Datalog

1. Datalog S. Costantini

2. Datalog base: notazione, esempi, semantica S. Costantini / Datalog e ASP

3. Datalog • A language for Deductive Databases • Extensional part: tables • Intensional part: rules S. Costantini / Datalog e ASP

4. Logic: Intuition S. Costantini / Datalog e ASP

5. Datalog Terminology: Atoms • An atom is a predicate, or relation name with variables or constants as arguments. E.g., in(alan,r123) part_of(r123,cs_building) p(X,a) S. Costantini / Datalog e ASP

6. Datalog Terminology: Atoms • Conventions: • Databases: Predicates begin with a capital, variables begin with lower-case. Example: P(x,’a’) where ‘a’ is a constant, i.e., a data item. • Logic Programming: Variables begin with a capital, predicates and constants begin with lower-case. Example: p(X,a). S. Costantini / Datalog e ASP

7. Datalog Terminology: Rules • The head of a rule is an atom; the body of a rule is the AND of one or more atoms. E.g., in(X,Y):- part_of(Z,Y), in(X,Z). where “:-” stands for “” S. Costantini / Datalog e ASP

8. Datalog Terminology: Rules • Rules without body: E.g., in(alan,r123). S. Costantini / Datalog e ASP

9. Datalog and Transitivity • Rule in(X,Y):- part_of(X,Z),in(Z,Y) defines in as the transitive closure of part_of S. Costantini / Datalog e ASP

10. Arithmetic Subgoals • In addition to relations as predicates, a predicate of the body can be an arithmetic comparison. • We write such atoms in the usual way, e.g.: x < y. S. Costantini / Datalog e ASP

11. Example: Arithmetic • A beer is “cheap” if there are at least two bars that sell it for under $2. cheap(Beer) :- sells(Bar1,Beer,P1), sells(Bar2,Beer,P2), P1 < 2.00, P2 < 2.00, Bar1 <> Bar2. S. Costantini / Datalog e ASP

12. Datalog Terminology: p  q, not r. or p :- q, not r. is a rule, where p is the head, or conclusion, or consequent, and q, not r is the body, or the conditions, or the antecedent. p, q and r are atoms. A rule without body, indicated as p . or p. Is called a unitrule, or a fact. This kind of rules ale also called Horn Clauses. S. Costantini / Datalog e ASP

13. Datalog Terminology: p  q, not r Atoms in the body can be called subgoals. Atoms which occur positively in the body are positive literals, and the negations are negative literals. S. Costantini / Datalog e ASP

14. Datalog Terminology: p  q, not r We say that p depends on q and not r. The same atom can occur in a rule both as a positive literal, and inside a negative literal (e.g. rule p  not p). S. Costantini / Datalog e ASP

15. Datalog Programs • A Datalog theory, or “program”, is a collection of rules. • In a program, predicates can be either • EDB = Extensional Database = facts. • IDB = Intensional Database = relation defined by rules. S. Costantini / Datalog e ASP

16. Expressive Power of Datalog • Without recursion, Datalog can express all and only the queries of core relational algebra. • But with recursion, Datalog can express more than these languages. • Yet still not Turing-complete. S. Costantini / Datalog e ASP

17. Example p(X) r(X),q(X). r(Z)  s(Z). s(a). s(b). q(b). • In every rule, each variable stands for the same value. • Thus, variables can be considered as “placeholders” for values. • Possible values are those that occur as constants in some rule/fact of the program itself. S. Costantini / Datalog e ASP

18. Datalog Its grounding can be obtained by: • considering the constants, here a,b. • substituting variables with constants in any possible (coherent) way) • E. g., the atom r(Z) is transformed by grounding over constants a, b into the two ground atoms r(a), r(b). S. Costantini / Datalog e ASP

19. Grounding • E. g., the atom r(Z) is transformed by grounding over constants a, b into the two ground atoms r(a), r(b). • Rule r(Z)  s(Z). is transformed into the two rules r(a)  s(a). r(b)  s(b). S. Costantini / Datalog e ASP

