Formal Logic

1
Formal Logic

• Representation Inference
• As a knowledge representation mechanism
• As state space search in And/Or graphs
• As a formalism for heuristic rules
• Propositional Calculus
• propositions, connectives
• symbols, sentences, well formed formula
• Predicate Calculus
• predicates, constants, variables, functions,
• clauses, connectives, qualifiers

2
Propositional Logic

• Propositions are statements about the world
• Propositions can either be true or false
• Simple propositions copper is a metal true
• wood is a metal false
• viviane is nice true
• Logical connectives
• And ? conjunction
• Or V disjunction
• Not negation
• If ... Then ... ? implication
• ? equivalence

3
Propositional Calculus

• The symbols of the propositional calculus are
• Propositional symbols P, Q, R, ...
• Thruth symbols true, false
• Connectives Not, And, Or, ?, ?
• Well Formed Formula or Legal Sentences are
  defined as
• A simple proposition is a formula
• If P is a formula then Not (P) is a formula
• If P and Q are formula then
• (P And Q) (P Or Q) (P ? Q) (P?Q)
• are formula

4
Propositional Calculus

• An Interpretation of a set of propositions is
  the assignment of a truth value (T or F) to each
  propositional symbol
• The interpretation or thruth value of a sentence
  is defined by the Thruth Tables for the
  connectives

5
Proving Equivalences

• Not(Not(P))?P
• (POrQ)?(Not(P)gtQ)
• (PgtQ)?(Not(Q)gtNot(P)) contraposition
• Not(POrQ)?(Not(P)AndNot(Q)) de Morgan
• Not(PAndQ)?(Not(P)OrNot(Q))
• (PAndQ)?(QAndP) commutative
• ((PAndQ)AndR)?(PAnd(QAndR)) associative
• (POr(QAndR))?(POrQ)And(POrR) distributive

6
A Proof (Not(P)OrQ)?(PgtQ)
7
Legal Inference

• Modus Ponens If P Then Q
• P Q
• If (Bill has cancer) Then (Bill feels bad)
• Bill has cancer)
  
• Bill feels bad
• ModusTollens If P Then Q
• Not Q
• Not P
• If (Bill has cancer) Then (Bill feels bad)
• Not (Bill feels bad)
  
• Not (Bill has cancer)

8
Illegal inference

• Abduction If P Then Q
• !!! not legal Q
• offers explanation P
• If (Bill has cancer) Then (Bill feels bad)
• Bill feels bad
  
• Bill has cancer

9
Proving Assertions

• today is Saturday P
• the current season is fall Q
• If (today is Saturday) If (P And Q)
• And (the current season is fall) Then R
• Then (there is a football game)
• there is a football game R is proven

10
Predicate Calculus Symbols

• Symbols
• Thruth Symbols true, false
• Constant Symbols 5, pipe-1, helen
• Variable Symbols X, Person, Day
• Function Symbols sin, father !arity
• Terms helen saturday
• Person X
• times(2,5) father(helen)
• times(X,5) father(Person)
• Predicate Symbols likes, part-of, free !arity

11
Predicate Calculus Sentences

• Propositions or Atomic Sentences
  likes(helen,bart) likes(helen,father(bart))
• likes(Person,bart) likes(helen,father(X))
• on (block-1,block-2) on (X,Y)
• Connectives Not, And, Or, ?, ?
• Variable Quantifiers ???universal quantification
• ? existencial quantification
• Sentences
• likes(helen,bart) likes(bart,(father(helen)))
• ??X likes(X,ice-cream) ?Y likes(Y,bart)
• ?X ?Y likes(X,Y) ? likes(X,(father(Y)))

12
Predicate Calculus Semantics

• An Interpretation of a set of predicate calculus
  sentences is an assignment of the entities in a
  domain of discourse to each constant, variable,
  predicate, and function symbol
• The thruth value of an atomic sentence is
  determined by the interpretation, for non-atomic
  sentences the Thruth Tables are used for the
  connectives, and in addition
• the value of ??X ltsentencegt is T if ltsentencegt
  is T for all possible assignments to X
• the value of ??X ltsentencegt is T if there is an
  assignment to X for which ltsentencegt is T

13
First Order Predicate Calculus

• First Order Predicate Calculus allows quantified
  variables to refer to objects in the domain of
  discourse and NOT to predicates or functions
• If it does not rain tomorrow, tom will go to the
  sea.
• weather(rain,tomorrow)?go(tom,sea)
• All basketball players are tall.
• ?X (basketball-player(X)?tall(X))
• Some people like anchovies.
• ?X (person(X) ? likes(X,anchovies))
• Nobody likes taxes.
• ?X(likes(X,taxes))
• Most natural language sentences can be written
  as first order predicate calculus sentences

