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Introduction to Logic Programming and Predicate Calculus

Explore the fundamental concepts of logic programming and predicate calculus, including symbolic logic, proposition, predicate calculus, compound propositions, quantifiers, and clausal forms.

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Introduction to Logic Programming and Predicate Calculus

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  1. Lectures 3,4 PROLOG LOGic PROgramming

  2. Logic: what is it? • Philosophical logic • fundamental questions • Mathematical logic • theory of correct reasoning • Computational logic • effectiveness of reasoning • Language of scientific thinking

  3. Logic Programming • Logic programming uses a form of symbolic logic and a logical inference process • Languages based on symbolic logic are called logic programming languages or declarative languages • Prolog is the most widely used logic language

  4. Predicate Calculus • Formal logic is the basis for logic programming • Key Points • Proposition: • a logical statement that may or may not be true • Symbolic logic used for three purposes • express propositions • express the relationships between propositions • describe how new propositions may be inferred • Predicate calculus is the form of symbolic logic used for logic programming

  5. Propositions

  6. Atomic propositions • Objects in logic programming propositions are • Constants: symbols that represent an object • Variables: symbols that can represent different objects at different times • Atomic propositions are the simplest propositions and consist of compound terms • Compound term: an element of a mathematical relation, written in form of function notation, which is an element of the tabular definition of a function

  7. Compound terms • Compound term has two parts • Functor: symbol that names the relation • An ordered list of parameters • Compound term with a single parameter is called a 1-tuple; with two – a 2-tuple, etc. • Example: man (jake) {jake} is a 1-tuple in relation man like (bob, steak) {bob, steak} is a 2-tuple in like

  8. Facts and queries • Simple terms in propositions – man, jake, bob, and steak – are constants • These propositions have no intrinsic semantics • like (bob, steak) could mean several things • Propositions are stated in two modes • Fact: one in which the proposition is defined to be true • Query: one in which the truth of the proposition is to be determined

  9. Logical operators

  10. Compound propositions • Compound proposition examples: a  b  c a  b  d equivalent to (a ( b))  d • Precedence of logical connectors:  highest precedence , ,  next ,  lowest precedence

  11. Quantifiers • Variables may appear in propositions – only when introduced by symbols called quantifiers • Note: The period separates the variable from the proposition

  12. Quantifiers • Examples of propositions with quantifiers • X.(woman(X)  human(X)) For any value of X, if X is a woman, then X is human • X.(mother(mary, X)  male (X)) There exists a value of X, such that mary is the mother of X and X is a male • Note: Quantifiers have a higher precedence than any of the logical operators

  13. Example of Logic Predicate Calculus • 0 is a natural number. natural (0) • 2 is a natural number. natural (2) • For all X, if X is a natural number, then so is the successor of X. X.natural (X)  natural (successor (X))

  14. Logic Statement to P.C. • A horse is a mammal. • A human is a mammal. • Mammals have four legs and no arms, or two legs and two arms. • A horse has no arms. • A human has arms. • A human has no legs.

  15. Solution:First Order Predicate Calculus • mammal (horse). • mammal (human). • X. mammal (X)  legs (X,4)  arms (X,0)  legs (X,2) arms (X,2) • arms (horse, 0). • arms (human, 2) or  arms (human, 0). • legs (human, 0).

  16. Clausal forms

  17. General statement • Redundancy is a problem with predicate calculus • there are many different ways of stating propositions that have the same meaning • not a problem for logicians • For computerized system, this is a problem • Clausal form is one standard form of propositions used for simplification: B1  B2  ...  Bn  A1  A2  ...  Am Meaning: If all As are true, then at least one B is true

  18. Characteristics • Existential quantifiers are not required • Universal quantifiers are implicit in the use of variable in the atomic propositions • Only the conjunction and disjunction operators are required • Disjunction appears on the left side of the casual form and conjunction on the right side • The left side is called the consequent • The right side is called the antecedent

