UIUC CS 497: Section EA Lecture

1
UIUC CS 497 Section EALecture 3

• Reasoning in Artificial Intelligence
• Professor Eyal Amir
• Spring Semester 2004

2
Last Time

• SAT checking using DPLL (instantiate, propagate,
  backtrack)
• Entailment/SAT checking using Resolution (create
  more and more clauses until KB is saturated)
• Formal verification uses mainly SAT checking such
  as DPLL, but also sometimes resolution

3
From Homework You Should Know

• Deduction theorem for FOL
• Language of FOL
• Soundness, completeness, and incompleteness
  theorems
• Models of FOL

4
Today

• Reasoning procedure for FOL
• Proving entailment using Resolution
• Application du jour Temporal Reasoning
• Applications we will not touch
• Spatial reasoning, formal verification,
  mathematics, planning, NLP,

5
First-Order Theories

• Signature L
• Function symbols (f(x,y))
• Predicate (relation) symbols (P(x,A))
• Constant symbols (A,B,C,)
• FOL language quantification over objects

6
Model Theory Reminder

• Structure/Interpretation ltU,Igt
• U Universe of elements
• I Mapping of
• Constant symbols to elements in U
• Predicate symbols to relations over U
• Function symbols to functions over U
• M T - M satisfies T
• T is a theory, i.e., a set of FOL sentences in
  language L for which M is an interpretation

-
7
Logical Entailment

-
b
a
?
Lman, woman, loves M1ltU1,I1gt U1Sue,Kim,Pat
I1manPat I1womanSue,Kim I1lovesltPat,
Kimgt,ltPat,Suegt
-
b
g

?
8
Clausal Form

• Every FOL formula is consistency-equivalent to
  conjunction of F.O. clauses.
• First-order clause
• Only universal quantifiers (which are implicit)
• Disjunction of literals (atoms or their negation)

9
Conversion to Clausal Form

• ? replaced by ?, ?
• Negations in front of atoms
• Standardize variables (unique vars.)
• Eliminate existentials (Skolemization)
• Drop all universal quantifiers
• Move disjunctions in (put into CNF)
• Rename vars. (standardize vars. apart)

10
Take a Breath

• Until now
• First-order logic basics
• How to convert a general FOL sentence to clausal
  form
• From now Resolution theorem proving
• Search in the space of proofs
• Later Temporal reasoning

11
Resolution Theorem Proving

• Given
• KB a set of first-order sentences
• Query Q a logical sentence
• Calling procedure
• Add ?Q to KB
• Convert KB into clausal form
• Run theorem prover. If we prove contradiction,
  return T.

12
Resolution Theorem Proving

• Add ?Q to KB
• Convert KB into clausal form
• Run theorem prover. If we prove contradiction,
  return T.
• Deduction theorem
• KB Q iff KB ? ?Q
  FALSE

-
-
13
Resolution Theorem Proving

• Add ?Q to KB
• Convert KB into clausal form
• Run theorem prover. If we prove contradiction,
  return T.
• Deduction theorem
• KB Q iff KB ? ?Q
  FALSE

-
-
14
First-Order Resolution

• Resolution inference rule
• C1 P(t1,,tk) ? C1(t1,,tk)
• C2 ?P(s1,,sk) ? C2(s1,,sk)
• mgu(ltt1,,tkgt,lts1,,skgt) r1,,rn
• --------------------------------------------
• C3 (C1 ? C2) r1,,rn

15
First-Order Resolution

• Resolution algorithm (saturation)
• While there are unresolved C1,C2
• Select C1, C2 in KB
• If C1, C2 are resolvable, resolve them into a new
  clause C3
• Add C3 to KB
• If C3
• return T.
• STOP

