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Lecture 9: Resolution in First Order Logic. Theorem Proving in First Order Logic Unification Resolution. Heshaam Faili [email protected] University of Tehran. FOL Decidability (1).

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Lecture 9 resolution in first order logic

Lecture 9: Resolution in First Order Logic

Theorem Proving in First Order Logic

Unification

Resolution

Heshaam Faili

[email protected]

University of Tehran


Fol decidability 1
FOL Decidability (1)

  • If a proof procedure is always guaranteed to find a proof of a particular sentence or of its negation, then the question of logical implication for the sentence is said to be decidable.

  • To see if KB |= c, we generate all possible inferences from KB, stopping when we get f or ~c.


Fol decidability 2
FOL Decidability (2)

  • In general, neither f nor ~c may be logically implied by KB.

  • In this case, the proof procedure will never stop.

  • The question of logical implication in this case is thus semidecidable:

  • If KB |= c or KB |= ~c , the proof procedure will eventually discover that. Otherwise, the procedure will run forever.


Fol inference rules for quantifiers
FOL -- Inference Rules for quantifiers

  • Universal and Existential Elimination

  • Existential Introduction


Resolution in fol
Resolution in FOL

  • Mechanical proof procedure with a single rule

  • Invented by J. A. Robinson in 1965

  • Three main steps: 1. Clause form: transform all clauses to uniform format 2. Unification: find a variable assignment so that two clauses can be resolved 3. Resolution: rule to obtain new conclusions


1 clause form
1. Clause Form

  • We want a uniform representation for our logical propositions, so that we can use a simple uniform proof procedure.

  • Clausal form expresses all logical propositions using literals (atomic sentences or negated atomic sentences) and clauses (sets of literals representing their disjunction).

  • Ex: all the following in a single form: a => b ~abb ~a ~(~a ~b)


  • Any set of sentences in predicate calculus can be converted to clausal form.

  • Eight-step procedure for conversion:

  • 1. Replace implications

  • 2. Distribute negations

  • 3. Standardize variables

  • 4. Replace existential

  • 5. Remove universals

  • 6. Distribute disjunctions

  • 7. Replace operators

  • 8. Rename variables

Clause Form Conversion


1 replace implications
1. to clausal form.Replace implications

Any sentence of the form: j => y becomes ~j  y at all levels

2. Distribute negations

Repeatedly do the following:

~~ y becomes y

~(j  y) becomes ~j  ~y

~(j  y) becomes ~j  ~y

~Vj becomes  V ~j ~Vj becomes V ~j


3. to clausal form.Standardize variables

Rename variables to ensure that each quantifier has its own unique variable.

Ex: (Xp(X))  (X q(X)) becomes (Xp(X))  (Y q(Y))

Consider: (YX p(X,Y))

The identity of X depends on the value of Y. We can replace X by a function of Y: (Yp(g(X),Y))where g is a Skolem function

4. Replace existential


Skolemization
Skolemization to clausal form.

  • Replace each occurrence of an existentially quantified variable by a Skolem function.

  • The function’s arguments are the universally quantified variables that are bound by universal quantifiers whose scope includes the scope of the existential quantifier being eliminated.

  • The Skolem functions must be new, i.e., not already present in any other sentences.

  • If the existential quantifier is not within a universal quantifier, use a Skolem function of no arguments, i.e., a constant.


Examples of skolemization
Examples of Skolemization to clausal form.

1.X p(X)becomes p(g)

2. YXp(X,Y)becomes Yp(g(Y),Y)

3.X Yp(X,Y)becomes Yp(g,Y)

4.YX (((q(X)  p(X,Y))ZW(r(W)))

becomes

Y (((q(f(Y))  p(f(Y),Y))Z(r(g(Y,Z)))

where f and g are Skolem functions


5 remove universal quantifiers
5. to clausal form.Remove universal quantifiers

Throw away "’s and assume all variables are universally quantified. "X "Y "Z q(h(Y)) p(h(Y),X) r(X,Z) becomes q(h(Y)) p(h(Y),X) r(X,Z)

6. Distribute disjunctions

Write the formula in conjunctive normal form (the conjunction of sets of disjunctions) with:

j  (y  q) becomes (j  y)  (j  q)


7. to clausal form.Replace operators

  • Replace the conjunction

  • S1 S2 …. Sn

  • with the set of clauses S1 , S2 , …, Sn

  • Convert each Si into a set of literals (that is, get rid of the disjunctions symbols “”, and write them in clause notation.