14
Predicate Calculus Example

• Given mother(eve,abel)
• mother(eve,cain)
• father(adam,abel)
• father(adam,cain)
• ? X ? Y father(X,Y) V mother(X,Y) ??parent(X,Y)
• ? X ?Y ??Z parent(Z,X) ? parent(Z,Y) ?
  sibling(X,Y)
• Inferred parent(eve,abel)
• parent(eve,cain)
• parent(adam,abel)
• parent(adam,cain)
• sibling(abel,cain)
• sibling(cain,abel)
• sibling(cain,cain) !!! logic is all form
• sibling(abel,abel) no meaning

15
Inference Rules

• Modus Ponens P ? Q
• P
• Q
• Modus Tolens P ? Q
• Not(Q)
• Not(P)
• Universal ?X P(X)
• Instantiation a is from the domain of X
• P(a)

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Inference Example

• Given (1) ?X man(X)?mortal(X)
• (2) man(bill)
• Inferred
• (3) man(bill)?mortal(bill) from (1) and (2) with
  UI
• (4) mortal(bill) from (3) and (2) with MP

17
Pattern Matching and Unification

• Unification is an algorithm for determining the
  substitutions needed to make expressions match
• Example match foo(X,a,goo(Y)) with
• foo(X,b,foo(Y)) no match
• foo(X,Y) no match
• moo(X,a,goo(Y)) no match
• foo(fred,a,goo(Z)) fred/X, Z/Y
• foo(W,a,goo(jack)) W/X, jack/Y)
• foo(Z,a,goo(moo(Z))) Z/X, moo(Z)/Y

18
Pattern Matching Algorithm

• (1) Constant/Constant Matches only when
  identical
• (2) Constant/Variable Matches
• a. unbound variable variable becomes bound to
  constant
• b. bound variable refer to (1)
• (3) Variable/Variable Matches
• a. two unbound variables always match !also in
  future
• b. bound/unbound variable refer to (2) a.
• c. two bound variables refer to (1)
• (4) Expression/Expression Matches
• only when function or predicate names and
  arities identical
• then match arguments one by one
• The scope of a variable is one sentence. Once a
  variable is bound, future unifications and
  inferences must take this binding into account

19
Logic Inference as State Space Search

• A set of proposition or predicate calculus
  sentences form a AND/OR or Hypergraph
• And nodes indicate problem decomposition, all
  subproblems must be solved (proved true)
• Or nodes indicate alternative problem-solving
  strategies, solving one (prove it to be true) is
  enough
• Both goal and data directed search is possible
• To implement it, extra bookkeeping in the
  algorithm is necessary

20
And/Or Graphs

• a
• b
• c
• a ? b ? d
• a ? c ? e
• b ? d ? f
• f ? g
• a ? e ? h
• b V z ? y
• y ? z ? g

g
f
h
x
y
d
e
a
z
b
c
21
The Dog Example

• Fred is a collie collie(fred)
• Sam is Fred's master master(fred,Sam)
• the day is saterday day(Saturday)
• it is cold on Saturday warm(Saturday)
• Fred is trained trained(Fred)
• Spaniels are good-dogs and so are trained collies
• ?X (spaniel(X) V (collie(X) ? trained(X))
  ?gooddog(X)
• A good dog will be with his master
• ?X,Y,Z (gooddog(X) ? master(X,Y) ? location (Y,Z)
  ? location(X,Z))
• If it is Saturday and warm then Sam is at the
  park
• day(Saturday) ??warm(Saturday) ?
  location(Sam,park)
• If it is Saturday and not warm then Sam is at the
  museum
• day(Saturday) ??warm(Saturday) ?
  location(Sam,museum)

22
The Financial Advisor

• individuals with inadequate savings should
  increase the amount saved, regardless their
  income
• individuals with adequate savings and adequate
  income should consider investment in stock market
• individuals with lower income but with adequate
  savings should consider to split their surplus
  income between savings and stock
• with
• adequate savings 5000 per dependent
• minsavings(X) 5000 X
• adequate income 15000 4000 per dependent
• minincome(X) 15000 (4000 X)

