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Declarative Programming

Declarative Programming. Motivation Warm Fuzzies What is Logic? ... Logic Programming? Foundations: Terms, Substitution, Unification, Horn Clauses Proof (Resolution) Examples: List Processing; Integer Issues Logic / Theorem Proving Search Strategies Declarative/Procedural, ...

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Declarative Programming

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  1. Declarative Programming • Motivation • Warm Fuzzies • What is Logic? ... Logic Programming? • Foundations: • Terms, Substitution, Unification, Horn Clauses • Proof (Resolution) • Examples: List Processing; Integer • Issues • Logic / Theorem Proving • Search Strategies • Declarative/Procedural, ... • Other parts of Prolog • “Impure'' Operators” --- NOT, ! • Utilities • ? Constraint Programming • ? Bayesian Belief Nets

  2. Given • { All men are mortal. • Socrates is a man. } • infer (conclude, reason that, ...) • Socrates is mortal. What is Logic? • Logic is formal system for reasoning • Reasoning is inferring new facts from old • What is role of Logic within CS? • Foundation of discrete mathematics • Automatic theorem proving • Hardware design/debugging • Artificial intelligence (Cmput366) • Components: • Syntax (What does it look like?) • Reasoning/ProofTheory (New facts from old) • Semantics (What should it do?)

  3. Logic Programming • Program  Logic FormulaExecution of program  Proving a theorem • User: 1. Specifies WHAT is true 2. Asks if something else followsProlog: Answers question. • By comparison, in Procedural Programming (C, JAVA, Fortran, ...) user must … • decide on data-structure • explicitly write procedures: search, match, substitute • write diff programs for father(X, tom) vs father(tom, Y)

  4. Given • { All men are mortal. • Socrates is a man. } • infer (conclude, reason that, ...) • Socrates is mortal. What is Logic? • Syntax: Set of Expressions ? Well-formed or not? • SEQUENCE of Symbols  • Accept: The boys are at home. at(X, home) :- boy(X). • Reject: boys.home the angrily democracy X(at), xY=, Boy(1X,( ) :- Accept Reject • Proof Process: Given Believed statements, Determine other Believed statements. { s1, ..., sn } ⊢P s • Semantics: Which expressions are Believed ? • John's mother is Mary.T • John's mother is Fred. F • Colorless green ideas sleep furiously.F

  5. Concept of PROLOG • PROgramming in LOGic • Sound Reasoning Engine • User asserts true statements. User asserts { All men are mortal. Socrates is a man. } • User poses query. • “Is Socrates mortal?” • “Who/what is mortal?” • Prolog provides answer (Y/N, binding). • “Yes” • “Socrates''

  6. Tying Prolog to Logic • Syntax: Horn Clauses(aka Rules + Facts) • Terms • Proof Process: Resolution • Substitution • Unification • Semantics • (From Predicate Calculus … + not )

  7. Terms • BNF: term ::= constant | variable | functor constant ::= atom starting w/lower case variable ::= atom starting w/upper case functor ::= constant (tlist )  tlist  ::= “” | term {, tlist } • Examples of term : a1 b fred constant X Yc3 Fred_4 variable married(fred) g(a, f(Yc3), b) functor • Ground Term  term with no variables f(q) g(f(w), w1(b,c)) are ground f(Q) g(f(w), w1(B,c)) are not

  8. Substitution • A Substitution is a set { t1/v1 t2/v2 ... tn/vn }where vi are distinct variables ti are terms that do not use any vj • Examples: +{ a/X } { a/X b/Y f(a,W)/Z } { W/X f(W)/Y W/Z } - { a/f(X) } { a/X b/X } { f(X)/X } { f(Y)/X g(q)/Y } - { f(g(Y))/X g(q)/Y }

  9. t -- a term  -- a substitution Applying a Substitution • Given“t” is term, result of applying  to t • Small Examples: • X {a/X} a • f(X) {a/X} f(a) • Example, using t = f( a, h(Y,b), X )

  10. Applying a Substitution • f( a, h(Y,b), X ) { b/X } =f( a, h(Y,b), b ) f( a, h(Y,b), X ) { b/X f(Z)/Y } = f( a, h(f(Z),b), b ) f( a, h(Y,b), X ) { Z/X f(Z,a)/Y } = f( a, h(f(Z,a),b), Z ) f( a, h(Y,b), X ) { Z/W } = f( a, h(Y,b), X ) •  need not include all variables in t • can include variables not in t.

