010'141 Engineering Mathematics II Lecture 20 The Predicate Calculus

1
010.141 Engineering Mathematics IILecture 20The
Predicate Calculus

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University

2
Outline

• Why First Order Logic?
• Syntax and semantics of First Order Logic
• Using First Order Logic
• Wumpus world in First Order Logic
• Knowledge engineering in First Order Logic

3
Pros and cons of propositional logic

• ? Propositional logic is declarative
• ? Propositional logic allows partial/
  disjunctive/ negated/ implicational information
• (unlike most data structures and databases)
• Propositional logic is compositional
• meaning of B1,1 ? P1,2 is derived from meaning of
  B1,1 and of P1,2
  
• ? Meaning in propositional logic is
  context-independent
  
• (unlike natural language, where meaning depends
  on context)
  
• ? Propositional logic has very limited expressive
  power
  
• (unlike natural language)
• E.g., cannot say "pits cause breezes in adjacent
  squares
• except by writing one sentence for each square

4
First-order logic

• Propositional logic assumes the world consists of
  facts
• First-order logic (like natural language) assumes
  the world contains
  
• Objects people, houses, numbers, colors,
  baseball games, wars,
  
• Relations red, round, prime, brother of, bigger
  than, part of, comes between,
• Functions father of, best friend, one more than,
  plus,
  

5
Syntax of FOL Basic elements

• Constants KingJohn, 2, SNU,...
• Predicates Brother, Sibling, gt,...
• (sibling means either brother or sister)
• Functions Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives ?, ?, ?, ?, ?
• Equality
• (can be treated as just a special predicate)
• Quantifiers ?, ?
• Note we follow standard mathematical
  terminology, so constants have capitals,
  variables are lower case
• This is the opposite convention to prolog

6
Atomic sentences

• Atomic sentence predicate (term1,...,termn)
  or term1 term2
• Term function (term1,...,termn)
  or constant or variable
• E.g.,
• Brother(KingJohn,RichardTheLionheart)
• gt (Length(LeftLegOf(Richard)), Length(LeftLegOf(Ki
  ngJohn)))

7
Complex sentences

• Complex sentences are made from atomic sentences
  using the same connectives as in the propositional
  calculus
• ?S, S1 ? S2, S1 ? S2, S1 ? S2, S1 ? S2,
• E.g.
• Sibling(KingJohn,Richard)?
• Sibling(Richard,KingJohn)
• gt(1,2) ? (1,2)
• gt(1,2) ? ? gt(1,2)
• (just as in propositional calculus, they dont
  have to be true!)

8
Models in first-order logic

• Sentences are true with respect to a model and an
  interpretation
• Model contains objects (domain elements) and
  relations among them
  
• Interpretation specifies meanings for
• constant symbols ? objects
• predicate symbols ? relations
• function symbols ? functional relationships
• An atomic sentence predicate(term1,...,termn) is
  true
• iff the objects referred to by term1,...,termn
• are in the relation referred to by predicate

9
Models for FOL Example

• Please dont rely on this (or Russel and Norvig)
  for accurate English history

10
Universal quantification

• ?ltvariablesgt ltsentencegt
• Says that ltsentencegt is true for all values of
  ltvariablesgt
• Everyone at SNU is smart
• ?x (At(x,SNU) ? Smart(x))
• Formally, ?x P is true in a model m iff for every
  possible value of x as an object in the model, P
  is true
• Roughly speaking, equivalent to the conjunction
  of instantiations of P
• (since the model might be infinite, it might be
  an infinite conjunction)
• At(KingJohn,SNU) ? Smart(KingJohn)
• ? At(Richard,SNU) ? Smart(Richard)
• ? At(SNU,SNU) ? Smart(SNU)
• ? ...

11
A common mistake to avoid

• English can be a little misleading
• Maybe Korean too?
• ? is the connective we most commonly use with ?
• Common mistake
• Using ? as the connective with ? when ? is
  intended
• ?x (At(x,SNU) ? Smart(x))
• means Everyone is at SNU and everyone is smart
• (even the people who arent at SNU)

12
Existential quantification

• ?ltvariablesgt ltsentencegt
• (there is a way of making sentence true by
  substituting values for variables)
• Someone at SNU is smart
• ?x At(x,SNU) ? Smart(x)
• Formally, ?x P is true in a model m iff P is true
  with x being some possible object in the model.
• Roughly speaking, it can be seen as equivalent to
  the disjunction of instantiations of P
  
• At(KingJohn,SNU) ? Smart(KingJohn)
• ? At(Richard,SNU) ? Smart(Richard)
• ? At(SNU,SNU) ? Smart(SNU)
• ? ...