  19. Examples • Proposition 1: likes (bob, trout)  likes (bob, fish)  fish (trout) • Meaning: if bob likes fish and a trout is a fish, then bob likes trout • Proposition 2: father(louis, al)  father(louis, violet)  father(al, bob)  mother(violet, bob)  grandfather(louis, bob) • Meaning: if al is bob’s father and violet is bob’s mother and louis is bob’s grandfather, then louis is either al’s father or violet’s father

  20. Resolution

  21. Proving theorems • One of the most significant breakthroughs in automatic theorem-proving was the discovery of the resolution principle by Robinson in 1965 • Resolution is an inference rule that allows inferred propositions to be computed from given propositions • Resolution was devised to be applied to propositions in clausal form

  22. Basic idea • The concept of resolution is the following: Given two propositions: P1  P2 Q1  Q2 • Suppose P1 is identical to Q2 and we rename them as T. Then T  P2 Q1  T • Since P2 implies T and T implies Q1, it is logically obvious that P2 implies Q1, that is Q1  P2

  23. Example • Consider the two propositions: older (joanne, jake)  mother (joanne, jake) wiser (joanne, jake)  older (joanne, jake) • We can conclude that wiser (joanne, jake)  mother (joanne, jake) • The mechanics of this resolution construction is one of cancellation of the common term when both left sides are combined and both right sides are combined

  24. Expanded Example • father(bob, jake)  mother(bob,jake)  parent (bob,jake) • grandfather(bob,fred)  father(bob,jake)  father(jake,fred) • Resolution process cancels common term on both the left and right sides: • mother(bob,jake)  grandfather(bob,fred)  parent (bob,jake)  father(jake,fred)

  25. Most important laws • Unification: The process of determining useful values for variables in propositions to find values for variables that allow the resolution process to succeed. • Instantiation: The temporary assigning of values to variables to allow unification. • Inconsistency detection: A critically important property of resolution is its ability to detect any inconsistency in a given set of propositions. This property allows resolution to be used to prove theorems.

  26. Horn clauses

  27. Basics • When propositions are used for resolution, only a restricted kind of clausal form can be used • Horn clauses • special kind of clausal form to simplify resolution • two forms: • single atomic proposition on the left side, or • an empty left side • left side of Horn clause is called the head • Horn clauses with left sides are called headed Horn clauses

  28. Relationships and facts • Headed Horn clauses are used to state relationships: likes(bob, trout)  likes (bob, fish)  fish(trout) • Headless Horn clauses are used to state facts: father(bob,jake) • Most propositions may be stated as Horn clauses

  29. Logic Programming

  30. Brief history • Resolution principle (Robinson 1965) • Prolog language (Colmerauer 1972) • Operational semantics (Kowalski 1974) • Prolog compiler (Warren 1977, 1983) • 5th generation project (Japan 1981--) • Constraint LP (Jaffar & Lassez 1987)

  31. Brief basics • Robert Kowalski says: Algorithm = Logic + Control • Prolog is the most popular logic programming language • A Prolog program is a collection of Horn clauses

  32. Declarative languages • Languages used for logic programming are called declarative languages • Programming with declarative languages is nonprocedural • programs do not state exactly how a result is to be computed but rather describe the form of the result • it is assumed that the computer can determine how the result is to be obtained • one needs to provide the computer with the relevant information and a method of inference for computing desirable results

  33. Important issues • Program • knowledge about the problem in the form of logical axioms • Call of the program • provide a problem description (as a logical statement) • Execution • attempt to prove the given statement (given the program) • Constructive proofs essential • can you prove sort([2,3,1], x)? • non-constructive answer: “Yes I can” • constructive answer: “Yes, with x = [1,2,3]”

  34. PROLOG

  35. Notation

  36. Example • /* simple rules for deductions */ sees(X,Y):-person(X),object(Y),open(eyes(X)),in_front_of(X,Y). sees(X,Y):- person(X),event(Y),watching(X,film_of(Y)). person(mother(nikos)). event(birthday(kostas)). event(graduation(nikos)). watching(mother(nikos),film_of(graduation(nikos))). person(kostas). open(eyes(kostas)). in_front_of(kostas,tv). object(tv). • /* questions */ /* does the mother of nikos see his graduation? */ ?-sees(mother(nikos),graduation(nikos)).

