1. Deductive Databases Jan. 2008Yangjun Chen ACS-39021 Outline What is a deductive database system? Some basic concepts Basic inference mechanism for logic programs Datalog programs and their evaluation

2. Deductive Databases Jan. 2008Yangjun Chen ACS-39022 What is a deductive database system? A deductive database can be defined as an advanced database augmented with an inference system. Database + Inference Deductive database By evaluating rules against facts, new facts can be derived, which in turn can be used to answer queries. It makes a database system more powerful.

3. Deductive Databases Jan. 2008Yangjun Chen ACS-39023 Some basic concepts from logic To understand the deductive database system well, some basic concepts from mathematical logic are needed. -term -n-ary predicate -literal -(well-formed) formula -clause and Horn-clause -facts -logic program

4. Deductive Databases Jan. 2008Yangjun Chen ACS-39024 -term A term is a constant, a variable or an expression of the form f(t 1, t 2,..., t n ), where t 1, t 2,..., t n are terms and f is a function symbol. -Example: a, b, c, f(a, b), g(a, f(a, b)), x, y, g(x, y) -n-ary predicate An n-ary predicate symbol is a symbol p appearing in an expression of the form p(t 1, t 2,..., t n ), called an atom, where t 1, t 2,..., t n are terms. p(t 1, t 2,..., t n ) can only evaluate to true or false. -Example: p(a, b), q(a, f(a, b)), p(x, y)

5. Deductive Databases Jan. 2008Yangjun Chen ACS-39025 -literal A literal is either an atom or its negation. -Example: p(a, f(a, b)),  p(a, f(a, b)) -(well-formed) formula -A well-formed (logic) formula is defined inductively as follows: - An atom is a formula. - If P and Q are formulas, then so are  P, (P  Q), (P  Q), (P  Q), and (P  Q). - If x is a variable and P is a formula containing x, then (  xP) and (  xP) are formulas.

6. Deductive Databases Jan. 2008Yangjun Chen ACS-39026 -clause -A clause is an expression of the following form:  A 1   A 2 ...   A n  B 1 ...  B m where A i and B j are atoms. -The above expression can be written in the following equivalent form: B 1 ...  B m  A 1 ...  A n or B 1,..., B m  A 1,..., A n antecedent consequent

7. Deductive Databases Jan. 2008Yangjun Chen ACS-39027 -clause  A  B AB 100 111 011 001 B  A AB 100 111 011 001 -Horn clause A Horn clause is a clause with the head containing only one positive atom. B m  A 1,..., A n

8. Deductive Databases Jan. 2008Yangjun Chen ACS-39028 -fact -A fact is a special Horn clause of the following form: B  with all variables in B being instantiated. (B  can be simply written as B.) -logic program A logic program is a set of Horn clauses.

9. Deductive Databases Jan. 2008Yangjun Chen ACS-39029 -Example (a logic program) Facts: supervise(franklin, john), supervise(franklin, ramesh), supervise(franklin, joyce) supervise(james, franklin), supervise(jennifer, alicia), supervise(jennifer, ahmad), supervise(james, jennifer). Rules: superior(X, Y)  supervise(X, Y), superior(X, Y)  supervise(X, Z), superior(Z, Y), subordinary(X, Y)  superior(Y, X). james franklin jennifer johnrameshjoycealicia ahmad

10. Deductive Databases Jan. 2008Yangjun Chen ACS-390210 Facts can be considered as the data stored as relations in a relational database.

11. Deductive Databases Jan. 2008Yangjun Chen ACS-390211 Basic inference mechanism for logic programs -interpretation of programs (rules + facts) There are two main alternatives for interpreting the theoretical meaning of rules: proof theoretic, and model theoretic interpretation - proof theoretic interpretation 1.The facts and rules are considered to be true statements, or axioms. facts - ground axioms rules - deductive axioms 2.The deductive axioms are used to construct proofs that derive new facts from existing facts.

