CS 363 Comparative Programming Languages

1. CS 363 Comparative Programming Languages Logic Programming Languages

2. Chapter 16 Topics • An Overview of Logic Programming • The Origins of Prolog • The Basic Elements of Prolog • Applications of Logic Programming CS 363 Spring 2005 GMU

3. Introduction • Logic (declarative) programming languages express programs in a form of symbolic logic • Runtime system uses a logical inferencing process to produce results • Declarative rather that procedural: • only specification of results are stated (not detailed procedures for producing them) CS 363 Spring 2005 GMU

4. Example: Sorting a List • Describe the characteristics of a sorted list, not the process of rearranging a list sort(old_list, new_list)  permute (old_list, new_list)  sorted (new_list) sorted (list)  for all j, 1 j < n, list(j)  list (j+1) CS 363 Spring 2005 GMU

5. Overview of Logic Programming • Declarative semantics • There is a simple way to determine the meaning of each statement • Simpler than the semantics of imperative languages • Programming is nonprocedural • Programs do not state now a result is to be computed, but rather the form of the result • Based on Propositional logic CS 363 Spring 2005 GMU

6. Propositional Logic A proposition is formed using the following rules: • Constants true and false are propositions • Variables (p,q,r,…) which have values true or false are propositions • Operators ^, v, , ≡, and ⌐ (conjunction, disjunction, implication, equilivance and negation) are used to form more complex propositions. p v q ^ r  ⌐ s v t CS 363 Spring 2005 GMU

7. Predicate Logic Expressions Include: • Propositions • Variables in domains such as integer, real • Boolean valued functions • Ex: prime(n), 0 <= x + y, • Quantifiers (forall, exists) CS 363 Spring 2005 GMU

8. Properties of Predicate Logic Expressions CS 363 Spring 2005 GMU

9. Horn Clauses A Horn clause has a head h (which is a predicate), and a body, which is a list of predicates p1, p2, … , pn. • Predicate p is true only if p1, p2, … , pn are all true. • Meaning: p1 ^ p2^ … ^ pn  h Many, but not all predicates can be translated into Horn clauses using predicate logic expression properties. CS 363 Spring 2005 GMU

10. The Origins of Prolog • University of Aix-Marseille • Natural language processing • University of Edinburgh • Automated theorem proving CS 363 Spring 2005 GMU

11. The Basic Elements of Prolog • Edinburgh Syntax • Term: a constant, variable, or structure • Constant: an atom or an integer • Atom: symbolic value of Prolog • Atom consists of either: • a string of letters, digits, and underscores beginning with a lowercase letter • a string of printable ASCII characters delimited by apostrophes CS 363 Spring 2005 GMU

12. The Basic Elements of Prolog • Variable: any string of letters, digits, and underscores beginning with an uppercase letter • Instantiation: binding of a variable to a value • Lasts only as long as it takes to satisfy one complete goal • Structure: represents atomic proposition functor(parameter list) CS 363 Spring 2005 GMU

13. Fact Statements • Used for the hypotheses (assumed information): • Ex: student(jonathan). sophomore(ben). brother(tyler, cj). CS 363 Spring 2005 GMU

14. Rule Statements • Give the rules of implication: • Consequence :- antecedent_expression • Right side: antecedent (if part) • May be single term or conjunction • Conjunction: multiple terms separated by logical AND operations (implied) • Left side: consequent (then part) • Must be single term CS 363 Spring 2005 GMU

15. Rule Statements Can use variables (universal objects) to generalize meaning: parent(X,Y):- mother(X,Y). parent(X,Y):- father(X,Y). sibling(X,Y):- mother(M,X), mother(M,Y),father(F,X), father(F,Y). CS 363 Spring 2005 GMU

16. Example mother(Susan,Emily). mother(Susan,Lucy). mother(Anne,Susan). mother(Anne,Mary). father(Jim,Emily). father(Jim,Lucy). father(Russell,Susan). father(Russell,Mary). parent(X,Y):-mother(X,Y). parent(X,y):-father(X,y). sibling(X,Y):-mother(M,X),mother(M,Y), father(F,X),father(F,Y). Facts Rules CS 363 Spring 2005 GMU

