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Chapter 6
Logical Reasoning
Xiu-jun GONG (Ph. D)School of Computer Science and Technology, Tianjin
University
http://cs.tju.edu.cn/faculties/gongxj/course/ai/
Outline Intro to logic reasoning Preparation for resolution
Stand form clauses (9 steps) Substitution and unify Control strategies
Resolution reasoning In propositional logic In Predicate logic In particular interrogative sentence
Summary
Logic is the Calculus of Computer Science
The central role of logic in computer science is comparable to the role of differential equations in the natural sciences
Intro to logic Logic is the study of the principles of valid
demonstration and inference. Consistency, which means that none of the
theorems of the system contradict one another. Soundness, which means that the system's rules
of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.
Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved in the system.
Classifications of logic Informal logic is the study of natural
language arguments. Formal logic is the study of inference with
purely formal content Symbolic logic is the study of symbolic
abstractions that capture the formal features of logical inference
Mathematical logic is an extension of symbolic logic into other areas(model theory, proof theory, set theory, and recursion theory)
Syntax Elements
Atoms: {A,B, …, T,F} Literal: atoms and a sign in front of them Connectives: , , , and Sentences: Syntax of well-formed formula (wff),
Any atom is a wff. If w1 and w2 are wffs, so are w1 w2, w1 w2, w1 w2, w1. There are no other wffs.
Universal sign: x Quantifier sign: x
Inference Reasoning is the cognitive process of
looking for reasons for beliefs, conclusions, actions or feelings.
Inference is the act or process of deriving a conclusion based solely on what one knows.
Logic reasoning Deduction: determining the conclusion Induction: determining the rule. Abduction: determining the precondition.
Semantics Semantics
associating elements of a logical language with elements of a domain of discourse
Interpretation the proposition associated with an atom (value)
Propositional Truth Table Given the values of atoms under some interpretation, use a
truth table to compute a value for any wff under that same interpretation.
Standard Form Clauses (1) Clause: a disjunction of literals.
Conjunctive Normal Form: CNF
Disjunctive Normal Form: DNF
CNFDNF
edba ~
)bf(~)edba(~)cb~a(
Standard Form Clauses (2) Prenex Normal Form: A wff of predicate
logic is in prenex normal form, iff all its quantifiers are clustered at the left no quantifier is negated the scope of each quantifier extends to the end
of the wff no two quantifiers quantify the same variable every quantified variable occurs in the matrix
of the wff
Standard Form Clauses (3) Skolem Standard Form:
A formula is in SSF if it is in conjunctive prenex normal form with only universal first-order quantifiers
Removing existential quantifiers from formal logic statements
Standard Form Clauses: Clauses in SSF
P(c)
Convert WFF to SFCs (1)1. Eliminate implication signs, >
2. Reduce the scopes of ~
3. Rename variables
qp~qp
QPQP
QPQP
~~)(~
~~)(~ laws sMorgan' DePP )(~~ :law negative double
)(~)()()(~
)(~)()()(~
XPXXPX
XPXXPXlawconversionQuantifies
)))Y,X(r~)Y,X(q)(Y()Y,X(p~)Y)((X(
)))Z,X(r~)Z,X(q)(Z()Y,X(p~)Y)((X(
Convert WFF to SFC (2)4. Forward quantifiers
5. Skolem transfer Eliminate existential quantifiers
))),(~),()((),(~))((( ZXrZXqZYXpYX
)))Z,X(r~)Z,X(q()Y,X(p)(~Z)(Y)(X(
)c(president)X(president)X(
)()()())()(( ZrYqXpZYX
))(())(()()( XgrXfqXpX
Convert WFF to SFC (3)6. Get CNF with Skolem standard form