23
A Logic System for the Financial Advisor

• savings_account(inadequate) ? investment(savings)
• savings_account(adequate) ? income(adequate) ?
  investment(stock)
• savings_account(adequate) ? income(inadequate) ?
  investment(combination)
• ?X amount_saved(X) ? ?Y (dependents(Y) ?
  greater(X,minsavings(Y))) ? savings_account(adequa
  te)
• ?X amount_saved(X) ? ?Y (dependents(Y) ?
  greater(X,minsavings(Y))) ? savings_account(inade
  quate)
• ?X earnings(X, steady) ? ?Y (dependents(Y) ?
  greater(X,minincome(Y))) ? income(adequate)
• ?X earnings(X, steady) ? ?Y (dependents(Y) ?
  greater(X,minincome(Y))) ? income(inadequate)
• ?X earnings(X, unsteady) ? income(inadequate)
• amount_saved(22000)
• earnings(25000,steady)
• dependents(3)

24
Alternative Styles for Rules

• P ? Q Q ? P
• If premise Then conclusion Q If P
• If condition Then action Q - P
• If antecedent Then consequent Q When P
• Sometimes constrained to
• no disjunction in the premise
• no negation in the conclusion
• no disjunction in the conclusion
• When variable quantifiers are omitted
• variables that appear on both sides of the rule
  are universally quantified.
• variables that appear in the premise of the rule
  only are existentially quantified

25
Prolog A family example

• parent(michel, nina)
• No.1 yes No more solutions
• parent(michel, X)
• No.1 X emma
• No.2 X nina No more solutions
• parent(X, nina)
• No.1 X viviane
• No.2 X michel No more solutions
• parent(X, Y)
• No.1 X viviane, Y nina
• No.2 X viviane, Y emma
• No.3 X maria, Y michel
• No.4 X julia, Y viviane
• No.5 X michel, Y emma
• No.6 X michel, Y nina
• No.7 X jozef, Y michel

• parent( X, Y)- mother( X, Y).
• parent( X, Y)- father( X, Y).
• mother( viviane, nina).
• mother( viviane, emma).
• father( michel, emma).
• father( michel, nina).
• father( jozef, michel).
• mother( maria, michel).
• father( jules, viviane).
• mother( julia, viviane).

26
Prolog A family example (2)

• grandparent(jozef, Y)
• No.1 Y emma
• No.2 Y nina
• No more solutions
• grandparent(X, nina)
• No.1 X maria
• No.2 X julia
• No.3 X jozef
• No.4 X jules
• No more solutions

• parent( X, Y)- mother( X, Y).
• parent( X, Y)- father( X, Y).
• mother( viviane, nina).
• mother( viviane, emma).
• father( michel, emma).
• father( michel, nina).
• father( jozef, michel).
• mother( maria, michel).
• father( jules, viviane).
• mother( julia, viviane).
• grandparent( X, Y)- parent( X, Z), parent( Z,
  Y).

27
Prolog The dog example

• location(fred, L)
• No.1 L park
• No more solutions
• location(billie, L)
• No.1 L home
• No more solutions

• collie( fred).
• master( fred, sam).
• today( saturday).
• warm( saturday).
• trained( fred).
• spaniel( billie).
• master( billie, viviane).
• location( viviane, home).
• location( sam, park)- today( saturday), warm(
  saturday).
• location( sam, museum)- today( saturday), not
  warm( saturday).
• gooddog( X)- collie( X), trained( X).
• gooddog( X)- spaniel( X).
• location( D, P)- gooddog( D), master( D, X),
  location( X, P).

28
Prolog The advisor example

• investment(savings)- saving_account(inadequate).
• investment(stock)- saving_account(adequate),
  income(adequate).
• investment(combination)- saving_account(adequate)
  ,
• 
  income(inadequate).
• income(inadequate)- earnings( X, unsteady).
• income(adequate)- earnings( X, steady),
  dependents( Y),
• X gt (15000
  (4000 Y)).
• income(inadequate)- earnings( X, steady),
  dependents( Y),
• X lt (15000
  (4000 Y)).
• saving_account(adequate)- amount_saved( X),
  dependents( Y),
• X gt
  (5000 Y).
• saving_account(inadequate)- amount_saved( X),
  dependents( Y),
• 
  X lt (5000 Y).

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Prolog The advisor example (cont)

• amount_saved(22000).
• earnings(25000, steady).
• dependents(3).
• amount_saved(22000).
• earnings(25000, steady).
• dependents(8).
• amount_saved(22000).
• earnings(30000, steady).
• dependents(2).

• investment(X)
• No.1 X combination
• No more solutions
• investment(X)
• No.1 X savings
• No more solutions
• investment(X)
• No.1 X stock
• No more solutions

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Control and Implementation of State Space Search

• General (Abstract) Search Algorithms
• Pattern Directed Search
• implements search in And/Or graphs
• separates problem solving knowledge from
  implementation and control
• Production Systems
• implements search
• models human problem solving
• separates knowledge and control
• separates general static problem solving
  knowledge from case data in working memory