17. Example speed(ford,100). speed(chevy,105). speed(dodge,95). speed(volvo,80). time(ford,20). time(chevy,21). time(dodge,24). time(volvo,24). distance(X,Y) :- speed(X,Speed), time(X,Time), Y is Speed * Time. Facts Rules CS 363 Spring 2005 GMU

18. Goal Statements • For theorem proving, theorem is in form of proposition that we want system to prove or disprove – goal statement • Same format as facts student(james). • Conjunctive propositions and propositions with variables also legal goals father(X,joe). CS 363 Spring 2005 GMU

19. Example mother(Susan,Emily). mother(Susan,Lucy). mother(Anne,Susan). mother(Anne,Mary). father(Jim,Emily). father(Jim,Lucy). father(Russell,Susan). father(Russell,Mary). parent(X,Y):-mother(X,Y). parent(X,y):-father(X,y). sibling(X,Y):-mother(M,X),mother(M,Y), father(F,X),father(F,Y). mother(Anne,Mary). • T mother(Anne,Emily). • No father(Jim,X). • X = Emily, X = Lucy Facts Rules Goals (queries) CS 363 Spring 2005 GMU

20. Simple Arithmetic • Prolog supports integer variables and integer arithmetic • is operator: takes an arithmetic expression as right operand and variable as left operand • A is B / 10 + C • Not the same as an assignment statement! For example, k is k + 1 is not solvable. CS 363 Spring 2005 GMU

21. Example speed(ford,100). speed(chevy,105). speed(dodge,95). speed(volvo,80). time(ford,20). time(chevy,21). time(dodge,24). time(volvo,24). distance(X,Y) :- speed(X,Speed), time(X,Time), Y is Speed * Time. Distance(ford,S). > S = 2000 Facts Rules Goal CS 363 Spring 2005 GMU

22. Inferencing Process of Prolog • Queries are called goals • If a goal is a compound proposition, each of the facts is a subgoal • To prove a goal is true, must find a chain of inference rules and/or facts. For goal Q: B :- A C :- B … Q :- P • Process of proving a subgoal called matching, satisfying, or resolution CS 363 Spring 2005 GMU

23. Inferencing Process • Bottom-up resolution, forward chaining • Begin with facts and rules of database and attempt to find sequence that leads to goal • works well with a large set of possibly correct answers • Top-down resolution, backward chaining • begin with goal and attempt to find sequence that leads to set of facts in database • works well with a small set of possibly correct answers • Prolog implementations use backward chaining CS 363 Spring 2005 GMU

24. Inferencing Process • When goal has more than one subgoal, can use either • Depth-first search: find a complete proof for the first subgoal before working on others • Breadth-first search: work on all subgoals in parallel • Prolog uses depth-first search • Can be done with fewer computer resources CS 363 Spring 2005 GMU

25. Inferencing Process • With a goal with multiple subgoals, if fail to show truth of one of subgoals, reconsider previous subgoal to find an alternative solution: backtracking • Begin search where previous search left off • Can take lots of time and space because may find all possible proofs to every subgoal CS 363 Spring 2005 GMU

26. Example mother(Susan,Emily). mother(Susan,Lucy). mother(Anne,Susan). mother(Anne,Mary). father(Jim,Emily). father(Jim,Lucy). father(Russell,Susan). father(Russell,Mary). parent(X,Y):-mother(X,Y). parent(X,y):-father(X,y). sibling(X,Y):-mother(M,X),mother(M,Y), father(F,X),father(F,Y). sibling(Emily,Y). • Y = Lucy Sibling(Susan,Anne). > no Facts Rules Goals (queries) CS 363 Spring 2005 GMU

27. sibling(Emily,Y). mother(M,Emily),mother(M,Y),father(F,Emily), father(F,Y). After searching the db: mother(Susan,Emily),mother(Susan,Y), father(Jim,Emily),father(Jim,Y). Searching again: mother(Susan,Emily),mother(Susan,Lucy), father(Jim,Emily),father(Jim,Lucy) So Y = Lucy CS 363 Spring 2005 GMU