13
Another common mistake to avoid

• Typically, ? is the commonest connective used
  with ?
• Common mistake
• using ? as the main connective with ?
• ?x At(x,SNU) ? Smart(x)
• is true if there is anyone, anywhere (not
  necessarily smart) who is not at SNU!
  

14
Properties of quantifiers

• ?x ?y is the same as ?y ?x
• ?x ?y is the same as ?y ?x
• ?x ?y is not the same as ?y ?x
• ?x ?t can_fool_at(x,t)
• Some people can always be fooled (same people
  all the time)
  
• ?t ?x can_fool_at(x,t)
• Always, there is someone who can be fooled
  (maybe different people)
  
• ?t ?x can_fool_at(x,t)
• Sometimes you can fool everyone
• ?x ?t can_fool_at(x,t)
• Everyone has a time when they can be fooled
• ??x ?t can_fool_at(x,t)
• You cant fool all of the people all of the time

15
Properties of quantifiers

• Quantifier duality each can be expressed using
  the other
  
• ?x Likes(x,IceCream)?? ??x ?Likes(x,IceCream)
• ?x Likes(x,Broccoli) ? ??x ?Likes(x,Broccoli)
• ? x ? t ?can_fool_at(x,t)
• Theres a person and a time where the person
  cant be fooled

16
Equality

• term1 term2 is true under a given
  interpretation if and only if term1 and term2
  refer to the same object
  
• E.g., definition of Sibling in terms of Parent
• ?x,y Sibling(x,y) ?
• ?(x y) ?
• ?m,f ? (m f) ? Parent(m,x) ? Parent(f,x) ?
• Parent(m,y) ? Parent(f,y)

17
Using First Order Logic kinship

• The kinship domain
• Brothers are siblings
• ?x,y Brother(x,y) ? Sibling(x,y)
• One's mother is one's female parent
• ?m,c (Mother(m,c) m) ? (Female(m) ?
  Parent(m,c))
  
• Sibling is symmetric
• ?x,y Sibling(x,y) ? Sibling(y,x)

18
Using First Order Logic Sets

• Defining Sets (Adjoin is a function that adds one
  thing to a set)
• ?s Set(s) ? ((s ) ? (?x,s2 Set(s2) ? (s
  Adjoin(x,s2))
  )
• ??x,s adjoin(x,s)
• The empty set cannot be made by adjoining to
  existing sets
• ?x,s x ? s ? s Adjoin(x,s)
• ?x,s x ? s ? ?y,s2 (s adjoin(y,s2) ? (x y ?
  x ? s2))
  
• ?s1,s2 s1 ? s2 ? (?x x ? s1 ? x ? s2)
• ?s1,s2 (s1 s2) ? (s1 ? s2 ? s2 ? s1)
• ?x,s1,s2 x ? (s1 ? s2) ? (x ? s1 ? x ? s2)
• ?x,s1,s2 x ? (s1 ? s2) ? (x ? s1 ? x ? s2)

19
Interacting with 1st Order Knowledge

• Answers to queries can now be more informative
• Suppose a wumpus-world agent is using First Order
  Logic (FOL) in a Knowledge Base (KB).
• It perceives a smell and a breeze (but no
  glitter) at t5
• Tell(KB,Percept(Smell,Breeze,None,5))
• Ask(KB,?a BestAction(a,5))
• I.e., does the KB entail some best action at t5?
  
• What we would like is an answer like
• Answer Yes, a/Shoot ? substitution (binding
  list)

20
Interacting with 1st Order Knowledge

• Given a sentence S and a substitution s,
• Ss denotes the result of plugging s into S e.g.,
• S Smarter(x,y)
• s x/Hillary,y/Bill
• Ss Smarter(Hillary,Bill)
• Ask(KB,S) returns some/all s such that KB Ss

21
Knowledge base for wumpus world

• Remember that the KB needs to handle different
  times
• Need to model both how to infer the state of the
  World, and what to do about it
• Perception
• ?t,s,b Percept(s,b,Glitter,t) ? Glitter(t)
• Reflex
• ?t Glitter(t) ? BestAction(Grab,t)
• This representation cant handle any history
  dependence
• So infinite loops are likely
• Requires the agent to build an internal model of
  the World

22
Situation Calculus

• The World is a sequence of situations
• snapshots, completely describing the World
• Effect axioms describe the effect of different
  actions
• ?x,s ? Holding(x, (Result(Release,s))
• If an agent releases something, then in the next
  situation, it is not holding it
• Frame axioms describe the actions which _dont_
  change particular predicates
• This way, we need to have axioms for every
  combination of state predicates and actions
• The Frame Problem
• Once seen as a key weakness of FOL

23
Successor State Axioms

• An alternative is to use successor state axioms
• For each predicate, describe the ways in which it
  can change, and the ways in which it stays the
  same
• ?a,x,s (Holding(x, (Result(a,s))
• ? (aGrab ? Present(x,s) ? Portable(x)
• v (Holding(x,s) ? a ? Release)
• Substantially solve the frame problem
• One successor state axiom is needed for each
• way in which Holding can become or remain true
• way it can become or remain false