  11. Composition of Substitutions • ○ is composition of substitutions , For any term t, t [○] = (t ). • Example:f(X) [{ Z/X} ○{ a/Z}] = (f(X) { Z/X}) {a/Z} = f(Z) { a/Z} = f(a) • ○ is typically a substitution Eg: • { a/X} ○ { b/Y} = { a/X, b/Y} • { Z/X} ○{ a/Z} = { a/X, a/Z}

  12. Unifiers • t1 and t2 are unified by  iff t1  = t2 . Then  is called a unifier t1 and t2 are unifiable • Examples: t1 t2 unifier term f(b,c) f(b,c) {} f(b,c) f(X,b) f(a,Y) { a/X b/Y } f(a,b) f(a,b) f(c,d) * f(a,b) f(X,X) * f(X,a) f(Y,Y) { a/X a/Y } f( a, a ) f(g(U),d) f(X,U) { d/U g(d)/X } f( g(d),d ) f(X) g(X) * f(X,g(X)) f(Y,Y) * f(X) f(Y) { Y/X } f(Y) • NB t1 and t2 are symmetric -- either can have variables

  13.  t1  = t2  = { X/Y } f(X) { Y/X } f(Y) { a/Y a/X } f(a) { g(b,Z))/Y g(b,Z)/X) } f(g(b,Z) ) { X/Y W/g) } f(X) Multiple Unifiers • Unifier for t1 = f(X) and t2 = f(Y) … • { X/Y }and{ Y/X }make sense, but{ a/Y a/X }has irrelevant constant { X/Y g/W }has irrelevant variable (W) • Adding irrelevant constants/bindings  unifiers •  best one??

  14. Quest for Best Unifier • Wish list: • No irrelevant constants So {X/Y} preferred over { a/Y, a/X } • No irrelevant bindings So {X/Y} preferred over { X/Y, f(4,Z)/W } • Spse 1 has constant where 2 has variable 1 = { a/X, a/Y },2 = {Y/X} Then  substitution  s.t. 2○ = 1  = {a/Y} : {Y/X}○{a/Y} = { a/X, a/Y } • Spse 1 has extra binding over 2 1 = {a/X, b/Y},2 = {a/X} Then  substitution  s.t. 2○ = 1  = {b/Y} : {a/X}○{b/Y} = {a/X, b/Y} • INFERIOR unifier = composition of Good Unifier + another substitution

  15. Most General Unifier •  is MGU for t1 and t2 iff • unifies t1 and t2 , and • : unifier of t1 and t2 , substitution  s.t. ○ =  Ie, for all terms t, t  = (t) • Eg,  ={X/Y}is mgu for f(X) and f(Y)Consider unifier  = {a/X a/Y}Use  = { Y/a }: f(X) = f(X) {a/X a/Y} = f(a) f(X)[○] = [f(X)] = [f(X) {Y/X}] = f(Y) {a/X a/Y} = f(a) Similarly,f(Y) = f(a) = f(Y) [○] ( is not a mgu, as¬’s.t.  ○’=  )

  16. W/X g(Z)/Y h(X)/Z h(W)/Z g(h(W))/Y MGU --- Example#2 A mgu for f( W, Z, g(Z) ) & f( X, h(X), Y ) {W/X} h(W)/Z g(h(W))/Y

  17. MGU (con't) • Notes: • If t1 and t2 are unifiable, then  mgu. • Can be more than 1 mgu but they differ only in variable names. • Not every unifier is a mgu. • A mgu uses constants only as necessary. • Implementation:  fast algorithm that computes a mgu of t1 and t2, if one exists; or reports failure. • Slow part is verifying legal substitution: none of vi appear in any tj. Avoid by resetting Prolog's occurscheck parameter.