C1 P(t1,,tk) ? C1(t1,,tk) C2 ?P(s1,,sk)
? C2(s1,,sk) mgu(ltt1,,tkgt,lts1,,skgt)
r1,,rn ----------------------------------------
---- C3 (C1 ? C2) r1,,rn
16
Resolution in Action
C1 P(t1,,tk) ? C1(t1,,tk) C2 ?P(s1,,sk)
? C2(s1,,sk) mgu(ltt1,,tkgt,lts1,,skgt)
r1,,rn ----------------------------------------
---- C3 (C1 ? C2) r1,,rn
On board
KB
Negated Query
17
Resolution in Action
C1 P(t1,,tk) ? C1(t1,,tk) C2 ?P(s1,,sk)
? C2(s1,,sk) mgu(ltt1,,tkgt,lts1,,skgt)
r1,,rn ----------------------------------------
---- C3 (C1 ? C2) r1,,rn
On board
KB
Negated Query
18
First-Order Resolution

• Resolution algorithm (saturation)
• While there are unresolved C1,C2
• Select C1, C2 in KB
• If C1, C2 are resolvable, resolve them into a new
  clause C3
• Add C3 to KB
• If C3
• return T.
• STOP

C1 P(t1,,tk) ? C1(t1,,tk) C2 ?P(s1,,sk)
? C2(s1,,sk) mgu(ltt1,,tkgt,lts1,,skgt)
r1,,rn ----------------------------------------
---- C3 (C1 ? C2) r1,,rn
19
First-Order Resolution Rule

• (2) If C1, C2 are resolvable, resolve them into a
  new clause C3
• If C1,C2 have two literals l1,l2 with same
  predicates (P) and opposite polarity, and
• If l1 P(t1,,tk), l2 ?P(s1,,sk), unifiable
• with mgu (most general unifier) r1,,rn,
• then

C1 P(t1,,tk) ? C1(t1,,tk) C2 ?P(s1,,sk)
? C2(s1,,sk) mgu(ltt1,,tkgt,lts1,,skgt)
r1,,rn ----------------------------------------
---- C3 (C1 ? C2) r1,,rn
20
Unification

• P(t1,,tk),?P(s1,,sk), unifiable with mgu (most
  general unifier) sr1,rk
• Substitution replace vars. by terms
• Term constant, variable, or a function of terms
• Composition of substitutions
• x/g(w,v) w/A,v/f(B,z)
• x/g(A,f(B,z),w/A,v/f(B,z)

x/B,y/z
x/B,y/z,x/w
x/B,y/z,z/w
(P(x) v Q(f(x)) v P(g(B,x)) v P(f(y))) x/B,y/z
(P(B) v Q(f(B)) v P(g(B,B)) v P(f(z)))
21
Unification

• Unification find a substitution s for
• C1 P(t1,,tk) ? C1(t1,,tk)
• C2 ?P(s1,,sk) ? C2(s1,,sk)
• such that P(t1,,tk)s P(s1,,sk)s

P(A,y,g(x,y))y/f(A) P(z,f(z),g(x,f(w)))z/A,w/
A sy/f(A),z/A,w/A
P(A,y,g(x,y))y/f(w) P(z,f(w),g(x,f(w)))z/A s
y/f(w),z/A
Most general unifier
22
Unification

• Substitution s1 more general than s2 if there is
  substitution ? such that
• s1 ? s2

P(A,y,g(x,y))y/f(A) P(z,f(z),g(x,f(w)))z/A,w/
A sy/f(A),z/A,w/A
P(A,y,g(x,y))y/f(w) P(z,f(w),g(x,f(w)))z/A s
y/f(w),z/A
Most general unifier
23
Unification

• Substitution s1 more general than s2 if there is
  substitution ? such that
• s1 ? s2

s2y/f(A),z/A,w/A
?w/A
s1y/f(w),z/A
Most general unifier
24
Finding the MGU

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,w))
25
Finding the MGU

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,w))
26
Finding the MGU

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,w))
27
Finding the MGU

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,w))
28
Finding the MGU

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,w))
29
Finding the MGU

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

Occurs check
F
x P(A,y,g(x,y)) y P(z,f(w),g(v,w))
part(3,x) y part(3,y) f(w)
part(1,x) P part(1,y) P
part(2,x) A part(2,y) z
part(4,x) g(x,f(w)) part(4,y) g(v,w)
30
Finding the MGU another example

