is decidable for relational clausal logic

Because the Herbrand universe is finite,

hence there is a finite number of models,

hence we could check all models of P to verify if they are a model of C

Full Clausal Logic

Full clausal logic

propositional

relational

definite

A :- B1, B2,..., Bn.

Full clausal logic

Syntax

- introduces function symbols (functors), with an arity, constants are 0-ary functors

functor : single word starting with lowercasevariable : single word starting with uppercaseterm : variable | functor[(term[,term]*)]predicate : single word starting with lowercase atom : predicate[(term[,term]*)]clause : head [:- body]head : [atom [; atom]* ]body : atom [, atom]*

plus(0,X,X).plus(s(X),Y,s(Z)) :- plus(X,Y,Z).

read s(X) as “successor of X”

Everybody loves somebody

Full clausal logic

SemanticsThe Herbrand universe of a program P is the set of ground terms that can be constructed from constants and functors.!! can be infinite !!{0, s(0), ...}

The Herbrand base βP of P is the set of all ground atoms that can be constructed using predicates in P and the ground terms in the Herbrand universe of P!! can be infinite !!

A Herbrand interpretation I of P is a subset of βP, consisting of ground atoms that are true

Full clausal logic

according to first ground clause, plus(0,0,0) has to be in any modelbut then the second clause requires the same of plus(s(0),0,s(0)) and the third clause of plus(s(s(0)),0,s(s(0)))...

An interpretation is a model for a program if it is a model for each ground instance of every clause in the program.

plus(0,0,0).plus(s(0),0,s(0)):-plus(0,0,0).plus(s(s(0)),0,s(s(0))):-plus(s(0),0,s(0))....plus(0,s(0),s(0)).plus(s(0),s(0),s(s(0))):-plus(0,s(0),s(s(0))).plus(s(s(0)),s(0),s(s(s(0)))):-plus(s(0),s(0),s(s(0)))....

Semantics

Full clausal logic

1.renaming variables so that the two atoms have none in common2.ensuring that the atoms’ predicates and arity correspond3.scanning the subterms from left to right to

(I)find first pair of subterms where the two atoms differ;(a) if neither subterm is a variable, unification fails;(b) else substitute the other term for all occurrences of the variable and remember the partial substitution;

(II) repeat until no more differences found

plus(s(0),X,s(X)) plus(s(Y),s(0),s(s(Y)))and

have most general unifier

{Y/0, X/s(0))}

found by

Proof Theory (mgu)

yields unified atom plus(s(Y),s(0),s(s(Y)))

Full clausal logic

repeatselect s = t ∈ Ecase s = t of

f (s1, . . . , sn) = f (t1, . . . , tn) (n ≥ 0) :replace s = t by {s1 = t1, . . . , sn = tn}

f (s1, . . . , sm) = g(t1, . . . , tn) (f/m �= g/n) :fail

X = X :remove X = X from E

t = X (t �∈ Var) :replace t = X by X = t

X = t (X ∈ Var ∧ X �= t ∧ X occurs more than once in E) :if Xoccurs in tthen failelse replace all occurrences of X in E (except in X = t) by t

esacuntil no change

Clausal logic

examples

{f (X , g(Y )) = f (g(Z ), Z )}⇒ {X = g(Z ), g(Y ) = Z}⇒ {X = g(Z ), Z = g(Y )}⇒ {X = g(g(Y )), Z = g(Y )}⇒ {X/g(g(Y )), Z/g(Y )}

{f (X , g(X ), b) = f (a, g(Z ), Z )}⇒ {X = a, g(X ) = g(Z ), b = Z}⇒ {X = a, X = Z , b = Z}⇒ {X = a, a = Z , b = Z}⇒ {X = a, Z = a, b = Z}⇒ {X = a, Z = a, b = a}⇒ fail

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Clausal logic

examples

{f (X , g(Y )) = f (g(Z ), Z )}⇒ {X = g(Z ), g(Y ) = Z}⇒ {X = g(Z ), Z = g(Y )}⇒ {X = g(g(Y )), Z = g(Y )}⇒ {X/g(g(Y )), Z/g(Y )}

{f (X , g(X ), b) = f (a, g(Z ), Z )}⇒ {X = a, g(X ) = g(Z ), b = Z}⇒ {X = a, X = Z , b = Z}⇒ {X = a, a = Z , b = Z}⇒ {X = a, Z = a, b = Z}⇒ {X = a, Z = a, b = a}⇒ fail

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Proof Theory: mgu using Martelli-Montanari algorithm

Full clausal logic

Proof Theory: occur check

loves(X,person_loved_by(X)). :- loves(Y,Y).

without occur check, atoms to be resolved upon unify under substitution

{Y/X, X/person_loved_by(X)}

and therefore resolving to the empty clause

BUT...try to print answer:

X=person_loved_by(person_loved_by(person_loved_by(...)))

moreover, not a logical consequence of the program

Full clausal logic

Clausal logic

occur check

{l(Y , Y ) = l(X , f (X ))}⇒ {Y = X , Y = f (X )}⇒ {Y = X , X = f (X )}⇒ fail

The last example illustrates the need for the “occur check” (which isnot done in most Prolog implementations)

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Martelli-Montanari algorithm SWI-Prolog

?- l(Y,Y) = l(X,f(X)).Y = f(**),X = f(**).?-

?- unify_with_occurs_check(l(Y,Y),l(X,f(X))).false.?-

Proof Theory: occur check

Full clausal logic

P⊦C ⇒ P⊧Cfull clausal logic is sound

P∪{C} inconsistent ⇒ P ∪ {C} ⊢ �full clausal logic is refutation-complete

The question “P⊧C?” is only semi-decidable.