20. Datalog Grounding of the program p(a)  r(a),q(a). p(b)  r(b),q(b). r(a)  s(a). r(b)  s(b). s(a). s(b). q(b). Semantics: Least Herbrand Model M = {s(a),s(b),q(b),r(a),r(b),p(b)} S. Costantini / Datalog e ASP

21. Datalog A shortcut of the grounded program p1  r1,q1. p2  r2,q2. r1  s1. r2  s2. s1. s2. q2. M = {s1,s2,q2,r1,r2,p2} S. Costantini / Datalog e ASP

22. Datalog semantics • Without negation (no negative literal in the body of rules): Least Herbrand Model • The head of a rule is in the Least Herbrand Model only if the body is in the Least Herbrand Model. S. Costantini / Datalog e ASP

23. Least Herbrand Model: Intuition S. Costantini / Datalog e ASP

24. Least Herbrand Model: Intuition In the example:Least Herbrand Model M0 = {in(alan,r123), part_of(r123,cs_building)} M1= {in(alan,r123), part_of(r123,cs_building), in(alan,cs_building)} S. Costantini / Datalog e ASP

25. Least Herbrand Model: How to find it p g,h. g  r,s. m  p,q. r  f. s. f. h. Step 1: facts M0 = {s,f,h} Step 2: M1 = M0 plus what I can derive from facts M1 = {s,f,h,r} Step 3: M2 = M1 plus what I can derive from M1 M2 = {s,f,h,r,g} Step 4: M3 = M2 plus what I can derive from M2 M3 = {s,f,h,r,g,p} If you try to go on, no more added conclusions: fixpoint S. Costantini / Datalog e ASP

26. Datalog con Negazione S. Costantini / Datalog e ASP

27. Negated Subgoals • We may put NOT in front of an atom, to negate its meaning. • Example: at_home(alan):- not at_work(alan). S. Costantini / Datalog e ASP

28. Negated Subgoals: Intuition • Given atom A, not A holds if A cannot be proved • Negation as finite failure S. Costantini / Datalog e ASP

29. Minimal Models • When there is no negation, a Datalog program has a unique minimal model (one that does not contain any other model). • But with negation, there can be several minimal models. S. Costantini / Datalog e ASP

30. Cos’è un modello minimo: è un insieme di atomi Data una regola A:- B1,...,Bn,not C1,...,not Cm • Un modello è un insieme di atomi che se NON contiene C1,...,Cm e contiene B1,...,Bn allora deve contenere A • Un modello minimo non è sovrainsieme di un altro modello S. Costantini / Datalog e ASP

31. Datalog semantics • With negation: several proposals, in deductive databases Answer Set Semantics. • Based on Answer Set Semantics: Answer Set Programming, new logic programming paradigm S. Costantini / Datalog e ASP

32. Safe Rules • A rule is safe if: • Each variable in the head • Each variable in an arithmetic subgoal, • Each variable in a negated subgoal, also appears in a nonnegated, subgoal in the body. • We allow only safe rules. S. Costantini / Datalog e ASP

33. Example: Unsafe Rules • Each of the following is unsafe and not allowed: • s(X) :- r(Y). • s(X) :- r(Y), not r(X). • s(X) :- r(Y), X < Y. • In each case, an infinity of X ’s can satisfy the rule, even if R is a finite relation. S. Costantini / Datalog e ASP

34. Non-Monotonicità • Aggiungere nuove conoscenze può portare a ritrarre precedenti conclusioni. • Vi possono essere conclusioni alternative tramite cicli sulla negazione. S. Costantini / Datalog e ASP

35. Non-Monotonicità = Problema? • No, se gestita con opportuni strumenti semantici • Non-Monotonic Reasoning (NMR) = campo di ricerca in • Intelligenza Artificiale • Database deduttivi • si propone di sfruttare la non-monotonicità S. Costantini / Datalog e ASP