12. Deductive Databases Jan. 2008Yangjun Chen ACS-390212 -Example: 1.superior(X, Y)  supervise(X, Y).(rule 1) 2.superior(X, Y)  supervise(X, Z), superior (Z, Y).(rule 2) 3.supervise(jennifer, ahmad). (ground axiom, given) 4.supervise(james, jennifer). (ground axiom, given) 5.superior(jennifer, ahmad). (apply rule 1 on 3) 6.superior(james, ahmad). (apply rule 2 on 4 and 5)

13. Deductive Databases Jan. 2008Yangjun Chen ACS-390213 - model theoretic interpretation 1.Given a finite or an infinite domain of constant values, assign to each predicate in the program every possible combination of values as arguments. 2.All the instantiated predicates contitute a Herbrand base. 3. An interpretation is a subset of the Herbrand base. 4.In the Herbrand base, each instantiated predicate evaluates to true or false in terms of the given facts and rules. 5.An interpretation is called a model for a specific set of rules and the corresponding facts if those rules are always true under that interpretation. 6.A model is a minimal model for a set of rules and facts if we cannot change any element in the model from true to false and still get a model for these rules and facts.

14. Deductive Databases Jan. 2008Yangjun Chen ACS-390214 -Example: 1.superior(X, Y)  supervise(X, Y).(rule 1) 2.superior(X, Y)  supervise(X, Z), superior(Z, Y).(rule 2) known facts: supervise(franklin, john), supervise(franklin, ramesh), supervise(franklin, joyce), supervise(james, franklin), supervise(jennifer, alicia), supervise(jennifer, ahmad), supervise(james, jennifer). For all other possible (X, Y) combinations supervise(X, Y) is false. domain = {james, franklin, john, ramesh, joyce, jennifer, alicia, ahmad}

15. Deductive Databases Jan. 2008Yangjun Chen ACS-390215 Interpretation - model - minimal model known facts: supervise(franklin, john), supervise(franklin, ramesh), supervise(franklin, joyce), supervise(james, franklin), supervise(jennifer, alicia), supervise(jennifer, ahmad), supervise(james, jennifer). For all other possible (X, Y) combinations supervise(X, Y) is false. derived facts: superior(franklin, john), superior(franklin, ramesh), superior(franklin, joyce), superior(jennifer, alicia), superior(jennifer, ahmad), superior(james, franklin), superior(james, jennifer), superior(james, john), superior(james, ramesh), superior(james, joyce), superior(james, alicia), superior(james, ahmad). For all other possible (X, Y) combinations superior(X, Y) is false.

16. Deductive Databases Jan. 2008Yangjun Chen ACS-390216 The above interpretation is also a model for the rules (1) and (2) since each of them evaluates always to true under the interpretation. For example, superior(X, Y)  supervise(X, Y) superior(franklin, john)  supervise(franklin, john) is true. superior(franklin, ramesh)  supervise(franklin, ramesh) is true.... … superior(X, Y)  supervise(X, Z), superior(Z, Y) superior(james, ramesh)  supervise(james, franklin), superior (franklin, ramesh) is true. superior(james, alicia)  supervise(james, jennifer), superior (jennifer, alicia) is true.

17. Deductive Databases Jan. 2008Yangjun Chen ACS-390217 The model is also the minimal model for the rule (1) and (2) and the corresponding facts since eliminating any element from the model will make some facts or instatiated rules evaluate to false. For example, eliminating supervise(franklin, john) from the model will make this fact no more true under the interpretation; eliminating superior (james, ramesh) will make the following rule no more true under the interpretation: superior(james, ramesh)  supervise(james, franklin), superior(franklin, ramesh)

18. Deductive Databases Jan. 2008Yangjun Chen ACS-390218 -Inference mechanism In general, there are two approaches to evaluating logical programs: bottom-up and top-down. -Bottom-up mechanism (also called forward chaining and bottom-up resolution) 1. The inference engine starts with the facts and applies the rules to generate new facts. That is, the inference moves forward from the facts toward the goal. 2. As facts are generated, they are checked against the query predicate goal for a match.

19. Deductive Databases Jan. 2008Yangjun Chen ACS-390219 -Example query goal: superior(james, Y)? rules and facts are given as above. 1.Check whether any of the existing facts directly matches the query. 2.Apply the first rule to the existing facts to generate new facts. 3.Apply the second rule to the existing facts to generate new facts. 4.As each fact is gnerated, it is checked for a match of the the query goal. 5.Repeat step 1 - 4 until no more new facts can be found. All the facts of the form: superior(james, a) are the answers.

