Deductive Database

Deductive Database

Relation Database Title Year Length Type Harry Porter 2001 180 Color Gone with the wind 1938 125 Black and White Snow White Color 1950 90

A different view: a set of atoms movie(“Harry Porter”, 2001, 180, “Color”) movie(“Gone with the wind”, 1938, 125, “Black and White”) movie(“Snow White”, 1950, 90, “Color”) Predicate name Parameters – order is important

Predicates & Atoms Predicate – relation (associated with a fixed number of arguments) Atoms – predicates followed by a list of arguments (arguments can be variables or constants) – grounded if all arguments are constants, e.g., movie(“Harry Porter”, 2001, 180, “Color”) – non-grounded if some arguments are variables, e.g., movie(X, Y, 180, “Color”)

Atoms & Rows movie(“Harry Porter”, 2001, 180, “Color”) movie(“Gone with the wind”, 1938, 125, “Black and White”) movie(“Snow White”, 1950, 90, “Color”) Predicate name Parameters – order is important Each relation instance can be represented by a set of grounded atoms Each row corresponds to an atom. An atom is true if it corresponds to a row in the table; it is false if it does not correspond to any row in the table.

Why Deductive Database? Declarative: clear understanding of what a database represents Expressiveness: there are queries that cannot be formulated using relational algebra can be easily expressed by datalog rules

Datalog Rule a :- a1, , an, not b1, , not bm where a, a1,.,an,b1, ,bm are atoms and not is the negation-as-failure operator (n, m can be 0). A program (or query, knowledge base) is a set of rules

Rule - Meaning Rule a :- a1, , an, not b1, , not bm says the following: “If a1, ,an are true and b1, ,bm can be assumed to be false then a must be true.” If n 0 and m 0, then a must be true (fact). If m 0, the rule is a definite rule.

Rule – Examples colorMovie(M) :- movie(M,X,Y,Z), Z “Color”. longMovie(M):- movie(M,X,Y,Z), Y 120. oldMovie(M):- movie(M,X,Y,Z), X 1940. movieWithSequel(M):- movie(M,X1,Y1,Z1), movie(M,X2,Y2,Z2), X1 X2.

Rule – Do-not-care variables colorMovie(M) :- movie(M, , ,Z), Z “Color”. longMovie(M):- movie(M, ,Y, ), Y 120. oldMovie(M):- movie(M,X, , ), X 1940. movieWithSequel(M):- movie(M,X1, , ), movie(M,X2, , ), X1 X2.

Rule – Negation as failure noColorMovie(M) :- movie(M, , ,Z), not Z “Color”. notYetProducedMovie(M) :- not movie(M,X,Y,Z). NOTE: should be careful when using ‘not’; the second rule is not a good one (unsafe). It can say that any movie is not yet produced !

Deductive Database & Relational Algebra Projection colorMovie(M) :- movie(M, , ,Z), Z “Color”. Selection oldMovie(M,X,Y,Z):- movie(M,X,Y,Z),X 1940. Join (two relations p(A,B), q(B,C)) r(X,Y,Z):- p(X,Y), q(Y,Z) Union (two relations p(A,B), q(A,B)) r(X,Y):- p(X,Y). r(X,Y):- q(X,Y). Set difference (two relations p(A,B), q(A,B)) r(X,Y):- p(X,Y), not q(X,Y). Cartesian product (two relations p(A,B), q(C,D)) c(X,Y,Z,W):- p(X,Y), q(Z,W).

Programs – Examples Given the following program q(1,2). q, r are “extensional predicates” p is “intensional predicate” q(1,3). Stored in Tables r(2,3). r(3,1). p(X,Y ):- q(X,Z), r(Z,Y ), not q(X,Y ). Question: what are the atoms of the predicate p which are defined by the program? Computed from the programs

Programs – Examples q(1,2). q(1,3). r(2,3). r(3,1). p(X,Y ):- q(X,Z), r(Z,Y ), not q(X,Y ). X, Y can be 1, 2, or 3 p(1,3):- q(1,2),r(2,3), not q(1,3). NO p(1,3):- q(1,3),r(3,1), not q(1,1). YES .

The meaning of a program For a program P, what are the extensional atoms defined by P? It is simple for the case where rules defining the extensional predicate is nonrecursive (previous example!)

Recursive Datalog Given: Mother-Child relation mother(Ana, Deborah) mother(Deborah, Nina) mother(Nina, Anita) Define: grand-mother relation? grandma(X,Y):- mother(X,Z),mother(Z,Y).

Recursive Datalog Given: Mother-Child relation mother(Ana, Deborah) mother(Deborah, Nina) mother(Nina, Anita) Define: grand-mother relation? grandma(X,Y):- mother(X,Z),mother(Z,Y). Answer: grandma(Deborah, Anita) and grandma(Ana,Nina) Question: relational algebra expression for grandma? POSSIBLE?

Recursive Datalog Given: Mother-Child relation mother(Ana, Deborah) mother(Deborah, Nina) mother(Nina, Anita) Define: ancestor relation? ancestor(X,Y):- mother(X,Y). ancestor(X,Y):- mother(X,Z), ancestor(Z,Y). Answer: ? (mother, grandma, and great-grandma(Ana, Anita)) Question: relational algebra expression for ancestor? POSSIBLE?