  18. MGU Procedure Recursive Procedure MGU( x, y ) If x=y then Return() If Variable(x) then Return( MguVar(x,y) ) If Variable(y) then Return( MguVar(y,x) ) If Constant(x) or Constant(y) then Return( False ) If Not(Length(x) = Length(y)) then Return( False ) g  [] For i = 0 .. Length(x) s  MGU( Part(x,i), Part(y,i) ) g  Compose( g, s ) x  Substitute( x, g ) y  Substitute( y, g ) Return( g ) End Procedure MguVar (v,e) If Includes(v,e) then Return( False ) Return( [v/e] ) End

  19. Prolog's Syntax • BNF:Horn ::= literal. | literal :- llist . llist ::= literal { , llist } literal ::= term • Examples: father(john, sue). father(odin, X). parent(X, Y) :- father(X, Y). gparent(X, Z) :- parent(X, Y), parent(Y, Z). • How to read as predicate calculus? father(john, sue)  X. father(odin, X). X,Y. father(X,Y)  parent(X,Y) X,Y,Z.parent(X,Y)} & parent(Y,Z)  gparent(X,Z)

  20. Relation to Predicate Calculus • In general:t. x1, …, xm. t[called “atomic formula”]t :- t1, …, tn. x1, …, xm. t1, …, tn  t.[called “(production) rule”] • Can write any Predicate Calculus Expression in Conjunctive Normal Form • Horn clauses are CNF, with ONE Positive Literal • Horn  CNF  Predicate Calculus expressions that canNOT be written as Horn Clauses. Axioms A v B

  21. What User Really Types > sicstus SICStus 3.11.2 (x86-linux-glibc2.3): Wed Jun 2 11:44:50 CEST 2004 Licensed to cs.ualberta.ca | ?- [user].% For user to enter ``Assert-fact'' mode. % consulting user... | on(aa,bb). | on(bb,cc). | above(X,Y) :- on(X,Y). | % Typing ^D exits ``assert'' mode. % consulted user in module user, 0 msec 608 bytes yes % Prolog's answer to most operations | ?- on(aa,bb).% User asks a question. yes % Prolog's answer | ?- on(aa,Y).% User's second question Y = bb ? % Prolog's answer: a binding list yes % Prolog's statement that there was answer | ?- on(X,Y). X = aa, Y = bb ? ; % User types “;” to ask for ANOTHER answer X = bb, Y = cc ? ; no % Prolog's no means there are no other answers | ?-

  22. What User Really Types -- II > sicstus SICStus 3.11.2 (x86-linux-glibc2.3): Wed Jun 2 11:44:50 CEST 2004 Licensed to cs.ualberta.ca | ?- [file1].% file ”file1” contains facts, rules % file1 consulted 120 byes 0.3333 sec. yes | ?- on(aa,b10). no % Prolog's answer: not derivable | ?- on(X,b10). no % Again: NO entailments | ?- above(bb,cc). yes % More than simple look-up! | ?- above(bb,W). W = cc ;% Prolog finds answer; user wants another no % but this is only one… | ?-

  23. Prolog's Proof Process • User provides • KB: Knowledge Base (List of Horn Clauses --- axioms) • : Query (aka Goal, Theorem) -- Literal1. Who is mortal? mortal(X). 2. Is Socrates mortal? mortal(socrates). • Prolog finds • a Proof of , from KB , if one exists & substitution for γ's variables: σ KB ⊢P { σ } KB1 ⊢Pmortal(X){ socrates/X } • Failure (otherwise)

  24. Proof Process (con’t) • Prolog finds ``Top-Down'' (refutation) Proof • … returns bindings • Actually returns LIST of σi s [one for each proof] • { socrates/X } { plato/X } { freddy/X } ...

  25. (1) success empty substitution Examples of Proofs: I (DB retrieval) • Using KB1 = • Query 1 :on(aa, bb) on(aa, bb) • Hence KB1 ├Pon(aa, bb){} ( Like Data Base retrieval )

  26. (1) Y=bb success –{bb/Y} Examples of Proofs: IIa (Variables) • Using KB1 = on(aa, Y) • Query 2 : on(aa, Y) • KB1 ├Pon(aa, Y) { bb/Y }