Meta-theory: soundness, completeness, decidability

Clausal Logic: Overview

Propositional Relational Full

Herbrand universe

Herbrand base

clause

Herbrand models

meta-theory

finite infinite

{p, q}

{a,f(a),f(f(a)),...}

{p(a,a), p(b,a),...}

{p(a,f(a)), p(f(a), f(f(a))),...}

p:-q p(X,Z):- q(X,Y),p(Y,Z)

p(X,f(X)):- q(X)

{}{p}{p,q}

{}{p(a,a)}{p(a,a),p(b,a),q(b,a)}...

finite number of finite models

{}{p(a,f(a)),q(a)}{p(f(a),f(f(a)), q(f(a))} ...

infinite number of finite or infinite models

soundrefutation-complete

decidable

soundrefutation-complete

decidable

sound (occurs check)refutation-complete

semi-decidable

Conversion to first-order predicate logic (1)

married;bachelor :- man,adult.haswife :- married.

becomes (man ∧ adult ⇒ married ∨ bachelor) ∧(married ⇒ haswife)

(¬man ∨ ¬adult ∨ married ∨ bachelor )∧ (¬married ∨ haswife)

reachable(X,Y,route(Z,R)):- connected(X,Z,L), reachable(Z,Y,R).

becomes ∀X∀Y∀Z∀R∀L : ¬connected(X,Z,L) ∨ ¬reachable(Z,Y,R) ∨ reachable(X,Y,route(Z,R))

A ⇒ B ≡ ¬A ∨ B

¬(A ∧ B) ≡ ¬A ∨ ¬B

nonempty(X) :- contains(X,Y).

becomes

∀X∀Y: nonempty(X) ∨ ¬contains(X,Y)

∀X: (nonempty(X) ∨ ∀Y¬contains(X,Y))

∀X: nonempty(X) ∨ ¬(∃Y:contains(X,Y))or

∀X: (∃Y:contains(X,Y)) ⇒ nonempty(X))or

Conversion to first-order predicate logic (2)

Conversion from first-order predicate logic (1)

1 eliminate ⇒ using A ⇒ B ≡ ¬A ∨ B.

2Put into negation normal form: negation only occurs immediately before propositions

∀X[brick(X)⇒(∃Y[on(X,Y)∧¬pyramid(Y)]∧ ¬∃Y[on(X,Y) ∧ on(Y,X)]∧ ∀Y[¬brick(Y)⇒¬equal(X,Y)])]

∀X[¬brick(X)∨(∃Y[on(X,Y)∧¬pyramid(Y)]∧ ¬∃Y[on(X,Y)∧on(Y,X)]∧ ∀Y[¬(¬brick(Y))∨¬equal(X,Y)])]

∀X[¬brick(X)∨(∃Y[on(X,Y)∧¬pyramid(Y)]∧ ∀Y[¬on(X,Y)∨¬on(Y,X)]∧ ∀Y[brick(Y)∨¬equal(X,Y)])]

A ⇒ B ≡ ¬A ∨ B

¬(A∧B) ≡ ¬A∨¬B ¬(A∨B) ≡ ¬A∧¬B

¬(¬A) ≡ A¬∀X [p(X)] ≡ ∃X [¬p(X)]¬(∃X [p(X)] ≡ ∀X [¬p(X)]

replace ∃ using Skolem functors (abstract names for objects, functor has to be new)

∀X[¬brick(X)∨([on(X,sup(X))∧¬pyramid(sup(X))]∧ ∀Y[¬on(X,Y)∨¬on(Y,X)]∧ ∀Y[brick(Y)∨¬equal(X,Y)])]

∀X[¬brick(X)∨(∃Y[on(X,Y)∧¬pyramid(Y)]∧ ∀Y[¬on(X,Y)∨¬on(Y,X)]∧ ∀Y[brick(Y)∨¬equal(X,Y)])]

standardize all variables apart such that each quantifier has its own unique variable

∀X[¬brick(X)∨([on(X,sup(X))∧¬pyramid(sup(X))]∧ ∀Y[¬on(X,Y)∨¬on(Y,X)]∧ ∀Z[brick(Z)∨¬equal(X,Z)])]

