1 / 20

COMP313A Programming Languages

COMP313A Programming Languages. Logic Programming (1). Lecture Outline. Conceptual foundations of Logic Programming The Basics of Logic Programming Predicate Calculus A little bit of logic programming Prolog. Conceptual Foundations. What versus how specification versus implementation

min
Download Presentation

COMP313A Programming Languages

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 <= 0{ return (-1); } high = size; low = 0; for(;;) { mid = (high + low) / 2; if (mid = low){ return (val != array[low]) ?-1:mid; } if (val < array[mid]) { high = mid; } else if (val > array[mid]) { 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) -> 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) -> 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) -> 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) ÞMammal(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 $and " • Negation Everyone dislikes rugbyº Noone likes rugby "x ØLikes (x, rugby) ºØ$x Likes(x, rugby) Everyone likes icecream º Noone dislikes icecream "x Likes (x, icecream) ºØ$x ØLikes(x, icecream)

  14. First-Order Predicate Calculus cont… • " is a conjunction over the universe of objects • $Is a disjunction over the universe of objects "x ØP ºØ$x P Ø "x P º$x ØP "x P ºØ$x ØP Ø "x ØP º$x P

  15. De Morgan’s Laws ØPÙØQ º (PÚQ) Ø(PÙQ) ºØPÚØQ PÙQ ºØ(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 isn’t 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

More Related