7. Eliminate universal quantifiers
))(())(()(()( XgrXfqXpX )))(())((()))(()(()( XgrXfqXgrXpX
)))(())((()))(()(( XgrXfqXgrXp
Convert WFF to SFC (4)8. Get clauses
9. Rename variables again
))}(())(()),(()({
)))(())((()))(()((
XgrXfqXgrXp
XgrXfqXgrXp
))}(())(()),(()({ ygryfqXgrXp
An example)))Y,X(r)Y,X(q)(Y(~)Y,X(p)Y)((X(
)))Y,X(r~)Y,X(q)(Y()Y,X(p~)Y)((X(
)))Y,X(r)Y,X(q)(~Y(~)Y,X(p)Y()(~X(
)))Z,X(r~)Z,X(q)(Z()Y,X(p~)Y)((X(
)))Z,X(r~)Z,X(q()Y,X(p)(~Z)(Y)(X(
)))Z,X(r~)Z,X(q())X(f,X(p)(~Z()X(
)))X(g,X(r~))X(g,X(q())X(f,X(p(~)X(
))))X(g,X(r~))X(f,X(p(~)))X(g,X(q))X(f,X(p)((~X(
)))X(g,X(r~))X(f,X(p(~)))X(g,X(q))X(f,X(p(~
))}(,(~))(,()),(,(~))(,({~ YgYrYfYpYgYrYfYp
Resolution reasoning of propositional logic Resolution is a rule of inference leading
to a refutation theorem-proving technique for sentences in propositional logic and first-order logic.
Procedure Choose two clauses that have exactly one pair
of literals that are complementary Produce a new clause by deleting the
complementary literals and combining the remaining literals
CB
BA
CA
General form of resolution'11 CPC '
22 CP~C
'2
'121 CC)C,C(R
'2
'1
'2
'1 CC)CP(~)CP(
Proof: Show that if C1 and C2 are true for explanation I, then R is true for I.
1. Under I, if P=false, then C1’ =true, so R=C1’VC2’=true
2. Under I, if P=true, then C2’ =true, so R=C1’VC2’=true
Resolution on propositional logic Convert all the expressions (facts, rules
and the negation of conclusion) into standard form clauses
Resolve on all the clauses until reaches contradict (nil)
resolution refutation
Example TQTSRQPP ,,, Conditions:
Conclusion: R
RQP ~~)2(
QS ~)3(
QT ~)4(
R~)6(
P)1(
)(~)(~)~(~ QTQSQTS
T)5(
QP ~~)7( Q~)8(
T~)9(NIL)10(
(6) (2)
(7) (1)
(8) (4)
(9) (5)
Q~P~
Q~
T~
NIL
(6) (2)
(1)
(4)
(5)
Example cont.
RQP ~~)2(
QS ~)3(
QT ~)4(
P)1(
T)5(
)2)(1(~)6( RQ
)5)(4()7( Q
)7)(6()8( R
Lucky enough
Substitution and Unify Preparing for predicate logic resolution Universal quantifications for all variables.
If two clauses have matching but complementary literals, it is possible to resolve them
))(,,( 11 kn k 1
literalClause
xyf /)(
Substitution Any substitution can be represented by a
set of ordered pairs}/,..../,/{ 2211 nn xtxtxt
variableaisxWhere i
functionor constant variable,)( abemayxy ii
on.substituti theof scope thethroughout variablethe
of occurrenceevery for dsubstitute is term/
i
i
t
xthatmeansxt ii
F predicateon ofon substituti of results theis F
Unify F1 and F2 are predicate formula ,θ is a
substitution. if F1θ=F2θ,then θ is the Unifier of F1 and F2 P(c,y,b)and P(x,a,b), θ={c/x,a/y} P(x,y,b)and P(z,u,b) θ={z/x,u/y} other else? P(f(y),a)and P(x,a) θ={f(y)/x}
Three types of unifier Constant/variable —— 变量实例化 Variable/variable —— 变量一致化 Function/variable —— 变量函数化
Most General Unifier: MGU σ is a MGU between F1 and F2 if for any θ be one of unifiers between F1 and F2, there exists a λ such that θ=σ·λ
i.e. F1 =P(x), F2 =P(f(y)) θ1 ={f(a)/x,a/y},θ2 ={f(b)/x,b/y} θ1 ,θ2 are unifiers , but not MGUs σ = {f(y)/x}is a MGU, where
θ1=σ· {a/y}, λ= {a/y} θ2=σ· {b/y}, λ= {b/y}
Algorithms looking for a MGU Steps
Find out the difference sets of variables between two clauses
Identify the substitutions for each set
Ex1. F1= P(x,y,z) F2= P(x,f(a),h(b)) Difference set: D1={y,f(a)}, D2={z,h(b)}
MGU= {f(a)/ y, h(b)/z}
Ex2. F1= P(x), F2= P(y) Difference set: D={x,y} MGU={x/y} or {y/x}
Resolution Reasoning of Predicate Logic F1 and F2 are clauses, P1 in F1 and ~ P2 in F2 are
literals such that P1 and P2 have a MGU μ, then These two clauses have a resolvent ρ. The resolvent is obtained by applying the substitution μ to
the union of F1 and F2, leaving out the complementary literals.