20. Deductive Databases Jan. 2008Yangjun Chen ACS-390220 -Example: 1.superior(X, Y)  supervise(X, Y).(rule 1) 2.superior(X, Y)  supervise(X, Z), superior(Z, Y).(rule 2) known facts: supervise(franklin, john), supervise(franklin, ramesh), supervise(franklin, joyce), supervise(james, franklin), supervise(jennifer, alicia), supervise(jennifer, ahmad), supervise(james, jennifer). For all other possible (X, Y) combinations supervise(X, Y) is false. domain = {james, franklin, john, ramesh, joyce, jennifer, alicia, ahmad} superior(james, Y)? applying the first rule: superior(james, franklin), superior(james, jennifer) Y = {franklin, jennifer} applying the second rule: Y = {John, Joyce, Ramesh, alicia, ahmad}

21. Deductive Databases Jan. 2008Yangjun Chen ACS-390221 -Top-down mechanism (also called back chaining and top-down resolution) 1. The inference engine starts with the query goal and attempts to find matches to the variables that lead to valid facts in the database. That is, the inference moves backward from the intended goal to determine facts that would satisfy the goal. 2. During the course, the rules are used to generate subgoals. The matching of these subgoals will lead to the match of the intended goal.

22. Deductive Databases Jan. 2008Yangjun Chen ACS-390222 -Example query goal: ?-superior(james, Y) rules and facts are given as above. Query: ?-superior(james, Y) Rule1: superior(james, Y)  supervise(james, Y) Rule2: superior(james, Y)  supervise(james, Z), superior(Z, Y) supervise(james, Z) superior(franklin, Y)superior(jennifer, Y) Y=franklin, jennifer Z=frankiln Z=jennifer

23. Deductive Databases Jan. 2008Yangjun Chen ACS-390223 Rule1: superior(franklin, Y)  supervise(franklin, Y) Rule1: superior(jennifer, Y)  supervise(jennifer, Y) Y= john, ramesh, joyce Y= alicia, ahmad

24. Deductive Databases Jan. 2008Yangjun Chen ACS-390224 Datalog programs and their evaluation 1.A Datalog program is a logic program. 2.In a Datalog program, each predicate contains no function symbols. 3.A Datalog program normally contains two kinds of predicates: fact-based predicates and rule-based predicates. fact-based predicates are defined by listing all the combinations of values that make the predicate true. Rule-based predicates are defined to be the head of one or more Datalog rules. They correspond to virtual relations whose contents can be inferred by the inference engine.

25. Deductive Databases Jan. 2008Yangjun Chen ACS-390225 Datalog programs and their evaluation Example: -All the programs discussed earlier are Datalog programs. superior(X, Y)  supervise(X, Y). superior (X, Y)  supervise(X, Z), superior (Z, Y). supervise(jennifer, ahmad). supervise(james, jennifer). -The following is a logic program, but not a Datalog program: p(X, Y)  q(f(Y), X)

26. Deductive Databases Jan. 2008Yangjun Chen ACS-390226 Datalog programs and their evaluation two important concepts: - safety of programs - predicate dependency graph

27. Deductive Databases Jan. 2008Yangjun Chen ACS-390227 Datalog programs and their evaluation -Safety of programs A Datalog program or a rule is said to be safe if it generates a finite set of facts. -Condition of unsafty A rule is unsafe if one of the variables in the rule can range over an infinite domain of values, and that variable is not limited to ranging over a finite predicate before it is instantiated. -Example: big_salary(Y)  Y > 60000. big_salary(Y)  Y > 60000, employee(X), salary(X, Y).

28. Deductive Databases Jan. 2008Yangjun Chen ACS-390228 Datalog programs and their evaluation -Example: ?-big_salary(Y) big_salary(Y)  Y > 60000. big_salary(Y)  Y > 60000, employee(X), salary(X, Y). The evaluation of these rules (no matter whether in bottom- up or in top-down fashion) will never terminate. The following is a safe rule: big_salary(Y)  employee(X), salary(X, Y), Y > 60000.