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,f(w)))
31
Finding the MGU another example

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,f(w)))
32
Finding the MGU another example

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,f(w)))
33
Finding the MGU another example

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,f(w)))
34
Finding the MGU another example

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,f(w)))
35
Finding the MGU another example

• Procedure MGU(x,y)
• If xy, return
• If x or y are vars., return MGUvar(x,y)
• If x or y are const.,or Len(x)/Len(y), return
  F.
• s ? For i ? 1 to Len(x)
• s ? MGU(part(i,x),part(i,y)) if sF, return F.
• s ? compose(s,s) x ? subst(x,s) y ? subst(y,s)
  
• Return s

x P(A,y,g(x,y)) y P(z,f(w),g(v,f(w)))
part(3,x) y part(3,y) f(w)
part(1,x) P part(1,y) P
part(2,x) A part(2,y) z
part(4,x) g(x,f(w)) part(4,y)
g(v,f(w))
sy/f(w),z/A,v/x
36
Correctness of FOL Resolution

• Soundness Resolution is sound for first-order
  inference
• Refutation Completeness Resolution will find the
  empty clause, if FALSE is entailed
• No guarantee of termination (saturation), if
  FALSE not entailed FOL semi-decidable

37
Simple Efficiency Improvements

• Subsumption between clauses
• If clause C subsumes clause D (C entails D), then
  we can remove D from the KB.
• P(A) P(A),P(B)
• P(A) P(A)
• P(f(x),A),Q(g(x),B) P(f(v),y),Q(g(v),y)
• Algorithm for checking subsumption?

-
-
-
P(A)
P(t)
-
38
Simple Efficiency Improvements

• Subsumption within the clause
• If literal a subsumes literal b (a entails b),
  then we can remove a from the clause
• But, notice the variables scope
• P(A),P(t)
• P(A),P(B),P(t)
• P(A,x),P(y,B)
• P(x),Q(x),P(A)

P(A)
P(A),P(B)
P(A,x),P(y,B)
P(x),Q(x),P(A)
39
Properties of Resolution

• Unifying two literals of length n
• O(n2) because of occurs check
• Finding two resolvable clauses from m clauses of
  length n
• O(m2n2) the simple bound
• Overall algorithm
• Semi-decidable
• Unbounded length of proof as function of n,m

40
Related to FOL Resolution

• Clause selection and restriction strategies for
  resolution (lecture 5, paper 19)
• Consequence finding (paper 3)
• Constraint Satisfaction Problem (paper 5)
• Reasoning with equality (paper 6)
• DPLL in FOL (paper 7)
• Decidable fragments of FOL (paper 8)

41
Summary So Far

• Resolution theorem proving allows us to find
  contradictions and explanation.
• The deduction theorem tells us how to ask queries
  from Resolution
• Next Temporal Reasoning

42
Situation Calculus

• A first-order language for describing the effects
  of actions and events over time
• Constants S0 initial state action constants
• Functions result(ltactiongt,ltsituationgt)
• Predicates fluents properties that change
  over time
• at(1, S0)
• at(x,s) ? at(x1,result(move_fwd,s))
• Query at(11,s) ? ?ans(s) ?

?at(11,s) ? ans(s)
43
Situation Calculus

• Requires axioms describing effects and
  non-effects of actions/events
• Can be used for planning, projection, diagnosis,
  filtering (tracking)
• Frame Problem the compact and natural-language-li
  ke specification of effects of actions
• Qualification Problem the preconditions of
  actions

44
Notations

• Substitutions
• fs s x1/t1,,xk/tk
• fs s x1/t1,,xk/tk

45
Next Time

• Description Logics
• Requires prior knowledge of
• FOL theorem proving
• Tableau theorem proving will be useful
• Frame systems will be useful