28. sibling(Susan,Anne). mother(M,Susan),mother(M,Anne), father(F,Susan),father(F,Anne). If we choose M = Anne, we can’t resolve the second mother clause (similar problem with father)  fail CS 363 Spring 2005 GMU

29. List Structures • Other basic data structure (besides atomic propositions we have already seen): list • List is a sequence of any number of elements • Elements can be atoms, atomic propositions, or other terms (including other lists) [apple, prune, grape, kumquat] [] (empty list) [X | Y] (head X and tail Y) CS 363 Spring 2005 GMU

30. Example • Definition of member function: member(Elem,[Elem | _ ). member(Elem,[ _ | List]) :- member (Elem,List). Anonymous variable CS 363 Spring 2005 GMU

31. member(a,[b,c,d]). member(a,[b,c,d]) member(a,[c,d]) member(a,[d]) member(a,[])  fails CS 363 Spring 2005 GMU

32. Example • Definition of append function: append([], List, List). append([Head | List_1], List_2, [Head | List_3]) :- append (List_1, List_2, List_3). CS 363 Spring 2005 GMU

33. How append works append([a,b],[c,d],X). append([a|b],[c,d],_). append([b],[c,d],_). append([],[c,d],[c,d]).  first rule! append([b],[c,d],[b,c,d]). append([a,b],[c,d],[a,b,c,d]). X = [a,b,c,d] CS 363 Spring 2005 GMU

34. Example • Definition of reverse function: reverse([], []). reverse([Head | Tail], List) :- reverse (Tail, Result), append (Result, [Head], List). CS 363 Spring 2005 GMU

35. Cut Operator (!) • Explicit control of backtracking • ! Is a goal that always succeeds but cannot be resatisified by backtracking • a,b,!,c,d – if a and b succeed, but c fails, the whole goal fails. CS 363 Spring 2005 GMU

36. Deficiencies of Prolog • Resolution Order control • Prolog always starts at the beginning of DB when searching • Prolog always starts trying to resolve the leftmost subgoal (depth first) •  rule/definition order can affect performance! CS 363 Spring 2005 GMU

37. Deficiencies of Prolog • Easy to write infinite loops (due to resolution order) • ancestor(X,X). • ancestor(X,Y) :- ancestor(Z,y),parent(X,Z). Reversing rules fixes the problem CS 363 Spring 2005 GMU

38. Deficiencies of Prolog • Closed World assumption – Prolog can only prove things to be true (based on the info) but it cannot be used to prove a goal is false • Leads to a problem with negation CS 363 Spring 2005 GMU

39. Deficiencies of Prolog • Problems with negation: Given: parent(bill,jake). parent(bill, shelley). sibling(X,Y) :- parent(M,X),parent(M,Y). For goal: sibling(X,y). Prolog will respond with: X=jake, Y=jake • sibling(X,Y) :- parent(M,X),parent(M,Y),not(X=Y). CS 363 Spring 2005 GMU

40. Deficiencies of Prolog • Problems with negation: Need to write the rule using negation: • sibling(X,Y) :- parent(M,X),parent(M,Y),not(X=Y). However, must be done carefully – not(not(goal)) is not the same as goal. CS 363 Spring 2005 GMU

41. Trace • Built-in structure that displays instantiations at each step • Tracing model of execution - four events: • Call (beginning of attempt to satisfy goal) • Exit (when a goal has been satisfied) • Redo (when backtrack occurs) • Fail (when goal fails) CS 363 Spring 2005 GMU

42. Applications of Logic Programming • Relational database management systems • Expert systems • Natural language processing • Education CS 363 Spring 2005 GMU

43. Conclusions • Advantages: • Prolog programs based on logic, so likely to be more logically organized and written • Processing is naturally parallel, so Prolog interpreters can take advantage of multi-processor machines • Programs are concise, so development time is decreased – good for prototyping CS 363 Spring 2005 GMU