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Jan. 2012 Yangjun Chen ACS-3902 1
Deductive Databases
Outline Chapter 25 – 3rd ed. (Chap. 24.4 – 4th, 5th ed.; 26.5, 6th ed.)
•What is a deductive database system?
•Some basic concepts
•Basic inference mechanism for logic programs
•Datalog programs and their evaluation
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Deductive Databases
•What is a deductive database system?
A deductive database can be defined as an advanced database augmented with an inference system.
Database + InferenceDeductivedatabase
By evaluating rules against facts, new facts can be derived, which in turncan be used to answer queries. It makes a database system more powerful.
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•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
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- term
A term is a constant, a variable or an expression of the form f(t1, t2, ..., tn), where t1, t2, ..., tn 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(t1, t2, ..., tn), called an atom, where t1, t2, ..., tn are terms. p(t1, t2, ..., tn) can only evaluate to true or false.
- Example: p(a, b), q(a, f(a, b)), p(x, y)
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- 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, (PQ), (PQ), (PQ), and (PQ).
- If x is a variable and P is a formula containing x, then (xP) and (xP) are formulas.
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- clause
- A clause is an expression of the following form:
A1 A2 ... An B1 ... Bm
where Ai and Bj are atoms.
- The above expression can be written in the following equivalent form:
B1 ... Bm A1 ... An
or
B1, ..., Bm A1 , ..., An
antecedentconsequent
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- clause
A BA B
1 0 0
1 1 10 1 1
0 0 1
B AA B
1 0 0
1 1 10 1 1
0 0 1
- Horn clauseA Horn clause is a clause with the head containing only one positive atom.
Bm A1 , ..., An
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- 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.
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- 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
john ramesh joyce alicia ahmad
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Facts can be considered as the data stored as relations in a relational database.
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•Basic inference mechanism for logic programs- interpretation of programs (rules + facts)
There are two main alternatives for interpreting the theoreticalmeaning of rules:
proof theoretic, andmodel theoretic interpretation
- proof theoretic interpretation1. The facts and rules are considered to be true statements,
or axioms.facts - ground axiomsrules - deductive axioms
2. The deductive axioms are used to construct proofs thatderive new facts from existing facts.
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- 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)
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- model theoretic interpretation1. Given a finite or an infinite domain of constant values,
assign to each predicate in the program every possiblecombination 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 trueunder that interpretation.
6. A model is a minimal model for a set of rules and facts ifwe cannot change any element in the model from true to false and still get a model for these rules and facts.
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- 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}
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Interpretation - model - minimal modelknown 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.
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The above interpretation is also a model for the rules (1) and (2) since eachof 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.
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The model is also the minimal model for the rule (1) and (2) and thecorresponding facts since eliminating any element from the modelwill make some facts or instatiated rules evaluate to false. For example,
eliminating supervise(franklin, john) from the model will make this factno more true under the interpretation;
eliminating superior (james, ramesh) will make the following rule nomore true under the interpretation:
superior(james, ramesh) supervise(james, franklin),superior(franklin, ramesh)
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- Inference mechanismIn general, there are two approaches to evaluating logicalprograms: 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 appliesthe rules to generate new facts. That is, the inferencemoves forward from the facts toward the goal.2. As facts are generated, they are checked against thequery predicate goal for a match.
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- 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.
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- 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}
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- Top-down mechanism (also called back chaining and top-down resolution)1. The inference engine starts with the query goal andattempts to find matches to the variables that lead tovalid 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 generatesubgoals. The matching of these subgoals will lead tothe match of the intended goal.
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- 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
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Rule1: superior(franklin, Y) supervise(franklin, Y)
Rule1: superior(jennifer, Y) supervise(jennifer, Y)
Y= john, ramesh, joyce Y= alicia, ahmad
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•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.
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•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)
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•Datalog programs and their evaluation
two important concepts:
- safety of programs
- predicate dependency graph
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•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).
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•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.
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•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.
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•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), Y40000,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)).
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•Datalog programs and their evaluation
- predicate dependency graphs
workson employee salary supervise
department project female male
main_poductx_emp president over_40K_emp superior
supervisor under_40K_supervisor
subordinate
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•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.
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•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
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•Datalog programs and their evaluation
- evaluation involving only rule-based predicate
2. Single rule evaluation
To evaluate a rule of the from:
p p1, ..., pn
we first compute the relations corresponding to p1, ..., pn 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).
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•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 Rp; in this case, one of the following expression is returned and the algorithm is terminated. (We use the notation $i to refer to the name of the i-th attribute of relation Rp.)
(a) If all arguments in p are distinc variables, the relational expression returned is Rp.
(b) If some arguments are constants or if the same variable appears in more than one argument position, the expression returned is
SELECT<condition>(Rp),
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where the <condition> is a conjunctive condition made up of a number of simple conditions connected by AND, and constructed as follows:
i. if a constant c appears as argument i, include a simple condition ($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 Si, i = 1, 2, ..., n, n > 0 exist with predicate p as their head. For each such rule Si, 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 Si over the common variables. Let the resulting relation from this join be Rs.