  • Ex: (p(X) q(Y)) (p(X)  ~r(X,Y))

  • becomes two clauses:

  • {p(X), q(Y)}

  • {p(X), ~r(X,Y)}


8. to clausal form.Rename variables

Change variable names so that no variable symbol appears in more than one clause:

Ex: {p(X), q(Y)} and {p(X), ~r(X,Y)}

becomes {p(X), q(Y)} and {p(Z), ~r(Z,W)}

we can do this because:

"X "Y(p(X) q(Y)) (p(X)  ~r(X,Y))

is equivalent to

"X "Y(p(X) q(Y)) "Z "W(p(Z)  ~r(Z,W))


Example of entire conversion 1
Example of entire conversion (1) to clausal form.

"X (p(X) => (("Yp(Y) => (p(f(X,Y))) 

(~"Y (~q(X,Y) p(Y))))

1. "X (~p(X) (("Y~p(Y)  (p(f(X,Y))) 

(~"Y (~q(X,Y) p(Y))))

2. "X (~p(X)  (("Y ~p(Y)  (p(f(X,Y)) 

(Y(q(X,Y)  ~p(Y))))

3. "X (~p(X)  (("Y ~p(Y)  (p(f(X,Y))) 

(W (q(X,W)  ~p(W))))


Example of entire conversion 2
Example of entire conversion (2) to clausal form.

4. "X (~p(X)  (("Y ~p(Y)  (p(f(X,Y))) 

(q(X,g(X))  ~p(g(X)))))

5. ~p(X)  ((~p(Y)  (p(f(X,Y))) 

(q(X,g(X))  ~p(g(X)))))

6. (~p(X)  ~p(Y))  (p(f(X,Y))) 

(~p(X)  (q(X,g(X))) 

(~p(X) ~p(g(X)))))


Example of entire conversion 3
Example of entire conversion (3) to clausal form.

7. {~p(X), ~p(Y)),p(f(X,Y))},

{~p(X) , q(X,g(X)))},

{~p(X), ~p(g(X))}

8. {~p(X1), ~p(Y1)), p(f(X1,Y1))},

{~p(X2) , q(X2,g(X2)))},

{~p(X3), ~p(g(X3))}


Unification substitution
Unification -- Substitution to clausal form.

  • Unification is the process of determining whether two expressions can be made identical by appropriate substitutions for their variables.

  • The substitution applies to variables of both expressions and makes them syntactically equivalent. The result is a substitution instance

  • Example of a unifying substitution

    S1: p(X,f(b),Z) S2: p(a,Y,W)

    U = {X/a Y/f(b), Z/W}

    S1[U] = S2[U] = p(a,f(b),W)substitution instance


Properties of the substitution
Properties of the substitution to clausal form.

1. Each variable is associated with at most one expression;

2. No variable with an associated expression occurs within any associated expression. That is, no “left side” appears on a “right side”:

U = {X/a, Y/f(X), Z/Y}

is not a legal substitution


Unification of sentences to clausal form.

A set of expressions {S1,S2,….Sn}

is unifiable if there is a substitution U that makes them identical:

S1[U] =S2 [U] =… = Sn [U]

Ex: U = {X/a, Y/b, Z/c} unifies the expressions p(a,Y,Z) and p(X,b,Z) and p(a,Y,c)


Unifiers are not unique! to clausal form.

  • Two expressions can have more that one unifier that makes them equivalent:

    S1 = p(X,Y,Y) and S2 = p(a,Z,Z)

    are unified by

    U1={X/a,Y/Z} and U2={X/a,Y/b, Z/b}

    S1[U1] = S2[U1] = p(a,Z,Z) S1[U2] = S2[U2] = p(a,b,b)


Partial order between unifiers
Partial order between unifiers to clausal form.

  • Some unifiers are more general than others:

    U1 is more general than U2 if there exists a unifier U3 such that U1U3 = U2

    Example:

    U1 = {X/f(Y),Z/W,R/c} is more general that

    U2 = {X/f(a),Z/b, R/c} since U3 = {Y/a,W/b}

    U1 U3 ={X/f(Y),Z/W,R/c}{Y/a,W/b} = U2

  • Unifiers form a partial order


Most general unifier mgu
Most general unifier (MGU) to clausal form.

  • For each set of sentences, there exists a most general unifier (mgu) that is unique up to variable renaming

    Ex: for S1 = p(X,Y,f(Z)) and S2 = p(a,W,R) U={X/a,Y/W,R/f(Z)} is the mgu

  • For the resolution procedure, we want to find the most general unifier of literals: a constant, variable, or a functional object of length n.


Basic functions for mgu procedure
Basic functions for MGU procedure to clausal form.