29. Deductive Databases Jan. 2008Yangjun Chen ACS-390229 Datalog programs and their evaluation A variable X is limited if (1) it appears in a regular (not built-in) predicate in the body of the rule. (built-in predicates:, , , =,  ) (2) it appears in a predicate of the form X = c or c = X, where c is a constant. (3) it appears in a predicate of the form X = Y or Y = X in the rule body, where Y is a limited variable. (4) Before it is instantiated, some other regular predicates containing it will have been evaluated. -Condition of safty: A rule is safe if each variable in it is limited. A program is safe if each rule in it is safe.

30. Deductive Databases Jan. 2008Yangjun Chen ACS-390230 Datalog programs and their evaluation - predicate dependency graphs For a program P, we construct a dependency graph G representing a refer to relationship between the predicates in P. This is a directed graph where there is node for each predicate and an arc from node q to node p if and only if the predicate q occurs in the body of a rule whose head predicate is p. Exampel: superior(X, Y)  supervise(X, Y), superior(X, Y)  supervise(X, Z), superior(Z, Y), subordinary(X, Y)  superior(Y, X), supervisor(X, Y)  employee(X), supervise(X, Y), over_40K_emp(X)  employee(X), salary(X, Y), Y  40000, under_40K_supervisor(X)  supervisor(X), not(over_40K_emp(X)), main_productx _emp(X )  employee(X), workson(X, productx, Y), Y  20, president(X)  employee(X), not(supervise(Y, X)).

31. Deductive Databases Jan. 2008Yangjun Chen ACS-390231 Datalog programs and their evaluation -predicate dependency graphs worksonemployeesalarysupervise departmentprojectfemalemale main_poductx_emppresidentover_40K_empsuperior supervisorunder_40K_supervisor subordinate  

32. Deductive Databases Jan. 2008Yangjun Chen ACS-390232 Datalog programs and their evaluation Evaluation of nonrecursive rules -If the dependency graph for a rule set has no cycles, the rule set is nonrecursive.

33. Deductive Databases Jan. 2008Yangjun Chen ACS-390233 Datalog programs and their evaluation -Evaluation of nonrecursive rules -evaluation involving only fact-based predicates ?-salary(X, 60000)  $1 (  $2 = “60000” (salary)) -evaluation involving only rule-based predicates 1.rule rectification h(X, c) ... h(X, Y) ...,Y=c h(X, X) ...h(X, Y) ..., Y=X

34. Deductive Databases Jan. 2008Yangjun Chen ACS-390234 Datalog programs and their evaluation -evaluation involving only rule-based predicate 2.Single rule evaluation To evaluate a rule of the from: p  p 1,..., p n we first compute the relations corresponding to p 1,..., p n and then the relation corresponding to p. 3.All the rules will be evaluated along the predicate dependency graph. At each step, each rule will be evaluated in terms of step (2).

35. Deductive Databases Jan. 2008Yangjun Chen ACS-390235 Datalog programs and their evaluation -The general bottom-up evaluation strategy for a nonrecursive query ?-p(x1, x2, …, xn) 1. Locate a set of rules S whose head involves the predicate p. If there are no such rules, then p is a fact-based predicate corresponding to some database relation R p ; in this case, one of the following expression is returned and the algorithm is terminated. (We use the notation $i to refer to the name of the i-th attribute of relation R p.) (a)If all arguments in p are distinc variables, the relational expression returned is R p. (b)If some arguments are constants or if the same variable appears in more than one argument position, the expression returned is SELECT (R p ),

36. Deductive Databases Jan. 2008Yangjun Chen ACS-390236 where the is a conjunctive condition made up of a number of simple conditions connected by AND, and constructed as follows: i.if a constant c appears as argument i, include a simple condition ($i = c) in the conjuction. ii.if the same variable appears in both argument location j and k, include a condition ($j = $k) in the conjuction. 2.At this point, one or more rules S i, i = 1, 2,..., n, n > 0 exist with predicate p as their head. For each such rule S i, generate a relational expression as follows: a.Apply selection operation on the predicates in the body for each such rule, as discussed in Step 1(b). b.A natural join is constructed among the relations that correspond to the predicates in the body of the rule S i over the common variables. Let the resulting relation from this join be R s.