Ρ=[(F1-{P1}) ∨(F1-{ ~ P2}) ] μ
Example Problem
Marcus is a man Caesar is a ruler A person trying to assassinate its ruler is not loyal Marcus want to assassinate Caesar
Step1 Convert to logic expression man (marcus) ruler (caesar) person(X) rule(Y) tryassassinate(X,Y)∧ ∧→~ loyalto(X,Y) tryassassinate (marcus, caesar) man(X) → person(X)
Show that Marcus is not loyal to Caesar
~loyalto(marcus,caesar)
Example cont. Step2, build standard form clauses (1) man(m)
(2) ruler(c)
(3) ~ person(X)∨ ~ ruler(Y)∨ ~ trya (X,Y)∨ ~ loyalto(X,Y)
(4) trya (m,c)
(5) ~ man (Z) person(Z)∨(6) loyalto(m,c)
Step3, resolution reasoning(7) ~ person(m)∨ ~ ruler(c)∨ ~ trya (m,c) (6) (3) {m/X, c/Y}
(8) ~ person(m)∨ ~ ruler(c) (7) (4)
(9) ~ person(m) (8) (2)
(10) ~ man (m) (9) (5) {m/Z}
(11) NIL (10)(1)
Example cont.
)m(man~
6 3
),(~)(~)(~ cmtryacrulermperson Yc,Xm
4
)c(ruler~)m(person~ 2
)m(person~ 5
Zm 1
NIL
Control Strategy (1) Problem
How to pick up clauses to resolve? Quick and completeness
Strategies1. Breadth-first resolution: completeness, time-
consuming2. Unit resolution: with priority with literals3. Input resolution: each resolvent at lease
involves a input clause
Control strategies (2)4. Linear resolution: Start from an initiate clause, then
resolve continue linearly.
(1) ~P ∨S
(2) P ∨Q
(3)~P ∨R
(4)~Q ∨R
(5)~R
(6)Q ∨R (2)(3)
(7)R (6)(4)
(8)NIL (7)(5)
The point is how to choose the initiate clause
Control strategies (3)5. Supported set resolution: for each resolvent, one of
clause must be in “supported set” The supported set includes negation of the conclusion and
its resolvents
(1) ~P ∨S
(2) P ∨Q
(3)~P ∨R
(4)~Q ∨R
(5)~R
R~ RP~
P~ QP
Q RQ~
R R~
NIL
Resolution for Particular Interrogative Sentence Answers for who , what , when , where , how
Known as: pompeian(x)->died(x,70)
pompeian(marcus)
Ask: died(marcus,?) Clause set:
(1) ~pompeian(x) ∨ died(x,70)
(2) pompeian(marcus)
(3) ~died(marcus,t) ∨died(marus,t)
(ever-true expression is added for conclusion)
Example cont.
3 1
txmarcus /79,
)79,marcus(died)marcus(pompeian~ 2
)79,marcus(died
(1) ~pompeian(x) ∨ died(x,70)
(2) pompeian(marcus)
(3) ~died(marcus,t) ∨died(marus,t)