  • Constant(exp) returns true if exp is a constant Ex: a, f(g(a,b,c)) are constants

  • Variable(exp) returns true if exp is a simple variable

  • Length(exp) returns the number of items in a function Ex: f(a,g(Y)) is an object of length 2.

  • MguVar(Var, exp) returns

    • false if Var is included in exp

    • {Var/exp} otherwise


Recursive procedure for mgu
Recursive procedure for MGU to clausal form.

functionMgu(exp1,exp2) returns unifier

if exp1 = exp2 return {}

if Variable(exp1) then return MguVar(exp1,exp2)

if Variable(exp2) then return MguVar(exp2,exp1)

if Constant(exp1) or Constant(exp1) return false

if not(Length(exp1) = Length(exp2) return false

unifier := {}

for i := 1 toLength(exp1) do

s := Mgu(Part(exp1,i),Part(exp2,i))

if s is false return false

unifier := Compose(unifier,s)

exp1 := Substitute(exp1,unifier)

exp2 := Substitute(exp2,unifier)

return unifier; end


Example of MGU trace (1) to clausal form.

Find the mgu unifier for

p(f(X,g(a,Y)), g(a,Y)) and p(f(X,Z),Z).

1. Mgu is called on p and p, returns {}

2. Mgu is called recursively on f(X, g(a,Y)) and f(X,Z)

3. Mgu is called on f and f, returns {}

4. Mgu is called on X and X, returns {}

5. Mgu is called on g(a,Y) and Z; since Z is a variable, it returns (via Mguvar) {Z/g(a,Y)} after checking that Z is not in g(a,Y).


Example of mgu trace 2
Example of MGU trace (2) to clausal form.

6. {Z/g(a,Y)} is composed with the previous (empty) substitution

7. The entire substitution is applied to both expressions, yielding f(X, g(a,Y))

8. Since i = 3, Mgu returns {Z/g(a,Y)}, which is then applied to the top level expressions. All other checks show equality. The result is

p(f(X,g(a,Y)),g(a,Y))


Performance issues
Performance Issues to clausal form.

  • Predicate Indexing: Storing the answers of some queries

    • Knows(John, x) : store all x (in a hash table) which John knows

    • Using subsumption lattice to be more efficient


Subsumption lattice
Subsumption Lattice to clausal form.


3 resolution
3. to clausal form.Resolution

Given a clause containing the literal j and another clause the literal ~j , we can infer the clause consisting of all the literals of both clauses without j and ~j.

Ex: 1. {p, q} KB

2. {~q, r} KB

3. {p, r} 1,2


Why does this work
Why does this work? to clausal form.

Consider the two clauses

{winter, summer}

{~winter, cold}

At any point, winter is either true or false. If it is true, then cold must be true to guarantee the truth of clause 2. If winter is false, then summer must be true to guarantee the truth of clause 1. So regardless of winter's truth value, either cold or summer must be true, i.e., we can conclude:

{cold, summer}


Other resolution examples
Other Resolution Examples to clausal form.

1. {p, q} KB

2. {~p, q} KB

3. {q} 1,2

1. {p} KB

2. {~p} KB

3. {} 1,2

merge the q's

1. {~p, q} KB

2. {p} KB

3. {q} 1,2

much like Modus Ponens

We can derive the empty clause, showing

that the KB has a contradiction


Resolution Rule to clausal form.

For clauses containing variables, we can resolve j in one clause with ~y in another clause, as long as j and y have a mgu U. The resulting clause is the union of the original 2 clauses, with j and ~y removed, and with U applied to the remaining literals.

{r,j}and {~y,d}

{r[U],d[U]}

where j [U] = y [U]


Examples of Resolutions to clausal form.

  • Ex1: 1. {p(X), q(X,Y)} KB

  • 2. {~p(a), r(b,Z)} KB

  • 3. {q(a,Y), r(b,Z)} 1,2

  • Ex2: Two clauses may resolve in more than one way since f and ~y may be chosen in different ways:

    1. {p(X,X), q(X), r(X)} KB

    2. {~p(a,Z), ~q(b)} KB

    3. {q(a), r(a),~q(b)} 1,2

    4. {p(b,b), r(b), ~p(a,Z)} 1,2


Resolution Deduction to clausal form.

A resolution deduction of a clause j from a data base KB is a sequence of clauses in which

1. j is an element of the sequence;

2. Each element is either a member of KB or the result of applying the resolution principle to clauses earlier in the sequence.