37. Deductive Databases Jan. 2008Yangjun Chen ACS-390237 c.If any built-in predicate X  Y was defined over the arguments X and Y, the result of the join is subjected to an additional selection: SELECT X  Y (R s ) d.Repeat Step 2(c) until no more built-in predicates apply. 3. Take the UNION of the expressions generated in Step 2 (if more than one rule exists with predicate p as its head.)

38. Deductive Databases Jan. 2008Yangjun Chen ACS-390238 Datalog programs and their evaluation Evaluation of recursive rules -If the dependency graph for a rule set has at least one cycle, the rule set is recursive. ancestor(X, Y)  parent(X, Y), ancestor(X, Y)  parent(X, Z), ancestor(Z, Y). -naive strategy -semi-naive strategy -stratified databases

39. Deductive Databases Jan. 2008Yangjun Chen ACS-390239 Datalog programs and their evaluation -some teminology for recursive queries -linearly recursive -left linearly recursive ancestor(X, Y)  ancestor(X, Z), parent(Z, Y) -right linearly recursive ancestor(X, Y)  parent(X, Z), ancestor(Z, Y) -non-linearly recursive sg(X, Y)  sg(X, Z), sibling(Z, W), sg(W, Y)

40. Deductive Databases Jan. 2008Yangjun Chen ACS-390240 Datalog programs and their evaluation -some teminology for recursive queries -extensional database (EDB) predicate An EDB predicate is a predicate whose relation is stored in the database - fact-based predicate. -intensional database (IDB) predicate An IDB predicate is a predicate whose relation is defined by logic rules - rule-based predicate. -Datalog equation A Datalog equation is an equation obtained by replacing “  ” and “  ” with “=” and “ ” in a rule, respectively. a(X, Y) = p(X, Y)   X,Y (p(X, Z) a(Z, Y))

41. Deductive Databases Jan. 2008Yangjun Chen ACS-390241 Datalog programs and their evaluation -some teminology for recursive queries -fixed point Consider a relation sequence: g 0, g 1, …, g i, g i+1,... E i (g 0 ) = E (E(... E(g 0 )... )) i If at some time we have E i (g 0 ) = E i+1 (g 0 ), then E i (g 0 ) is the fixed point of the function E(...). It is also the least fixed point of E(...). If there exits some g such that g = E(g), g is called the fixed point. The least among all fixed points of E(...) is called the least fixed point. -evaluation of fixed points g 0 = , g i+1 = E(g i ),

42. Deductive Databases Jan. 2008Yangjun Chen ACS-390242 Datalog programs and their evaluation -some teminology for recursive queries -fixed point Example: a(X, Y) = p(X, Y)   X,Y (p(X, Z) a(Z, Y)) p = {(f, j), (f, r), (f, jo), (je, a), (je, ah), (ja, f), (ja, je)} a 0 = { } a 1 = {(f, j), (f, r), (f, jo), (je, a), (je, ah), (ja, f), (ja, je)} a 2 = {(f, j), (f, r), (f, jo), (je, a), (je, ah), (ja, f), (ja, je), (ja, j), (ja, r), (ja, jo), (ja, a), (ja, ah)} a 3 = a 2 least fixed point The least fixed point of the above equation is also called the transitive closure of p.

43. Deductive Databases Jan. 2008Yangjun Chen ACS-390243 Datalog programs and their evaluation -evaluation of recursive queries -naive strategy 1.The naive evaluation method is a bottom-up strategy which computes the least model of a Datalog program. 2.It is an iterative strategy and at each iteration all rules are applied to the set of tuples produced thus far to generate all implicit tuples. 3.This iterative process continues until no more new tuples can be produced.

