Logic Programming

1. Logic Programming Tasanawan Soonklang

2. Programming paradigms • Imperative • Object-oriented • Functional • Logic Procedural programming Non-procedural programming or Declarative programming

3. Non-proceduralprogramming • So far programming has been algorithmic. • Procedural: statements (C, C++, FORTRAN) • Functional: expressions (Postscript, ML, LISP) • Now declarative languages – logic programming • Programs do not state now a result is to be computed, but rather the formof the result

4. Title Logic Programming • Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. • Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. • Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. • Refers loosely to the use of • facts and rules to represent information. • deduction to answer queries. • algorithm = logic + control • We supply logic part, Programming Language supplies control part.

5. Imperative vs. Logic fact(0,1) :- true. fact(0,1). fact(N,F) :- N>0, N1 is N-1, fact(N1,F1), F is N * F1. int fact(int n) { inti = 1; for (int j = n; j>1; --j) i = i * j; return I; } ?- fact(3,W). W=6 Click to see the animation

6. Computing • Deals with relations rather than functions. • Search facts in order looking for matches. • Use rules for a logical inferencing process to produce results Premise • programming with relations is more flexible than with functions.

7. to city departure time arrival time Reserve fight Information • fight number • from city fight ( fight_number, from_city, to_city, departure_time, arrival_time ) fight(fight_number, “Bangkok”, “Los Angeles”, departure_time, arrival_time) fight(fight1,“Bangkok”, X, depart1,arrival1), fight(fight2,X,“Los Angeles”,depart2,arrival2), depart2 >= arrive1+30

8. Propositional logic Zeroth-order logic Proving theorems Predicate logic First-order logic Logic

9. Proposition • A declarative sentence. e.g. Woff is a dog. e.g. Jim is the husband of Mary. • A proposition is represented by a logical symbol. P: Woff is a dog. R: Jim is the husband of Mary.

10. Proposition • A logical statement that may or may not be true • Consists of • objects • relationships of objects to each other

11. Symbolic Logic • Logic which can be used for the basic needs of formal logic: • Express propositions • Express relationships between propositions • Describe how new propositions can be inferred from other propositions • Particular form of symbolic logic used for logic programming called predicatecalculus

12. Object Representation • Objects in propositions are represented by simple terms: either constants or variables • Constant - a symbol that represents an object • Variable - a symbol that can represent different objects at different times • Different from variables in imperative languages

13. Compound Terms • Atomic propositions consist of compound terms • Compound term: one element of a mathematical relation, written like a mathematical function • Mathematical function is a mapping • Can be written as a table

14. Parts of a Compound Term • Compound term composed of • Functor - function symbol that names the relationship • Ordered list of parameters (tuple) • Examples: student(jon) like(seth,OSX) like(nick,windows) like(jim,linux)

15. Forms of a Proposition • Fact -proposition is assumed to be true • Query -truth of proposition is to be determined • Compound proposition: • Have two or more atomic propositions • Propositions are connected by operators

16. Logical Operators

17. Quantifiers

18. Forms of a Proposition • Too many ways to state the same thing • Use a standard form for propositions • Clausal form • B1  B2  …  Bn A1  A2  …  Am • means if all the As are true, then at least one B is true • Antecedent - right side • Consequent - left side

19. Predicate Calculus and Proving Theorems A use of propositions is to discover new theorems that can be inferred from known axioms and theorems Resolution - an inference principle that allows inferred propositions to be computed from given propositions

20. Resolution Unification: finding values for variables in propositions that allows matching process to succeed Instantiation: assigning temporary values to variables to allow unification to succeed After instantiating a variable with a value, if matching fails, may need to backtrack and instantiate with a different value

21. Proof by Contradiction Hypotheses: a set of pertinent propositions Goal: negation of theorem stated as a proposition Theorem is proved by finding an inconsistency

22. Theorem Proving • Basis for logic programming • When propositions used for resolution, only restricted form can be used • Horn clause - can have only two forms • Headed: single atomic proposition on left side • Headless: empty left side (used to state facts) • Most propositions can be stated as Horn clauses

23. First Order Logic (FOL) • Allows the following to be modeled • Objects • properties of objects • relations among the objects • Like propositional logic, FOL has sentences • Additionally it has terms which allow the representation of objects

24. Terms • A term is a logical expression which refers to an object. • Elements of a term • Constant Symbols e.g. A, B, John • Predicate Symbols :- refer to a relation • Function Symbols :- refer to a relation which is a function

25. Sentences • Atomic Sentences - state a fact e.g. company(gordon). e.g. location(gordon,usa). • Complex Sentences - formed from: • atomic sentences • logical connectives • quantifiers

26. Quantifier • Universal Quantifier () • All cats are mammals. • x (Cat(x)  Mammal(x)) • Extensile Quantifier () • Spot has a sister who is a cat. • x ( Sister(x,Spot) Cat(x))

27. Prolog • Programming Logic • Prolog is based upon First Order Logic (FOL) • A Prolog program consists of a Knowledge Base composed of: • facts • rules • All facts and rules must be expressed as Horn Clauses

28. Applications • Relational DBMS • Artificial Intelligence • Expert systems • Natural language processing • Automatic theorem proving

29. The Origins of Prolog • University of Aix-Marseille • Natural language processing • University of Edinburgh • Automated theorem proving

30. Syntax rules Predicates (functors) must start with lower-case letter. Constants begin with a lower-case letter or number. Variables begin with an upper-case letter or an (_).

31. Syntax rules All clauses have a head and a body. head :- body. The symbol :- is read if All sentences (clauses) must end with a period.

32. 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

33. Edinburgh syntax • 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)

34. Fact statements Used for the hypotheses Headless Horn clauses female(shelley). male(bill). father(bill,jake).

35. Rule statements • Used for the hypotheses • Headed Horn clause • Right side: body (if part) • May be single term or conjunction • Left side: Head(then part) • Must be single term • Conjunction: multiple terms separated by logical AND operations (implied)

36. Rule statements ancestor(mary,shelley):- mother(mary,shelley). Can use variables (universalobjects) to generalize meaning: parent(X,Y):- mother(X,Y). parent(X,Y):- father(X,Y). grandparent(X,Z):- parent(X,Y), parent(Y,Z). sibling(X,Y):- mother(M,X), mother(M,Y), father(F,X), father(F,Y).

37. Goal statements For theorem proving, theorem is in form of proposition that we want system to prove or disprove Same format as headless Horn man(fred) Conjunctive propositions and propositions with variables also legal goals father(X,mike)

38. Simple Prolog facts A database of facts: inclass(john, cmsc330). inclass(mary, cmsc330). inclass(george, cmsc330). inclass(jennifer, cmsc330). inclass(john, cmsc311). inclass(george, cmsc420). inclass(susan, cmsc420). Queries: Prolog can confirm these facts: ?-inclass(john, cmsc330). “yes” ?- inclass(susan, cmsc420). “yes” ?- inclass(susan, cmsc330). “no”

39. Simple Prolog facts & rules A database of facts: dog(woff). barks(woff). barks(spot). wags_tail(woff). A database of rules: dog(X) :- barks(X), wags_tail(X). Queries: ?- dog(woff) => yes ?- dog(spot) => no ?- dog(Y) => Y = woff

40. Inferencing Process Facts and rules are Knowledge base (KB) Prolog uses a goal directed search of the KB Depth first search is used Query clauses are used as goals and searched left to right KB clauses are searched in the order they occur in the KB Goals are matched to the head of clauses Terms must unify based upon variable substitution before they match

41. Inferencing Process • 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

42. Approaches • 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

43. Subgoal Strategies • 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

44. Backtracking 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

45. Unification • Can use a form of substitution called unification to derive other relationships. inclass(susan, X). • Prolog searches database and responds “X=cmsc420.” • Hitting Enter key, Prolog says “No” since no other fact. inclass(john, Y). • Prolog has following responses: “Y=cmsc330.” “Y=cmsc311.” “no.”

46. Unification • Can define more complex queries: takingboth(X):- inclass(X, cmsc330), inclass(X, cmsc311). ?-takingboth(john) yes ?-takingboth(Y) Y=john; no

47. 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 / 17 + C Not the same as an assignment statement!

48. Data Data: Integers: 1, 2, 3, 4 Reals: 1.2, 3.4, 6.7 Strings: 'abc', '123' Facts: lower case names Variables: Upper case names Lists: [a, b, c, d]

49. 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.

50. List Structures basic data structure Listis 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)