Computing the intensional predicates in recursive Datalog ancestor Check if any rule satisfies: three rules – get – ancestor(Ana, Deborah) – ancestor(Deborah, Nina) – ancestor(Nina, Anita) Repeat the second steps until ancestor does not change: – ancestor(Ana, Nina) – ancestor(Deborah, Anita) – ancestor(Ana, Anita)

Program with Negation-as-failure Different situations: – There might be only one solution – There might be several solutions – There might be no solution

Recursive Program – Another Example Given a table of flight schedule of UA and AA Airline From To Departure Arrival UA SF DEN 930 1230 AA SF DEN 900 1430 UA DEN CHI 1500 1800 UA DEN DAL 1400 1700 AA DAL NY 1500 1930 AA DAL CHI 1530 1730 AA CHI NY 1900 2200 UA CHI NY 1830 2130

reachable(X,Y) – Y can be reached from X flight(Airline,From,To,Departure,Arrival) reachable(X,Y) :- flight(A,X,Y,D,R). reachable(X,Z) :- flight(A,X,Y,D,R), reachable(Y,Z). Computing reachable – similar to ancestor

reachableUA(X,Y) – Y can be reached by UA from X reachableUA(X,Y) :- flight(ua,X,Y,D,R). reachableUA(X,Z) :- flight(ua,X,Y,D,R), reachableUA(Y,Z). Easy: only considers UA flights

reachableUA(X,Y) – Y can be reached by UA’s flight from X reachableUA(X,Y) :- flight(ua,X,Y,D,R). reachableUA(X,Z) :- flight(ua,X,Y,D,R), reachableUA(Y,Z). Easy: only considers UA flights

reachableUA(X,Y) & reachableAA(X,Y) reachableUA(X,Y) :- flight(ua,X,Y,D,R). reachableUA(X,Z) :- flight(ua,X,Y,D,R), reachableUA(Y,Z). reachableAA(X,Y) :- flight(aa,X,Y,D,R). reachableAA(X,Z) :- flight(aa,X,Y,D,R), reachableAA(Y,Z).

reachableOnlyAA(X,Y) reachableUA(X,Y) :- flight(ua,X,Y,D,R). reachableUA(X,Z) :- flight(ua,X,Y,D,R), reachableUA(Y,Z). reachableAA(X,Y) :- flight(aa,X,Y,D,R). reachableAA(X,Z) :- flight(aa,X,Y,D,R), reachableAA(Y,Z). reachableOnlyAA(X,Y):- reachableAA(X,Y), not reachableUA(X,Y). reachableOnlyAA(X,Y) – Unique set of atoms satisfying this properties

A different situation Suppose that the extensional predicate is r and r(0) is the only atom which is true Consider the program p(X) :- r(X), not q(X) q(X) :- r(X), not p(X) {p(0)} or {q(0)} could be used as ‘the answer’ for this program

Yet another situation Suppose that the extensional predicate is r and r(0) is the only atom which is true Consider the program r(X) :- not r(X) Is r(0) true or false? Is r(1) true or false?

Minimal Model of Positive Program Given a program P without negation-as-failure TP(X) {a a :- a1, ,an P, {a1, ,an} X} If P is a positive program, the sequence TP( ), TP(TP( )), converges again a set of atoms, which is called the minimal model of P

mother(a, d). mother(d, n). mother(n, t) ancestor(X,Y):- mother(X,Y). ancestor(X,Y):- mother(X,Z), ancestor(Z,Y). T0 TP( ) {mother(a, d), mother(d, n), mother(n, t)} T1 TP(TP( )) T0 {ancestor(a, d), ancestor(d, n), ancestor(n, t)} T2 TP(TP(TP( ))) T1 {ancestor(a, n), ancestor(d, n)} T3 TP(TP(TP(TP( )))) T2 {ancestor(a, t)}

Stable Models Given a program P and a set of atoms S PS is a program obtained from S by – Removing all rules in P which contain some not a in the body and a S – Removing all not a from the remaining rules S is a stable model of P if S is the minimal model of the program PS

Example Given the program P: p :- not q. q :- not p. S {q} PS consists of a single rule q. The minimal model of PS is {q}. So, {q} is a stable model of P S {p} PS consists of a single rule p. The minimal model of PS is {p}. So, {p} is a stable model of P

Example Given the program P: p :- not q. q. S {q} PS consists of a single rule q. The minimal model of PS is {q}. So, {q} is a stable model of P S {q, p} PS consists of a single rule q. The minimal model of PS is {q}. So, {q, p} is not a stable model of P

Example Given the program P: p :- not p. S {} PS consists of a single rule p. The minimal model of PS is {p}. So, {} is not a stable model of P S {p} PS is empty. The minimal model of PS is {}. So, {p} is not a stable model of P

Entailment Given a program P and an atom p. Question: Is p true given P (or: P p)? Answer – YES if p appears in every stable model of P – NO if p does not appear in at least on stable model of P There are systems for computing the stable models.