44. Deductive Databases Jan. 2008Yangjun Chen ACS-390244 Datalog programs and their evaluation -naive strategy Consider the following equation system: R i = E i (R 1,..., R i,..., R n ) (i = 1,..., m) which is formed by replacing the  symbol with an equality sign in a Datalog program. Algorithm Jacobi naive strategy input:A system of algebraic equations and EDB. output:The values of the variable relations: R 1,..., R i,..., R n. for i = 1 to n do R i :=  ; repeat Con := true; for i = 1 to n do S i := R i ; for i = 1 to m do {R i := E i (S 1,..., S i,..., S n ); if R i  S i then {Con := false; S i := R i ;}} until Con = true;

45. Deductive Databases Jan. 2008Yangjun Chen ACS-390245 Datalog programs and their evaluation -naive strategy sg(X, Y)  sg(X, W), sibling(W, Z), sg(Z, Y) sibling(X, Y)  parent(X, W), sibling(W, Z), parent(Y, Z) sg = E 1 (sg, sibling) sibling = E 2 (sibling) sg(X, Y) =  X,Y (sg(X, W) sibling(W, Z) sg(Z, Y)) sibling(X, Y) =  X,Y (parent(X, W) sibling(W, Z) parent(Y, Z)) sg R1R1 sibling R2R2 R 1 = E 1 (R 1, R 2 ) R 2 = E 2 (R 2 )

46. Deductive Databases Jan. 2008Yangjun Chen ACS-390246 Datalog programs and their evaluation -naive strategy Example: ancestor(X, Y)  parent(X, Y), ancestor(X, Y)  parent(X, Z), ancestor(Z, Y). Parent = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank)} bert alicegeorge derekpat frank

47. Deductive Databases Jan. 2008Yangjun Chen ACS-390247 Datalog programs and their evaluation -naive strategy Example: A(X, Y) =  X,Y (P(X, Z) A(Z, Y))  P(X, Y) step 0:A 0 =  step 1:A 1 = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank)} step 2:A 2 = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank), (bert, derek), (bert, pat), (alice, frank)} step 3: A 3 = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank), (bert, derek), (bert, pat), (alice, frank), (bert, frank)} step 4: A 4 = A 3

48. Deductive Databases Jan. 2008Yangjun Chen ACS-390248 Datalog programs and their evaluation -naive strategy Algorithm Gauss-Seidel naive strategy Jacobi: k-th iteration: R 1 (k) = E 1 (R 1 (k-1),..., R i (k-1),..., R n (k-1)), … R i (k) = E i (R 1 (k-1),..., R i (k-1),..., R n (k-1)), … R n (k) = E n (R 1 (k-1),..., R i (k-1),..., R n (k-1)). Gauss-Seidel: k-th iteration: R 1 (k) = E 1 (R 1 (k-1),..., R i (k-1),..., R n (k-1)), … R i (k) = E i (R 1 (k),..., R i (k-1),..., R n (k-1)), … R n (k) = E n (R 1 (k),..., R i (k),..., R n (k-1)).

49. Deductive Databases Jan. 2008Yangjun Chen ACS-390249 Datalog programs and their evaluation -evaluation of recursive queries -semi-naive strategy 1.The semi-naive evaluation method is a bottom-up strategy. 2.It is designed to eliminate redundancy in the evaluation of tuples at different iterations. Let R i(k) be the temporary value of relation R i at iteration step k. The differential of R i between step k and step k - 1 is defined as follows: D i(k) = R i(k) - R i(k-1) For a linearly recursive rule set, D i(k) can be substituted for R i in the k-th iteration of the naïve algorithm. 3.The result is obtained by the union of the newly obtained term R i and that obtained in the previous step.

50. Deductive Databases Jan. 2008Yangjun Chen ACS-390250 Datalog programs and their evaluation -evaluation of recursive queries -semi-naive strategy Algorithm seminaiv strategy input:A system of algebraic equations and EDB. output:The values of the variable relations: R 1,..., R i,..., R n. for i = 1 to n do R i :=  ; for i = 1 to m do D i :=  ; repeat Con := true; for i = 1 to n do {D i := E(D 1,..., D i,..., D n ) - R i ; R i := D i  R i ; if D i   then Con := false; } until Con is true;

51. Deductive Databases Jan. 2008Yangjun Chen ACS-390251 Datalog programs and their evaluation -evaluation of recursive queries -semi-naive strategy Example: Step 0:D 0 = , A 0 =  ; Step 1:D 1 = P = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank)} A 1 = D 1  A 0 = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank)} Step 2: D 2 = {(bert, derek), (bert, pat), (alice, frank)} A 2 = D 2  A 1 = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank), {(bert, derek), (bert, pat), (alice, frank)}

52. Deductive Databases Jan. 2008Yangjun Chen ACS-390252 Datalog programs and their evaluation -evaluation of recursive queries -semi-naive strategy Example: Step 3: D 3 = {(bert, frank) A 3 = D 3  A 2 = {(bert, alice), (bert, george), (alice, derek), (alice, part), (derek, frank), {(bert, derek), (bert, pat), (alice, frank), (bert, frank)} Step 3: D 4 = . The advantage of the semi-naive method is that at each step a differential term D i is used in each equation instead of the whole R i. In this way, the time complexity of a computation is decreased drastically.

