Advanced Data Modeling Minimal Models

1. Advanced Data ModelingMinimal Models Steffen Staab TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Overview • Logic as query language. • Grounding. • Minimal Herbrand models. • Completion.

3. Logic as query language Given: • first-order formula A[x1, …, xn] • Herbrand interpretation I • This first-order formula can be considered as a definition of a relation RA on T nΣ as follows: (t1, … , tn)  RA := I |= A[t1, … , tn]

4. Clauses as definitions We say that a clause p(t1, … , tm) :- L1 , … , Ln defines the relation symbol p. Let C be a set of clauses and p be a relation symbol. We call the definition of p in C the set of all clauses in C that define p.

5. Principles of semantics A deductivedatabaseis a setofclauses. This setofclausesisregardedas a collectionofdefinitionsofrelations. The semanticsdefinesthemeaningofthesedefinitionsbyassociatingwiththem an interpretation, or a classofinterpretations. Query answeringisbased on thesemantics.

6. Two key assumptions • theuniquenameassumption: eachnamedenotes a uniqueobject. • theclosedworldassumption: • a negative statementA holdsifthecorresponding positive oneA does not hold. Bothassumptionsare not supportedby (Tarskian) modelsfor first-order logics (why not?)  Onesolution: minimal models

7. Minimal Herbrand Models Let I be a Herbrand model of a set of formulas S. We call I a minimal Herbrand model of S if it is minimal w.r.t. the subset relation, i.e. for every Herbrand model I´of S of the same signature we have I  I´ I is called the least Herbrand model of S if for every Herbrand model I´ of S of the same signature we have I  I´.

8. Does every set of formulas S have a least Herbrand model?

9. Does every set of formulas S have a least Herbrand model? Counterexample for normal clauses: person(a). man(X) :- person(X), not woman(X). woman(X) :- person(X), not man(X).

10. Does every set of formulas S have a least Herbrand model? What about definite clauses?

11. Instance and variant Let E, E´ be a pair of terms or formulas. E´ is an instance of E, denoted E > E´, if there exists a substitution θ: such that Eθ: = E´. ground instance: instance that is ground, E´ is a variant of E if E´ is an instance of E and E is an instance of E´.

12. Examples • P(x,a) is instance of P(x,y) because of P(x,y)[y|a] • P(b,a) is a ground instance • P(x,y) and P(u,v) are variants of each other, because of • [x|u, y|v] and • [u|x, v|y]

13. Grounding • Let C be a set of clauses and Σ be any signature containing all symbols used in C. The grounding of C w.r.t. Σ, denoted C* is the set of all ground instances of the signature Σ of clauses in C. • Lemma. Let I be a Herbrand interpretation and C be a set of clauses. Then I |= C if and only if I |= C*.

14. General proof scheme calculus First order formula First order Consequences (all possible ways of) grounding lifting calculus Propositional formula Propositional Consequences

15. Proof

16. Logic with equality • Additional atomicformulass = t, where s, t areterms. • Abbreviation: x y := (x = y). • Unlikeotherrelations, thesemanticsof s = t ispredefined in all Herbrandinterpretations: I |= s = t if s coincideswith t.

17. Example valid formulas • f(x1, … , xn) = f(y1, … yn) ! x1 = y1Æ … Æ xn = yn • f(x1, … , xn)  g(y1, … , yn) • f(x1, … ,xn)  c • d  c • A[t] $8x(x = t ! A[x])

18. Semantics of definitions Consider a definition of a relation r What is the meaning of this definition? Is there a largest Herbrand model? Especially in the light of negation as failure?

19. Idea of completion man(hans). man(adam). person(eva). woman(X) :- person(X), not man(X). Is eva a woman? She might be a man and we just don‘t know! Minimal model says that she is not a woman! (no minimal model) Trick: Completion: man(X)  (X=hans X=adam). woman(X)  X=eva. I.e. hans and adam are the only men.

20. Idea of Completion man(hans). man(X) :- lovesBeer(X,Y). Completion: man(X)  X=hans (Y: X=U lovesBeer(U,Y)) A man is a man onlyif he ishansorif he lovessomebrandofbeer.