36. DATALOG senza Negazione • Semantica del Least Herbrand Model • Paradigma procedurale della ricerca per tentativi ripetuti • Linguaggio Prolog (numerosi interpreti fra i quali SICSTUS-Prolog, SWI-Prolog) S. Costantini / Datalog e ASP

37. DATALOG + Negazione • Semantica degli Answer Sets • Paradigma dell’Answer Set Programming (ASP) • Negazione, insiemi-risposta alternativi • Numerosi solver S. Costantini / Datalog e ASP

38. Esempio NMR persona(anna). persona(carlo). persona(giorgio). malato(giorgio). a_casa(X):- persona(X), not in_ufficio(X). in_ufficio(X):- persona(X), not a_casa(X). :- in_ufficio(X),malato(X). % constraint S. Costantini / Datalog e ASP

39. Risultato atteso • Anna e Carlo sono o a casa oppure in ufficio. • Giorgio non può essere in ufficio perchè è malato e quindi sarà a casa. S. Costantini / Datalog e ASP

40. NMR in pratica • Vi sono vari linguaggi logici che consentono non-monotonicità. • Sono dotati di motori inferenziali (inference engines, o solvers) in grado di gestirla. • Possibili soluzioni inesistenti o altrernative. • Uno di essi: smodels S. Costantini / Datalog e ASP

41. Risultato di smodels (inference engine per NMR) • Answer: 1 malato(giorgio) a_casa(giorgio) a_casa(carlo) in_ufficio(anna) • Answer: 2 malato(giorgio) a_casa(giorgio) in_ufficio(carlo) in_ufficio(anna) S. Costantini / Datalog e ASP

42. Risultato di smodels (inference engine per NMR) • Answer: 3 malato(giorgio) a_casa(giorgio) in_ufficio(carlo) a_casa(anna) • Answer: 4 malato(giorgio) a_casa(giorgio) a_casa(carlo) a_casa(anna) S. Costantini / Datalog e ASP

43. Datalog con negazione: nuovo Paradigma di Programmazione Logica Answer Set Programming(ASP)basato su Answer Set Semantics S. Costantini / Datalog e ASP

44. Uno sguardo sull’Answer Set Programming (ASP) S. Costantini / Datalog e ASP

45. Example: 3-coloring in ASP Assigning colors red/blue/green to vertices of a graph, so as no adjacent vertices have the same color. node(0..3). col(red). col(blue). col(green). edge(0,1). edge(1,2). edge(2,0). edge(1,3). edge(2,3). S. Costantini / Datalog e ASP

46. 3-coloring in Answer Set Programming color(X,red) | color(X,blue) | color(X,green) :- node(X). :- edge(X,Y), col(C), color(X,C), color(Y,C). % I constraint (regole con conclusione vuota) % specificano cosa *non può* essere vero S. Costantini / Datalog e ASP

47. Come “eseguire” il programma? • Si usa un “Motore Inferenziale”, ad esempio SMODELS (ma ce ne sono diversi) • Esso fornisce le soluzioni al problema dato come “insiemi risposta, appunto “Answer Set” … S. Costantini / Datalog e ASP

48. Come “eseguire” il programma? • Sintassi: lparse < 3col.txt | smodels 0 • lparse effettua il parsing del programma dato e ne produce il grounding • smodels calcola le risposte (smodels 0 le chiede tutte) … … S. Costantini / Datalog e ASP

49. Cosa si ottiene nell’esempio? Answer1 color(0,red), color(1,blue), color(green), color(3,red) Answer2 color(0,red), color(1,green), color(blue), color(3,red) … % e tutte le altre colorazioni, ognuna come % “Answer Set” … S. Costantini / Datalog e ASP

50. Answer Set Semantics: su cosa si basano gli Answer Set e come si ottengono S. Costantini / Datalog e ASP