COMP313A Programming Languages

1
COMP313A Programming Languages

• Logic Programming (1)

2
Lecture Outline

• Conceptual foundations of Logic Programming
• The Basics of Logic Programming
• Predicate Calculus
• A little bit of logic programming
• Prolog

3
Conceptual Foundations

• What versus how
• specification versus implementation
• Declarative Programming
• Programmer declares the logical properties that
  describe the property to be solved
• From this a solution is inferred
• Inference engine

4
An example
Searching for an element in a list Predicate
is_in(x,L) true whenever element x is in the list
L.
For all elements x and lists L is_in(x,L) IFF L
x or L L1 . L2
and (is_in (x,L1) or is_in(x,
L2))
5
Example continuedImplementation

• Need to know how to split a list into right and
  left sublists
• How to order the elements stored in the list

6
A solution in C
int binary_search(const int val, const int size,
const int array) int high, low, mid if size
lt 0 return (-1) high size low
0 for() mid (high low) / 2 if
(mid low) return (val ! arraylow)
?-1mid if (val lt arraymid) high
mid else if (val gt arraymid) low
mid else return mid
7
A Declarative Solution

• Given an element x and a list L, to prove that x
  is in L,
• proceed as follows
• Prove that L is x
• Otherwise split L into L1 . L2 and prove one of
  the following
• (2.1) x is in L1, or
• (2.2) x is in L2

8
A sorting example
A predicate sort(X,Y) Sort(X,Y) is true if the
nonempty list Y is the sorted version of X Use
two auxiliary predicates permutation(X,Y) and
is_sorted(Y)
For all integer lists X,Y sort(X,Y) iff
permutation(X,Y) and sorted(Y)
9
Logic and Logic Programming

• First-order predicate calculus
• Logic statements
• Examples
• John is a man. man(John).
• John is a human. human(John).
• Tristan is the son of Margaret.
  son(Margaret,Tristan).
• A horse is a mammal. loathes(Margaret,
  Heavy_Metal).
• 0 is a natural number . natural(0).
• Mammals have four legs and no arms or two legs
  and two arms.
• For all X, mammal (x) -gt legs(x,4) and arms(x,0)
  or legs(x,2) and arms(x,2).
• Humans have two legs and two arms.
• For all X, human(x) -gt legs(x,2) and arms(x,2).
• If x is a natural number then so is the successor
  of x.
• For all x, natural(x) -gt natural(successor(x)).

10
First-Order Predicate Calculus

• Constants
• Predicates
• Functions
• Variables that stand for as yet unamed quantities
• Atomic sentences
• Connectives construct more complex sentences
• Quantifiers
• Punctuation
• Arguments to predicates can only be terms
  variables, constants and functions

11
First-Order Predicate Calculus cont

• Quanitifiers
• Universal, existential
• Express properties of entire collections of
  objects
• Universal quantifiers make statements about every
  object, "x
• A cat is a mammal
• "x Cat(x) Þ Mammal(x)
• Cat(Spot) Þ Mammal(Spot) Ù
• Cat(Rebecca) Þ Mammal(Rebecca) Ù
• Cat(Felix) Þ Mammal(Felix) Ù
• Cat(Richard) Þ Mammal(Richard) Ù
• Cat(John) Þ Mammal(John) Ù

12
First-Order Predicate Calculus cont

• Existential Quantifiers make statements about
  some objects, x
• Spot has a sister who is a cat
• x Sister(x, Spot) Ù Cat(x)
• (Sister(Spot, Spot) Ù Cat(Spot)) Ú
• (Sister(Rebecca, Spot) Ù Cat(Rebecca)) Ú
• (Sister(Felix, Spot) Ù Cat(Felix)) Ú
• (Sister(Richard, Spot) Ù Cat(Richard)) Ú
• (Sister(John, Spot) Ù Cat(John)) Ú

13
First-Order Predicate Calculus cont

• Connections between and "
• Negation
• Everyone dislikes rugby º Noone likes rugby
• "x ØLikes (x, rugby) º Øx Likes(x, rugby)
• Everyone likes icecream º Noone dislikes icecream
• "x Likes (x, icecream) º Øx ØLikes(x, icecream)

14
First-Order Predicate Calculus cont

• " is a conjunction over the universe of objects
• Is a disjunction over the universe of objects
• "x ØP º Øx P
• Ø "x P º x ØP
• "x P º Øx ØP
• Ø "x ØP º x P

15
De Morgans Laws

• ØPÙØQ º (PÚQ)
• Ø(PÙQ) º ØPÚØQ
• PÙQ º Ø(ØPÚØQ)
• Ø(ØPÙØQ) º PÚ Q

16
Using First-Order Predicate Calculus

• Marcus was a man
• Marcus was a Pompeian
• All Pompeians were Romans
• Caesar was a ruler
• All Romans were either loyal to Caesar or hated
  him

17

• Everyone is loyal to someone
• People only try to assassinate rulers they are
  not loyal to
• Marcus tried to assassinate Caesar
• Was Marcus loyal to Caesar?
• Prove Ø loyalto(Marcus, Caesar)

18

• Turn the following sentences into formulae in
  first order predicate logic
• John likes all kinds of food
• Apples are food
• Chicken is food
• Anything anyone eats and isnt killed by is food
• Bill eats peanuts and is still alive
• Sue eats everything Bill eats
• Prove that John likes peanuts using backward
  chaining

19
A little bit of Prolog

• Objects and relations between objects
• Facts and rules
• parent(pam, bob). parent(tom,bob).
• parent(tom, liz). parent(bob, ann).
• parent(bob, pat). parent(pat, jim).
• ? parent(bob, pat).
• ? parent(bob, liz).
• ? parent(bob, ben).
• ? parent(bob, X).
• ? parent(X, Y).

20
Prolog

• grandparent (X,Y) - parent(X, Z), parent(Z, Y).
• For all X and Y
• X is the grandparent of Y if
• X is a parent of Z and
• Z is a parent of Y
• sister (X,Y) - parent(Z, X), parent(Z, Y),
  female(X)
• For all X and Y
• X is the sister of Y if
• Z is the parent of both X and Y and
• X is a female