Notes
Slide Show
Outline
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FOL and Prolog
  • First Order Logic
  • Chapter 8
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Outline
  • Why FOL?
  • Syntax and semantics of FOL
  • Using FOL
  • Wumpus world in FOL
  • Knowledge engineering in FOL
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Pros and cons of propositional logic
  • J Propositional logic is declarative
  • J Propositional logic allows partial/disjunctive/negated information
    • (unlike most data structures and databases)
  • J Propositional logic is compositional:
    • meaning of B1,1 Ù P1,2 is derived from meaning of B1,1 and of P1,2
  • J Meaning in propositional logic is context-independent
    • (unlike natural language, where meaning depends on context)
  • L 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
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First-order logic
  • Whereas propositional logic assumes the world contains 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, …
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Syntax of FOL: Basic elements
  • Constants KingJohn, 2, NUS,...
  • Predicates Brother, >,...
  • Functions Sqrt, LeftLegOf,...
  • Variables x, y, a, b,...
  • Connectives Ø, Þ, Ù, Ú, Û
  • Equality =
  • Quantifiers  ", $
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Atomic sentences
  • Atomic sentence = predicate (term1,...,termn) or term1 = term2


  • Term            = function (term1,...,termn) or constant or variable


  • E.g.,
  • Brother(KingJohn,RichardTheLionheart)
  • Length(LeftLegOf(Richard)) = Length(LeftLegOf(KingJohn))
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Complex sentences
  • Complex sentences are made from atomic sentences using connectives
  • ØS, S1 Ù S2, S1 Ú S2, S1 Þ S2, S1 Û S2,


  • E.g. Sibling(KingJohn,Richard) Þ Sibling(Richard,KingJohn)
  •      >(1,2) Ú ≤ (1,2)
  •      >(1,2) Ù Ø >(1,2)
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Truth 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 referents for
    • constant symbols → objects
    • predicate symbols → relations
    • function symbols → functional relations


  • An atomic sentence predicate(term1,...,termn) is true
  • iff the objects referred to by term1,...,termn
  • are in the relation referred to by predicate
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Models for FOL: Example
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Universal quantification
  • "<variables> <sentence>


  • Everyone at NUS is smart:
  • "x At(x,NUS) Þ Smart(x)


  • "x P is true in a model m iff P is true with x being each possible object in the model


  • Roughly speaking, equivalent to the conjunction of instantiations of P
      • At(KingJohn,NUS) Þ Smart(KingJohn)
      • Ù At(Richard,NUS) Þ  Smart(Richard)
      • Ù At(NUS,NUS) Þ Smart(NUS)
      • Ù ...
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A common mistake to avoid
  • Typically, Þ is the main connective with "
  • Common mistake: using Ù as the main connective with ":
    • "x At(x,NUS) Ù Smart(x)
    • means “Everyone is at NUS and everyone is smart”
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Existential quantification
  • $<variables> <sentence>


  • Someone at NUS is smart:
  • $x At(x,NUS) Ù Smart(x)


  • $x P is true in a model m iff P is true with x being some possible object in the model


  • Roughly speaking, equivalent to the disjunction of instantiations of P
    • At(KingJohn,NUS) Ù Smart(KingJohn)
    • Ú At(Richard,NUS) Ù Smart(Richard)
    • Ú At(NUS,NUS) Ù Smart(NUS)
    • Ú ...
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A common mistake to avoid (2)
  • Typically, Ù is the main connective with $


  • Common mistake: using Þ as the main connective with $:
  • $x At(x,NUS) Þ Smart(x)
  • is true if there is anyone who is not at NUS!
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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 "y Loves(x,y)
    • “There is a person who loves everyone in the world”
  • "y $x Loves(x,y)
    • “Everyone in the world is loved by at least one person”

  • Quantifier duality: each can be expressed using the other
  • "x Likes(x,IceCream) Ø$x ØLikes(x,IceCream)
  • $x Likes(x,Broccoli) Ø"x ØLikes(x,Broccoli)
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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) Û [Ø(x = y) Ù  $m,f Ø (m = f) Ù Parent(m,x) Ù Parent(f,x) Ù Parent(m,y) Ù  Parent(f,y)]
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Using FOL
  • The kinship domain:
  • Brothers are siblings
    • "x,y Brother(x,y) Þ Sibling(x,y)
  • One's mother is one's female parent
    • "m,c Mother(c) = m Û (Female(m) Ù Parent(m,c))
  • “Sibling” is symmetric
    • "x,y Sibling(x,y) Û Sibling(y,x)
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Using FOL
  • The set domain:
  • "s Set(s) Û (s = {} ) Ú ($x,s2 Set(s2) Ù s = {x|s2})
  • Ø$x,s {x|s} = {}
  • "x,s x Î s Û s = {x|s}
  • "x,s x Î s Û [ $y,s2} (s = {y|s2} Ù (x = y Ú x Î s2))]
  • "s1,s2 s1 Í s2 Û ("x x Î s1 Þ x Î s2)
  • "s1,s2 (s1 = s2) Û (s1 Í s2 Ù s2 Í s1)
  • "x,s1,s2 x Î (s1 Ç s2) Û (x Î s1 Ù x Î s2)
  • "x,s1,s2 x Î (s1 È s2) Û (x Î s1 Ú x Î s2)
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Interacting with FOL KBs
  • Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:
    • Tell(KB,Percept([Smell,Breeze,None],5))
    • Ask(KB,$a BestAction(a,5))


  • I.e., does the KB entail some best action at t=5?


  • Answer: Yes, {a/Shoot}  ← substitution (binding list)
  • Given a sentence S and a substitution σ,
  • Sσ denotes the result of plugging σ into S; e.g.,
    • S = Smarter(x,y)
    • σ = {x/Hillary,y/Bill}
    • Sσ = Smarter(Hillary,Bill)

  • Ask(KB,S) returns some/all σ such that KB╞ σ
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KB for the wumpus world
  • Perception
    • "t,s,b Percept([s,b,Glitter],t) Þ Glitter(t)


  • Reflex
    • "t Glitter(t) Þ BestAction(Grab,t)
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Deducing hidden properties
  • "x,y,a,b Adjacent([x,y],[a,b]) Û
  • [a,b] Î {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}


  • Properties of squares:
  • "s,t At(Agent,s,t) Ù Breeze(t) Þ Breezy(s)
  • Squares are breezy near a pit:
    • Diagnostic rule - infer cause from effect
      • "s Breezy(s) Þ $r Adjacent(r,s) Ù Pit(r)
    • Causal rule - infer effect from cause
      • "r Pit(r) Þ ["s Adjacent(r,s) Þ Breezy(s) ]
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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
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The electronic circuits domain
  • One-bit full adder
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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 vocabulary
    • Alternatives:
      • Type(X1) = XOR
      • Type(X1, XOR)
      • XOR(X1)
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The electronic circuits domain
  • Encode general knowledge of the domain
    • "t1,t2 Connected(t1, t2) Þ Signal(t1) = Signal(t2)
    • "t Signal(t) = 1 Ú Signal(t) = 0
    • 1 ≠ 0
    • "t1,t2 Connected(t1, t2) Þ Connected(t2, t1)
    • "g Type(g) = OR Þ Signal(Out(1,g)) = 1 Û $n Signal(In(n,g)) = 1
    • "g Type(g) = AND Þ Signal(Out(1,g)) = 0 Û $n Signal(In(n,g)) = 0
    • "g Type(g) = XOR Þ Signal(Out(1,g)) = 1 Û Signal(In(1,g)) ≠ Signal(In(2,g))
    • "g Type(g) = NOT Þ Signal(Out(1,g)) ≠ Signal(In(1,g))
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The electronic circuits domain
  • Encode the specific problem instance
    • Type(X1) = XOR Type(X2) = XOR
    • Type(A1) = AND Type(A2) = AND
    • Type(O1) = OR


    • Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))
    • Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(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))
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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 Ù Signal(In(2,C1)) = i2 Ù Signal(In(3,C1)) = i3 Ù Signal(Out(1,C1)) = o1 Ù Signal(Out(2,C1)) = o2


  • Debug the knowledge base
    • May have omitted assertions like 1 ≠ 0
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Summary
  • First-order logic:
    • objects and relations are semantic primitives
    • syntax: constants, functions, predicates, equality, quantifiers


  • Increased expressive power: sufficient to define wumpus world
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PROgramming in LOGic
  • A crash course in Prolog


  • Slides edited from William Clocksin’s versions at Cambridge Univ.
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What is Logic Programming?
  • A type of programming consisting of facts and relationships from which the programming language can draw a conclusion.
    • In imperative programming languages, we tell the computer what to do by programming the procedure by which program states and variables are modified.
    • In contrast, in logical programming, we don’t tell the computer exactly what it should do (i.e., how to derive a conclusion). User-provided facts and relationships allow it to derive answers via logical inference.