Resolution procedure with to clausal form.

nondeterministic choices

functionResolution(KB) returns answer

while not(Empty_Clause(KB)) do

c1:= Choose_Clause(KB)

c2:= Choose_Clause(KB)

res := Choose_Resolvents(c1,c2)

KB := KB U {res}

end

return true

end


Ex resolution deduction of the empty clause
Ex: Resolution deduction to clausal form.of the empty clause

1. { p } KB

2. {~p, q } KB

3. { ~q, r } KB

4. { ~r } KB

5. {q} 1, 2

6. { ~q} 3, 4

7. { } 5, 6


{ to clausal form.p}

{~p,q}

{~q,r}

{~r}

{q}

{~q}

{~p,r}

{r}

{r}

{ }

{~p}

{~p}

All possible resolutions (to 3 levels)

A “resolution trace” is a linear form of the graph


Resolution Refutation to clausal form.

  • As for Propositional Logic, we will use refutation resolution as the single rule for proving sentences about a KB

  • We will use it to prove the unsatisfiability of a set of clauses, i.e., they contain a contradiction if we can derive the empty clause.

  • Resolution refutation: to prove c, prove that KB U {~c} |= 0 like a “proof by contradiction”

  • Rule is refutation-complete.


Answering questions with refutation resolution
Answering questions with Refutation-Resolution to clausal form.

  • True/False questions: we want to know if a conclusion follows from KB:

    Ex: father(art, jon)

    father(bob, kim)

    father(X, Y) => parent(X, Y) Is art the parent of jon?

  • Queries: fill-in-the blank: we want to know also what instantiation makes it true: Who is the parent of jon?


True false by refutation resolution
True/False by Refutation Resolution to clausal form.

1. {father(art, jon)} KB

2. {father(bob, kim)} KB

3. {~father(X,Y), parent(X,Y)} KB

4. {~parent(art,jon)}

5. {parent(art,jon)} 1, 3

6. {parent(bob, kim)} 2, 3

7. {~father(art, jon)} 3, 4

8. { } 4, 5

9. { } 1, 7

negated goal clause


Fill in the blank green s method
Fill-in-the-blank (Green’s method) to clausal form.

  • To obtain one instantiation that makes the conclusion true (if any), form a disjunction of the negation of the goal c and its “answer literal”: a term of form

    ans(X1,X2,…Xn)

    where the variables X1,X2,…Xn are the free variables in c. Ex: Add to the previous KB the clause {~parent(X, jon), ans(X)}

  • Resolution halts when it derives a clause containing only the ans literal


Example of fill-the-blank derivation to clausal form.

1. {father(art, jon)} KB

2. {father(bob, kim)} KB

3. {~father(X,Y), parent(X,Y)} KB

4. {~parent(Z,jon), ans(Z)} c

5. {parent(art, jon)} 1, 3

6. {parent(bob, kim)} 2, 3

7. {~father(w,jon), ans(W)} 3, 4

8. {ans(art) } 4, 5

9. {ans(art) } 1, 7


Properties of Fill-in-the-blank to clausal form.

  • The answer may not be unique: several different answers may result from the proof, depending on the clauses that were chosen for resolution.

  • We may also get an answer of the form {ans(a), ans(b) } where one of the answers is right, but the clause doesn’t tell us which one.

    1.{father(art,jon), father(bob,jon)} KB 2. {~ father(X,jon),ans(X)} c 3.{father(bob,jon),ans(art)}1, 2 4. {ans(art),ans(bob)}2, 3


Forward chaining
Forward Chaining to clausal form.


Example of forward chaining
Example of Forward Chaining to clausal form.


Completeness of fc
Completeness of FC to clausal form.

  • k : number of predicate argument

  • p: number of predicates

  • n: number of constant symbols

    • pnk distinct ground facts

    • Propositional logic with above facts


Efficient fc
Efficient FC to clausal form.

  • Problems of simple FC

    • Inner loop involve pattern matching

    • Algorithm rechecks every rule in each loop

    • Might generate many irrelevant facts

  • Should be addressed these problems


Pattern matching and csp
Pattern Matching and CSP to clausal form.

  • Every finite-domain CSP can be expressed as a single definite clause together with some associated ground facts.


Incremental fc
Incremental FC to clausal form.

  • Every new fact inferred on iteration t must be derived from at least — one new fact inferred on iteration t - 1.

  • Retain the partial matching of any rule inorder to use it next iterations


Irrelevant facts
Irrelevant facts to clausal form.

  • 3 ways to avoid:

    • Use backward chaining

    • Restrict subset of rules

    • Use magic set , some information about the goal in order to ignore irrelevant facts …


Backward chaining
Backward Chaining to clausal form.


Bc example
BC example to clausal form.


Logic programming
Logic Programming to clausal form.

Algorithm = Logic + Control

  • Prolog = Programming in Logic


? to clausal form.


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