53. Deductive Databases Jan. 2008Yangjun Chen ACS-390253 Datalog programs and their evaluation -evaluation of recursive queries -The magic-set rule rewriting technique motivation: 1.During a bottom-up evaluation, too many irrelevant tuples are evaluated. For example, to evaluate the query sg(john, Z)? using the following rules: sg(X, Y)  flat(X, Y), sg(X, Y)  up(X, Z), sg(Z, W), down(W, Y), a bottom-up method will generate all sg-tuples and then makes a selection operation to the anwsers. 2. Using the constants appearing in the query to restrict computation.

54. Deductive Databases Jan. 2008Yangjun Chen ACS-390254 Datalog programs and their evaluation -evaluation of recursive queries -The magic-set rule rewriting technique sg(X, Y)  magic_sg(X),flat(X, Y), sg(X, Y)  magic_sg(X), up(X, Z), sg(Z, W), down(W, Y), magic_sg(Z)  magic_sg(X), up(X, Z), magic_sg(john). Modified rules Magic rules Two-phase evaluation: 1st phase:evaluate magic rules to generate a magic set. 2nd phase:evaluate modified rules, by which that magic set is used to restrict the computation.

55. Deductive Databases Jan. 2008Yangjun Chen ACS-390255 Datalog programs and their evaluation -evaluation of recursive queries -Stratified databases A stratified database is a Datalog program containing negated predicates. Example: Suppose that a supplier might wish to backorder items that are not in the warehouse. It would be convenient to write: backorder(X)  item(X),  warehouse(X). Its logically equivalent form is backorder(X), warehouse  item(X). But this rule has a different meaning : if X is an item, then backorder it or it is stored in the warehouse. This is not what we want.

56. Deductive Databases Jan. 2008Yangjun Chen ACS-390256 Datalog programs and their evaluation -evaluation of recursive queries -Stratified databases -Prolblem: recursion via negation p(X)   q(X), q(X)  p(X). -To avoid the recursion via negation, we introduce the concept of stratification, which is defined by the use of a level l mapping. level l mapping: assign each literal in the program an integer such that if B  A 1, …, A n and A i is positive, then l(A i )  l(B) for all i, 1  i  n. If A i is negative, then l(B) < l(A i ) for all i, 1  i  n.

57. Deductive Databases Jan. 2008Yangjun Chen ACS-390257 Datalog programs and their evaluation -evaluation of recursive queries -Stratified databases -Prolblem: recursion via negation p(X)   q(X), q(X)  p(X). -To avoid the recursion via negation, we introduce the concept of stratification, which is defined by the use of a level l mapping. level l mapping: assign each literal in the program an integer such that if B  A 1, …, A n and A i is positive, then l(A i )  l(B) for all i, 1  i  n. If A i is negative, then l(B) < l(A i ) for all i, 1  i  n.

58. Deductive Databases Jan. 2008Yangjun Chen ACS-390258 Datalog programs and their evaluation -evaluation of recursive queries -Stratified databases -If you can assign integers to all the literals in a programusing a level mapping, then this program is stratifiable. p(X)   q(X), q(X)  p(X). In fact, we cannot find a level mapping for any program which contains recursion via negation. -Evaluation of a stratified database. Evaluate the literals in the program from low level to the high level.

59. Deductive Databases Jan. 2008Yangjun Chen ACS-390259 Datalog programs and their evaluation -evaluation of recursive queries -Stratified databases -However, you cannot find any level mapping for the following program: Example: path(X, Y)  edge(X, Y), path(X,Y)  edge(X, Z), path(Z, Y), acyclic_path(X, Y)  path(X,Y),  path(Y, X). We can many label mappings for this program. The following are just two of them: