Logic Programming Languages

1. Logic Programming Languages

2. Objective • To introduce the concepts of logic programming and logic programming languages • To introduce a brief description of a subset of prolog

3. Introduction • Logic programs are declarative programs • Specify the desired results - true • State the fact Programming Language Symbolic logic Logical Inferencing Process Result

4. Introduction • The major difference between logic programming and other programming languages (imperative and functional) • Every data item that exist in logic programming has written in specific representation (symbolic logic) • Prolog is a logic programming that widely used logic language

5. Introduction • Prolog specified the way of computer carries out the computation and it is divided to 3 parts: • logical declarative semantic of prolog • new fact prolog can infer from the given fact • explicit control information supplied by the programmer

6. Symbolic representation: Predicate Calculus • mathematical representation of formal logic • is a particular form of symbolic logic that is used for logic programming Predicate Calculus FOPL Higher-order PL Symbolic Logic Logic Formalism Proposition

7. Symbolic representation: Predicate Calculus Predicate Calculus FOPL Higher-order PL • symbolic logic used for the three basic need of formal logic • to express propositions • to express the relationships between propositions • to describe how new propositions can be inferred from other propositions that are assumed to be true Symbolic Logic Logic Formalism Proposition

8. Symbolic representation: Predicate Calculus Predicate Calculus FOPL Higher-order PL Symbolic Logic • Formal logic was developed to provide a method for describing proposition. Logic Formalism Proposition

9. Symbolic representation: Predicate Calculus Predicate Calculus FOPL Higher-order PL Symbolic Logic Logic Formalism • Proposition is a logical statement also known as fact • consist of object and relationships of object to each other Proposition

10. Proposition • Object: • Constant represents an object, or • Variable represent different objects at different times • Simple proposition called as atomic propositions, consist of compound terms – one element of mathematic relation which written in a form that has the appearance of mathematical function notation. Example (constants): single parameter (1-tuple): man(jake) double parameter (2-tuples): like(bob,steak)

11. Proposition • Object: • Constant represents an object, or • Variable represent different objects at different times • Simple proposition called as atomic propositions, consist of compound terms – one element of mathematic relation Example: single parameter (1-tuple): man(jake) double parameter (2-tuples): like(bob,steak) functor shows the names the relation

12. Proposition • Object: • Constant represents an object, or • Variable represent different objects at different times • Simple proposition called as atomic propositions, consist of compound terms – one element of mathematic relation Example: single parameter (1-tuple): man(jake) double parameter (2-tuples): like(bob,steak) list of parameter

13. Proposition • Two modes for proposition: • proposition defined to be true (fact), and • the truth of the proposition is something that is to be determined (queries) • Compound propositions have two or more atomic proposition, which are connected by logical operator (is the same way logic expression in imperative languages)

14. Logic operators

15. Compound propositions Example: a  b  c a  b  d (a  (b))  d Precedence:       higher lower

16. Variables in Proposition • Variable known as quantifiers • Predicate calculus includes two quanifiers, X – variable, and P – proposition

17. Variables in Proposition Example X.(woman(X)  human(X)) X.(mother(mary,X)  male(X))

18. Variables in Proposition Example X.(woman(X)  human(X))  for any value of X, if X is a woman, then X is a human (NL: woman is a human) X.(mother(mary,X)  male(X))

19. Variables in Proposition Example X.(woman(X)  human(X))  for any value of X, if X is a woman, then X is a human (NL: woman is a human) X.(mother(mary,X)  male(X))  there exist a value of X such that mary is the mother of X and X is a male (NL: mary has a son)

20. Clausal Form • Simple form of proposition, it is a standard form for proposition without loss of generality • Why we need to transform PC into CF? • too many different ways of stating propositions that have the same meaning Example: X.(woman(X)  human(X)) X.(man(X)  human(X))

21. Clausal Form • General syntax for CF B1 B2 … Bn A1 A2 … Am  if all the As are true, then at least one B is true Example: human(X)  woman(X)  man(X) likes(bob, trout)  likes(bob, fish)  fish(trout)

22. Clausal Form Example: likes(bob, trout)likes(bob, fish)  fish(trout) • Characteristics of CF: • Existential quantifiers are not required • Universal quantifiers are implicit in the use of variables in the atomic propositions • No operator other than conjunction and disjunction are required consequent antecedent

23. Clausal Form Example: likes(bob, trout)  likes(bob, fish)  fish(trout)  if bob likes fish and trout is a fish, then bob likes trout 

24. Clausal Form Example: father(louis, al)  father(louis, violet)  father(al,bob)  mother(violet, bob)  grandfather(louis, bob)  if al is bob’s father and violet is bob’s mother and louis is bob’s grandfather, louis is either al’s father or violet’s father

25. Proving Theorems • Method to inferred the collection of proposition • use a collections of proposition to determine whether any interesting or useful fact can be inferred from them • Introduced by Alan Robinson (1965)

26. Proving Theorems • Alan Robinson introduced resolution in automatic theorem proving • resolution is an inference rule that allows inferred proposition to be computed from given propositions • resolution was devised to be applied to propositions in clausal form

27. Proving Theorems • Idea of resolution: P1  P2 and Q1  Q2 which given P1 is identical to Q2  Q1  P2

28. Proving Theorems Example: older(joanne, jake)  mother(joanne, jake) wiser(joanne, jake)  older(joanne, jake)  wiser(joanne, jake)  mother(joanne, jake)

29. Proving Theorems Example: father(bob, jake)  mother(bob, jake)  parent(bob, jake) gfather(bob, fred)  father(bob, jake)  father(jake, fred) • gfather(bob, fred)  mother(bob, jake)  parent(bob, jake)  father(jake, fred)

30. Proving Theorems • Process of determining useful values for variables during resolution – unification • Unification • Hypotheses : original propositions • Goal: presented in negation of the theorem • Proposition in unification must be presented in Horn Clauses

31. Proving Theorems • Horn Clauses: • Headed Horn Clauses Example: likes(bob, trout)  likes(bob, fish)  fish(trout) • Headless Horn Clauses Example: father(bob, jake)

32. Applications of Symbolic Computation • Relational databases • Mathematical logic • Abstract problem solving • Understanding natural language • Design automation • Symbolic equation solving • Biochemical structure analysis • Many areas of artificial intelligent