21. Completion. Step 1. Replace every clause by an equivalent one such that the arguments of r are x1, …, xn: Given: r(t1, … , tn) :- G Replace by: r(x1, … , xn) :- x1 = t1 … xn = tn G

22. Completion. Step 2. If there are variables y1, … , yk occurring in a body but not in the head, apply  to these variables, i.e., Given r(x1, … , xn) :- G Modify to r(x1, … , xn) :-  y1 …  yk G

23. Completion. Step 3. If there are several definitions, replace them by one Given r(x1, … ,xn) :- G1 … r(x1, … , xn) :- Gm Replace by r(x1, … , xn) :- G1 … Gm

24. Completion. Step 4. Replace :- by $: Given r(x1, … , xn) :- G1Ç … Ç Gm Replace by r(x1, … , xn) $ G1Ç … Ç Gm The formula r(x1, … , xn) $ G1Ç … Ç Gm is called the completed definition of the original set of clauses.

25. Example r(u,v):-p(u,1,z). r(v,u):-p(2,u,z). r(x,y) :- x=uy=v p(u,1,z). r(x,y) :- x=v y=up(2,v,z). r(x,y) :- z,u,v x=uy=v p(u,1,z). r(x,y) :- z,u,v x=uy=v p(2,v,z). r(x,y)  (z,u,v x=uy=v p(u,1,z)) (z,u,v x=uy=v p(2,v,z))

26. Properties • All steps preserve Herbrand models, except for the last one. • Why? • Gives a unique semantics to non-recursive definitions • What about recursive definitions? • Logic programming semantics and first-order semantics coincide for definite programs

27. Recursive definitions odd(1). even(f(X)) :- odd(X). odd(f(X)) :- even(X). Completion: odd(X) X=1 Y(X=f(Y) even(Y)). even(X) Y(X=f(Y) odd(Y)).

28. Recursive definitions person(adam). person(eva). woman(X) :- person(X), not man(X). man(X) :- person(X), not woman(X). Completion: woman(X) Y(Y=X person(Y) not man(Y)). man(X) Y(Y=X person(Y) not woman(Y)). Semantics not unique in logicprogramming: Models are I={woman(adam),woman(eva),person(adam),person(eva)} I={man(adam),man(eva), person(adam),person(eva)} I={woman(adam),man(eva), person(adam),person(eva)} I={man(adam),woman(eva), person(adam),person(eva)} Whatisthesemantics in firstorderlogics?

29. Recursive definitions person(adam). person(eva). woman(X) :- person(X), not man(X). man(X) :- person(X), not woman(X). Completion: woman(X) Y(Y=X person(Y) not man(Y)). man(X) Y(Y=X person(Y) not woman(Y)). Semantics not unique in logicprogramming: Models are I={woman(adam),woman(eva),person(adam),person(eva)} I={man(adam),man(eva), person(adam),person(eva)} I={woman(adam),man(eva), person(adam),person(eva)} I={man(adam),woman(eva), person(adam),person(eva)} Whatisthesemantics in firstorderlogics? Models are: womanI={}=manI

30. Simple characterization of completion Let C be a definition of r, I be a Herbrand model of the corresponding completed definition, and r(t1, … , tn) be a ground atom. Then I |= r(t1, … , tn) , (r(t1, … , tn) :- L1, … , Lm) C* (I |= L1 …  Ln):

31. Immediate consequence operator TC(I) := { A | thereexists(A :- G)C* such that I |= G } Fixpoint: an interpretation such that TC(I) = I.

32. Definite clauses have the least model Let C be a setof definite clauses. Define I0 := {} In+1 := TC(In), for all n0, I:= i=0Ii ThenIisthe least fixpointof TCand also the least Herbrandmodelof C.

33. Non-recursive databases Let C be a non-recursivedatabaseand K be an arbitraryinterpretation. Define I0 := K In+1 := TC(In), for all n 0, I := i=0Ii ThenIistheonlyfixpointof TC. Moreover, forsomenwehaveI= In.

34. Non-recursive sets of clauses • Let C be a set of clauses. • Its dependency graph consists of pairs p ! r such that p occurs in the body of a clause which defines r in C. • A set of clauses is non-recursive if the dependency graph contains no cycles.

35. Non-recursive sets of clauses Let C be a set of clauses. Its dependency graph consists of pairs p ! r such that p occurs in the body of a clause which defines r in C. A set of clauses is non-recursive if the dependency graph contains no cycles. Person(a). Woman(X) :- Person(X), Not Man(X). Man(X) :- Person(X), Not Woman(X). Person Man Woman

36. Non-recursive sets of clauses Dependencygraphof C consistsofpairs p !r such that p occurs in thebodyof a clausewhichdefinesr in C. C is non-recursiveifthedependencygraphcontainsnocycles. A relation p depends on a relationq in C ifthereexists a pathoflength1 fromqto p in thedependencygraphof C. A setofclausesis non-recursiveifandonlyifnorelationdepends on itself.