∀X[¬brick(X)∨([on(X,sup(X))∧¬pyramid(sup(X))]∧ ∀Y[¬on(X,Y)∨¬on(Y,X)]∧ ∀Y[brick(Y)∨¬equal(X,Y)])]

move ∀ to the front

∀X∀Y∀Z[¬brick(X)∨([on(X,sup(X))∧¬pyramid(sup(X))]∧ [¬on(X,Y)∨¬on(Y,X)]∧ [brick(Z)∨¬equal(X,Z)])]

convert to conjunctive normal form using A∨(B∧C) ≡ (A∨B)∧(A∨C)

∀X∀Y∀Z[(¬brick(X)∨[on(X,sup(X))∧¬pyramid(sup(X))])∧ (¬brick(X)∨[¬on(X,Y)∨¬on(Y,X)])∧ (¬brick(X)∨[brick(Z)∨¬equal(X,Z)])]

∀X∀Y∀Z[¬brick(X)∨([on(X,sup(X))∧¬pyramid(sup(X))]∧ [¬on(X,Y)∨¬on(Y,X)]∧ [brick(Z)∨¬equal(X,Z)])]

∀X∀Y∀Z[((¬brick(X)∨on(X,sup(X)))∧(¬brick(X)∨¬pyramid(sup(X))))∧ (¬brick(X)∨[¬on(X,Y)∨¬on(Y,X)])∧ (¬brick(X)∨[brick(Z)∨¬equal(X,Z)])]

∀X∀Y∀Z[[¬brick(X)∨on(X,sup(X))]∧ [¬brick(X)∨¬pyramid(sup(X))]∧ [¬brick(X)∨¬on(X,Y)∨¬on(Y,X)]∧ [¬brick(X)∨brick(Z)∨¬equal(X,Z)]]

A∨(B∨C) ≡ A∨B∨C

split the conjuncts in clauses (a disjunction of literals)

∀X∀Y∀Z[[¬brick(X)∨on(X,sup(X))]∧ [¬brick(X)∨¬pyramid(sup(X))]∧ [¬brick(X)∨¬on(X,Y)∨¬on(Y,X)]∧ [¬brick(X)∨brick(Z)∨¬equal(X,Z)]]

convert to clausal syntax (negative literals to body, positive ones to head)

∀X ¬brick(X)∨on(X,sup(X))∀X ¬brick(X)∨¬pyramid(sup(X))∀X∀Y ¬brick(X)∨¬on(X,Y)∨¬on(Y,X)∀X∀Z ¬brick(X)∨brick(Z)∨¬equal(X,Z)

on(X,sup(X)) :- brick(X).:- brick(X), pyramid(sup(X)).:- brick(X), on(X,Y), on(Y,X).brick(X) :- brick(Z), equal(X,Z).

∀X: (∃Y:contains(X,Y))⇒ nonempty(X))

∀X: ¬(∃Y:contains(X,Y)) ∨ nonempty(X))1 eliminate ⇒

2 put into negation normal form ∀X: (∀Y:¬contains(X,Y)) ∨ nonempty(X))

5 move ∀ to the front ∀X∀Y: ¬contains(X,Y) ∨ nonempty(X)

8 convert to clausal syntax nonempty(X) :- contains(X,Y)

3 replace ∃ using Skolem functors

4 standardize variables

6 convert to conjunctive normal form

7 split the conjuncts in clauses

Definite Clausal Logicmarried(X);bachelor(X) :- man(X), adult(X).man(peter). adult(peter). man(paul).:-married(maria). :-bachelor(maria). :-bachelor(paul).

clause is used from right to left

clause is used from left to right

both literals from head and body are resolved

married(X);bachelor(X) :- man(X), adult(X). man(peter).

married(X);bachelor(X) :- man(peter), adult(peter). adult(peter).

married(peter);bachelor(peter)

married(X);bachelor(X) :- man(X), adult(X). :-married(maria).

bachelor(maria) :- man(maria), adult(maria). :-bachelor(maria).

:-man(maria);adult(maria)

married(X);bachelor(X) :- man(X), adult(X). man(paul).

married(paul);bachelor(paul) :- adult(paul). :-bachelor(paul).

married(paul):-adult(paul)

Syntax and Proof Procedure

A :- B1,...,Bn

full clausal logic clauses are restricted: at most one atom

in the head

from right to left:! procedural interpretation

“prove A by proving each of Bi“

rules out indefinite conclusions fixes direction to use clauses

for efficiency’s sake

Full clausal logic

Lost Expressivity

married(X); bachelor(X) :- man(X), adult(X).man(john). adult(john).

We can no longer express

which had two minimal models

{man(john),adult(john),married(john)}{man(john),adult(john),bachelor(john)}{man(john),adult(john),married(john),bachelor(john)}

first model is minimal model of general clause

married(X) :- man(X), adult(X), not bachelor(X).

second model is minimal model of general clause

bachelor(X) :- man(X), adult(X), not married(X).

Full clausal logic

Propositional Clausal Logic - VUBai.vub.ac.be/~ydehauwe/decl_prog/2_Logic.pdf · Semantics The...

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