24
Deducing hidden properties

• Describing next-to relationships
• ?x,y,a,b Adjacent(x,y,a,b) ?
• a,b ? x1,y, x-1,y,x,y1,x,y-1
• Properties of squares
• ?s,t At(Agent,s,t) ? Breeze(t) ? Breezy

25
Deducing hidden properties

• Squares are breezy near a pit
• Diagnostic rule---infer cause from effect
• ?s Breezy(s) ? ?r Adjacent(r,s) ? Pit(r)
• Often used to model the pragmatic knowledge of
  experts
• Causal rule---infer effect from cause
• ?r Pit(r) ? ?s Adjacent(r,s) ? Breezy(s)
• Model-based reasoning

26
Practical Difficulties with FOL

• Frame Problem
• Described previously
• Ameliorated by successor state axioms
• But still require two axioms for each state
  predicate
• Most reasoning is about what stays the same
• Completely avoiding the frame problem requires
• Either special purpose logics
• Higher order logics with special axioms
• Axioms specify which state predicates are
  affected by which action predicates

27
Practical Difficulties with FOL

• Qualification Problem
• Difficult to describe all the ways in which a
  particular action might fail to work
• Ramification Problem
• Difficult to describe all the consequences of a
  given action

28
Knowledge engineering in FOL

• Identify the task
• Assemble the relevant knowledge
• Decide on a vocabulary of predicates, functions,
  and constants
  
• Encode general knowledge about the domain
• Encode a description of the specific problem
  instance
  
• Pose queries to the inference procedure and get
  answers
  
• Debug the knowledge base

29
The electronic circuits domain

• One-bit full adder

30
The electronic circuits domain

• Identify the task
• Does the circuit actually add properly? (circuit
  verification)
  
• Assemble the relevant knowledge
• Composed of wires and gates Types of gates (AND,
  OR, XOR, NOT)
  
• Irrelevant size, shape, color, cost of gates
• Decide on a representation language
• Alternatives
• Type(X1) XOR
• Type(X1, XOR)
• XOR(X1)

31
The electronic circuits domain

• Encode general knowledge of the domain
• ?t1,t2 Connected(t1, t2) ? Signal(t1)
  Signal(t2)
• ?t Signal(t) 1 ? Signal(t) 0
• 1 ? 0
• ?t1,t2 Connected(t1, t2) ? Connected(t2, t1)
• ?g Type(g) OR ? Signal(Out(1,g)) 1 ? ?n
  Signal(In(n,g)) 1
  

32
The electronic circuits domain

• Encode general knowledge of the domain (cont.)
• ?g Type(g) AND ? Signal(Out(1,g)) 0 ? ?n
  Signal(In(n,g)) 0
  
• ?g Type(g) XOR ? Signal(Out(1,g)) 1 ?
  Signal(In(1,g)) ? Signal(In(2,g))
  
• ?g Type(g) NOT ? Signal(Out(1,g)) ?
  Signal(In(1,g))
  

33
The electronic circuits domain

• Encode the specific problem instance
• Type(X1) XOR Type(X2) XOR
• Type(A1) AND Type(A2) AND
• Type(O1) OR

34
The electronic circuits domain

• Encode the specific problem instance(cont.)
• Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),I
  n(1,X1))
• Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),I
  n(1,A1))
• Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),
  In(2,X1))
• Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),
  In(2,A1))
• Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1)
  ,In(2,X2))
• Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1)
  ,In(1,A2))

35
The electronic circuits domain

• Pose queries to the inference procedure
• What are the possible sets of values of all the
  terminals for the adder circuit?
  
• ?i1,i2,i3,o1,o2 Signal(In(1,C1)) i1 ?
  Signal(In(2,C1)) i2 ? Signal(In(3,C1)) i3
  ? Signal(Out(1,C1)) o1 ? Signal(Out(2,C1))
  o2
• Debug the knowledge base
• Its easy to accidentally leave out assertions
  like 1 ? 0
  

36
Summary

• First-order logic
• objects and relations are semantic primitives
• syntax constants, functions, predicates,
  equality, quantifiers
  
• Increased expressive power
• Sufficient to define wumpus world
• Practical difficulties in describing and
  reasoning about the World
• Especially about things that dont change or
  change systematically
• The World stays the same unless something
  specifically changes it
• Things that are attached to other things usually
  move with them
• A need for reasoning about default behaviours

37
Summary

• Why First Order Logic?
• Syntax and semantics of First Order Logic
• Using First Order Logic
• Wumpus world in First Order Logic
• Knowledge engineering in First Order Logic

38
?????