  27. (1) X=aa, Y=bb (2) X=bb, Y=cc success {aa/X,bb/Y} success {bb/X,cc/Y} Examples of Proofs: IIb (Variables) • Using KB1 = • Query 3 : on( X, Y ) Either is sufficient! • KB1 ├Pon(X, Y){ aa/X, bb/Y } ... on(aa,bb) • KB1 ├Pon(X, Y){ bb/X, cc/Y } ... on(bb,cc)

  28. Attempts Ie, KB1 ├Pon(aa,bb) Ie, KB1 ├Pon(X,b10) for any value of X. Examples of Proofs: III (Failure) • Using KB1 = • Query 4 : on(aa, b10) • Query 5 : on(X, b10)

  29. (3) X=bb, Y=cc on(bb,cc) (2) success Examples of Proofs: IVa (Rules) • Using KB1 = • Query 6 : above(bb, cc) Ie, KB1 ├Pabove(bb,cc)

  30. (3) X=bb, Y=W on(bb, W) (2) success –{cc/W} Examples of Proofs: IVb (Rules) • Using KB1 = • Query 7 : above(bb, W) Ie, KB1 ├Pabove(bb,W){cc/W}

  31. (3) X=bb, Y=W (4) X=aa, Y=cc on(aa,cc) on(aa,Z)&above(Z,cc) (1) Z=bb above(bb,cc) Examples of Proofs: V (big) • Using KB2 = • Query 7 : above(aa, cc)

  32. (4) X=aa, Y=cc (3) X=bb, Y=cc on(aa,cc) on(aa,Z),above(Z,cc) (1) Z=bb above(bb, cc) (4) X=bb, Y=cc (3) X=bb, Y=cc on(bb,cc) on(bb,Z), above(Z, cc) (2) (2) Z=cc success above(cc, cc) (3) (4) X=cc, Y=cc on(cc,cc) on(cc,Z),above(Z,cc) Examples of Proofs: V (big) above(aa,cc)

  33. Examples of Proofs: VI (many answers) • Using KB3 = • Query 9 : above( X, Y) • Answers • { X=aa, Y=bb }above( aa, bb)(3), (1) • { X=bb, Y=cc }above( bb, cc) (3), (2) • { X=aa, Y=cc }above( aa, cc)(4), (1), (3), (2) • { X=c1, Y=c2 }above( c1, c2)(5)

  34. Prolog's Proof Process • A goal is either • a sequence of literals (conjunction) , • the special goal ``success '' (eg, ) on(X,Y) p(X,5), q(X) success • Sequence of (sub)goals  G1, G2, ..., Gn is a top-down proof of G1 (from KB) iff • Gn = success, and • Gi is a SUBGOAL (in KB) of Gi-1, i=2,3,... n

  35. Subgoals Subgoals of G = { g1, ..., gr} in KB: Rule 1: If  atomic axiom ``t '' in KB where t and gi have mgu , then { g1 , ..., gi-1, gi+1, ..., gr } is a subgoal of G . (If r=1, then ``success'' is subgoal of G .) Rule 2: If  rule ``t :- t1, ..., t_k '' in KB where t and gi have mgu , then { t1, ..., tk, g1, ..., gi-1, gi+1, ..., gr } is a subgoal of G .

  36. Example of Subgoals -- I Subgoals of are •  = {X/A, Y/B}using Rule 2, (3) •  = {X/A, Y/B}using Rule 2, (4) •  = {c1/A, c2/B}using Rule 1, (5) • KB3 = above(A,B) on(A,B) on(A,Z), above(Z, B) success

  37. Example of Subgoals -- II Subgoals of are •  = {aa/A, bb/Z1}using Rule 3, (1) [first literal] •  = {bb/A, bb/Z1}using Rule 3, (2) [first literal] •  = {c1/Z1, c2/B}using Rule 3, (5) [second literal] •  = {X/Z1, B/Y}using Rule 4, (3) [second literal] •  = {X/Z1, Y/B}using Rule 4, (4) [second literal] above(A,Z1), above(Z1, B) above(bb,B) above(cc, B) on(A,c1) on(Z1,B), on(A, Z1) on(Z1,Z), above(Z,B), on(A, Z1)

  38. Variable bindings Unifier Comments wrt Prolog's Proof Procedure • found during proof • Prolog returns these overall mgu's 1-by-1 . • Prolog uses specific strategy called “SLD Resolution”

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