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Datalog

Datalog. S. Costantini. Logic As a Query Language. If-then logical rules have been used in many systems. Recursive rules extend relational algebra --- have been used to add recursion to SQL-99. Datalog. A language for Deductive Databases Extensional part: tables Intensional part: rules.

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Datalog

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  1. Datalog S. Costantini

  2. Logic As a Query Language • If-then logical rules have been used in many systems. • Recursive rules extend relational algebra --- have been used to add recursion to SQL-99. S. Costantini / Logical Query Languages

  3. Datalog • A language for Deductive Databases • Extensional part: tables • Intensional part: rules S. Costantini / Logical Query Languages

  4. Logic: Intuition S. Costantini / Logical Query Languages

  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 / Logical Query Languages

  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 / Logical Query Languages

  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 / Logical Query Languages

  8. Datalog Terminology: Facts • Rules without body called “facts”: E.g., in(alan,r123). • Facts constitute the extensional part of a deductive database, i.e., define tables. S. Costantini / Logical Query Languages

  9. Alternative notation • Explicitly list attributes • E.g., facts: in(Object:alan,Place:r123) • E.g., rules: in(Object:X,Place:Y):- part_of(Place: Z, Place:Y), in(Object:X, Place:Z). S. Costantini / Logical Query Languages

  10. Datalog semantics • 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 / Logical Query Languages

  11. 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 / Logical Query Languages

  12. Least Herbrand Model: Intuition S. Costantini / Logical Query Languages

  13. 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 / Logical Query Languages

  14. 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 / Logical Query Languages

  15. 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 / Logical Query Languages

  16. 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) AND Sells(bar2,beer,p2) AND p1 < 2.00 AND p2 < 2.00 AND bar1 <> bar2 S. Costantini / Logical Query Languages

  17. 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 / Logical Query Languages

  18. Negated Subgoals: Intuition • Given atom A, not A holds if A cannot be proved • Negation as finite failure S. Costantini / Logical Query Languages

  19. 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 / Logical Query Languages

  20. Example: Unsafe Rules • Each of the following is unsafe and not allowed: • S(x) <- R(y) • S(x) <- R(y) AND NOT R(x) • S(x) <- R(y) AND x < y • In each case, an infinity of x ’s can satisfy the rule, even if R is a finite relation. S. Costantini / Logical Query Languages

  21. Datalog Programs • A Datalog program is a collection of rules. • In a program, predicates can be either • EDB = Extensional Database = stored table. • IDB = Intensional Database = relation defined by rules. S. Costantini / Logical Query Languages

  22. 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 / Logical Query Languages

  23. Datalog semantics • With negation: several proposals, in deductive databases answer set semantics. S. Costantini / Logical Query Languages

  24. 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 / Logical Query Languages

  25. Non-Monotonicità • Aggiungere nuove conoscenze può portare a ritrarre precedenti conclusioni. • Vi possono essere conclusioni alternative tramite cicli sulla negazione. S. Costantini / Logical Query Languages

  26. 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 / Logical Query Languages

  27. 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 / Logical Query Languages

  28. DATALOG + Negazione • Semantica degli Answer Sets • Paradigma dell’Answer Set Programming (ASP) • Negazione, insiemi-risposta alternativi • Numerosi solver S. Costantini / Logical Query Languages

  29. 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 / Logical Query Languages

  30. 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 / Logical Query Languages

  31. 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 / Logical Query Languages

  32. 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 / Logical Query Languages

  33. 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 / Logical Query Languages

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