  37. General characteristics • A program uses terms such as constant symbols, variables, or functional terms • Queries may include conjunctions, disjunctions, variables, and functional terms • A program consists of a sequence of sentences, implicitly conjoined • All variables have implicit universal quantification • Prolog uses a negation as failure operator: a goal not P is assumed proved if the system fails to prove P

  38. Basic elements

  39. Terms • A Prolog term is a constant, a variable, or a structure • A constant is either an atom or an integer • Atoms are the symbolic values of Prolog • Atoms begin with a lowercase letter • Variables begin with an uppercase letter • not bound to types by declarations • binding of a value and type is an instantiation • Instantiations last only through completion of goal • Structures represent the atomic propositions of predicate calculus • form is functor (parameter list)

  40. Fact statements • Fact statements are used to construct hypotheses (database) from which new information may be inferred • Fact statements are headless Horn clauses assumed true • Examples: male(bill). female(mary). male(jake). father(bill, jake). mother(mary, jake).

  41. Rule statements • Rule statements are headed Horn clauses for constructing the database • The RHS is the antecedent (if), and the LHS is the consequent (then) • Consequent is a single term because it is a Horn clause • Conjunctions may contain multiple terms that are separated by logical ANDs denoted by commas, e.g.: female(shelley), child (shelley).

  42. Rule statements • General form of the Prolog headed Horn clause consequence_1 :- antecedent_expression • Example: ancestor(mary, shelley) :- mother(mary, shelley). • Demonstration of the use of variables: 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). universal objects are X, Y, Z, M, and F

  43. Goal statements • Because goal and some nongoal statements have the same form (headless Horn clauses), it is imperative to distinguish between the two • Interactive Prolog implementations do this by simply having two modes, indicated by different prompts: one for entering goals and one for entering fact and rule statements • Some versions of Prolog use ?- for goals

  44. Goal statements • Facts and rules are used to prove or disprove a theorem that is in the form of a proposition (called goal or query) • Syntactic form of Prolog goal statement is identical to headless Horn clauses, e.g. man (fred). to which the system will respond yes or no • Conjunctive propositions and propositions with variables are also legal goals, e.g. father ( X, mike). • When variables are present, the system identifies the instantiations of the variables that make the goal true

  45. Example • Database: likes(george,kate). likes(george,susie). likes(george,wine). likes(susie,wine). likes(kate,gin). likes(kate,susie).

  46. Example • Query: ?-likes(george,X). X = kate ; X = susie ; X = wine ; no

  47. Example • Database: likes(george,kate). likes(george,susie). likes(george,wine). likes(susie,wine). likes(kate,gin). likes(kate,susie). friends(X,Y) :- likes(X,Z), likes(Y,Z). • Query: ?- friends(george,susie) yes

  48. Inference process

  49. General scheme • Goals (queries) may be compound propositions; each of facts (structures) is called a subgoal • The inference process must find a chain of rules/facts in the database that connect the goal to one or more facts in the database • If Q is the goal, then either Q must be found as fact in the database or the inference process must find a sequence of propositions that give that result

  50. Matching • The process of proving (or satisfying) a subgoal by a proposition-matching process • Consider the goal or query: man (bob). • If the database includes the fact man(bob), the proof is trivial • If the database contains the following fact and inference father (bob). man (X) :- father (X) then Prolog would need to find these and infer the truth. This requires unification to instantiate X temporarily to bob

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