  • Prolog is the most widely used logic programming language.
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Prolog Features
  • Prolog uses logical variables. These are not the same as variables in other languages. Programmers can use them as ‘holes’ in data structures that are gradually filled in as computation proceeds.
  • Unification is a built-in term-manipulation method that passes parameters, returns results, selects and constructs data structures.
  • Basic control flow model is backtracking.
  • Program clauses and data have the same form.
    • A Prolog program can also be seen as a relational database containing rules as well as facts.
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Example: Concatenate lists a and b
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Outline
  • General Syntax
  • Terms
  • Operators
  • Rules
  • Queries
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Syntax
  • .pl files contain lists of clauses
  • Clauses can be either facts or rules



  • male(bob).
  • male(harry).
  • child(bob,harry).
  • son(X,Y):-
  • male(X),child(X,Y).
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Complete Syntax of Terms
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Compound Terms
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Examples of operator properties
  • Prolog has shortcuts in notation for certain operators (especially arithmetic ones)


  • Position Operator Syntax Normal Syntax
  • Prefix: -2 -(2)
  • Infix: 5+17 +(17,5)


  • Associativity: left, right, none.
  •  X+Y+Z    is parsed as  (X+Y)+Z
  • because addition is left-associative.


  • Precedence: an integer.
  • X+Y*Z   is parsed as  X+(Y*Z)
  • because multiplication has higher precedence.
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Rules
  • Rules combine facts to increase knowledge of the system


  • son(X,Y):-
  • male(X),child(X,Y).


  • X is a son of Y if X is male and
    X is a child of Y
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Interpretation of Rules
  • Rules can be given a declarative reading or a procedural reading.
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Queries
  • Prolog is interactive; you load a KB and then ask queries
  • Composed at the ?- prompt
  • Returns values of bound variables and yes or no


  • ?- son(bob, harry).
  • yes
  • ?- king(bob, france).
  • no
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Another example
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Quantifiers
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Points to consider
  • Variables are bound by Prolog, not by the programmer
    • You can’t assign a value to a variable.


  • Successive user prompts  ; cause the interpreter to return all terms that can be substituted for X.
    • They are returned in the order found.
    • Order is important


  •  PROLOG adopts the closed-world assumption:
    • All knowledge of the world is present in the database.
    • If a term is not in the database assume is false.
    • Prolog’s ‘yes’ = I can prove it, ‘no’ = I can’t prove it.
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Queries
  • Can bind answers to questions to variables
  • Who is bob the son of? (X=harry)
  • ?- son(bob, X).
  • Who is male? (X=bob, harry)
  • ?- male(X).
  • Is bob the son of someone? (yes)
  • ?- son(bob, _).
    • No variables bound in this case!
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Lists
  • The first element of a list can be separated from the tail
  • using operator |


  • Example:


  • Match the list [tom,dick,harry,fred] to


  • [X|Y]          then X = tom and Y = [dick,harry,fred]
  • [X,Y|Z]       then X = tom, Y = dick, and Z = [harry,fred]
  • [V,W,X,Y,Z|U] will not match
  • [tom,X|[harry,fred]] gives X = dick
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Example: List Membership
  • We want to write a function member that works as follows:


  • ?- member(a,[a,b,c,d,e])
  • yes
  • ?- member(a,[1,2,3,4])
  • no
  • ?- member(X,[a,b,c])
  • X = a
  • ;
  • X = b
  • ;
  • X = c
  • ;
  • no
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Function Membership Solution
  • Define two predicates:


  • member(X,[X|T]).
  • member(X,[Y|T]) :- member(X,T).


  • A more elegant definition uses anonymous variables:


  • member(X,[X,_]).
  • member(X,[_|T]) :- member(X,T).


  • Again, the symbol _ indicates that the contents of that variable is unimportant.
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Notes on running Prolog
  • You will often want to load a KB on invocation of Prolog
  • Use “consult(‘mykb.pl’).” at the “?-” prompt.
  • Or add it on the command line as a standard input “pl < mykb.pl”


  • If you want to modify facts once Prolog is invoked:
  • Use “assert(p).”
  • Or “retract(p).” to remove a fact
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Prolog Summary
  • A Prolog program is a set of specifications in FOL. The specification is known as the database of the system.
  • Prolog is an interactive language (the user enters queries in response to a prompt).
  • PROLOG adopts the closed-world assumption


  • How does Prolog find the answer(s)?  We return to this next week in Inference in FOL