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c. If any built-in predicate XY was defined over the arguments X and Y, the result of the join is subjected to an additional selection:
SELECT XY(Rs)
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.)
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•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
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•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 recursivesg(X, Y) sg(X, Z), sibling(Z, W), sg(W, Y)
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•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))
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•Datalog programs and their evaluation
- some teminology for recursive queries
- fixed point
Consider a relation sequence: g0, g1, …, gi, gi+1, ...
E i(g0) = E (E( ... E(g0) ... ))
i
If at some time we have E i(g0) = E i+1(g0),
then E i(g0) is the fixed point of the
function E(...). It is also the least fixedpoint 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
g0 = ,
gi+1 = E(gi),
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•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)}a0 = { }a1 = {(f, j), (f, r), (f, jo), (je, a), (je, ah), (ja, f), (ja, je)}a2 = {(f, j), (f, r), (f, jo), (je, a), (je, ah), (ja, f), (ja, je),
(ja, j), (ja, r), (ja, jo), (ja, a), (ja, ah)}a3 = a2 least fixed point
The least fixed point of the above equation is also called thetransitive closure of p.
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•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.
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•Datalog programs and their evaluation
- naive strategy
Consider the following equation system:
Ri = Ei(R1, ..., Ri, ..., Rn) (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: R1, ..., Ri, ..., Rn.for i = 1 to n do Ri := ;repeat
Con := true;for i = 1 to n do Si := Ri;for i = 1 to m do {Ri := Ei(S1, ..., Si, ..., Sn);
if Ri Si then {Con := false; Si := Ri;}}
until Con = true;
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•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 = E1(sg, sibling)sibling = E2(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 R1
sibling R2
R1 = E1(R1, R2)R2 = E2(R2)
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•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
alice george
derek pat
frank
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•Datalog programs and their evaluation
- naive strategy
Example:
A(X, Y) = X,Y(P(X, Z) A(Z, Y)) P(X, Y)
step 0: A0 =
step 1: A1 = {(bert, alice), (bert, george), (alice, derek), (alice,
part), (derek, frank)}
step 2: A2 = {(bert, alice), (bert, george), (alice, derek), (alice,
part), (derek, frank), (bert, derek), (bert, pat), (alice, frank)}
step 3: A3 = {(bert, alice), (bert, george), (alice, derek), (alice,
part), (derek, frank), (bert, derek), (bert, pat), (alice, frank),
(bert, frank)}
step 4: A4 = A3
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•Datalog programs and their evaluation
- naive strategy
Algorithm Gauss-Seidel naive strategy
Jacobi:
k-th iteration:R1(k) = E1(R1 (k-1), ..., Ri (k-1), ..., Rn (k-1)),
… … Ri(k) = Ei(R1 (k-1), ..., Ri (k-1), ..., Rn (k-1)),
… … Rn(k) = En(R1 (k-1), ..., Ri (k-1), ..., Rn (k-1)).
Gauss-Seidel:
k-th iteration:R1(k) = E1(R1 (k-1), ..., Ri (k-1), ..., Rn (k-1)),
… … Ri(k) = Ei(R1 (k), ..., Ri (k-1), ..., Rn (k-1)),
… … Rn(k) = En(R1 (k), ..., Ri (k), ..., Rn (k-1)).
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•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 Ri(k) be the temporary value of relation Ri at iteration step k.The differential of Ri between step k and step k - 1 is defined asfollows:
Di(k) = Ri(k) - Ri(k-1)
For a linearly recursive rule set, Di(k) can be substituted for Ri in the k-th iteration of the naïve algorithm.
3. The result is obtained by the union of the newly obtained term Ri and that obtained in the previous step.
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•Datalog programs and their evaluation
- evaluation of recursive queries
- semi-naive strategy
Algorithm seminaiv strategyinput: A system of algebraic equations and EDB.output: The values of the variable relations: R1, ..., Ri, ..., Rn.
for i = 1 to n do Ri := ;for i = 1 to m do Di := ;
repeatCon := true;for i = 1 to n do {Di := E(D1, ..., Di, ..., Dn) - Ri;
Ri := Di Ri; if Di then Con := false;}
until Con is true;
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•Datalog programs and their evaluation
- evaluation of recursive queries
- semi-naive strategy
Example:
Step 0: D0 = , A0
= ;
Step 1: D1 = P = {(bert, alice), (bert, george), (alice, derek), (alice, part),
(derek, frank)}
A1 = D1 A0 = {(bert, alice), (bert, george), (alice, derek), (alice,
part), (derek, frank)}
Step 2: D2 = {(bert, derek), (bert, pat), (alice, frank)}
A2 = D2 A1 = {(bert, alice), (bert, george), (alice, derek), (alice,
part), (derek, frank), {(bert, derek), (bert, pat),(alice, frank)}
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•Datalog programs and their evaluation
- evaluation of recursive queries
- semi-naive strategy
Example:
Step 3: D3 = {(bert, frank)
A3 = D3 A2 = {(bert, alice), (bert, george), (alice, derek), (alice,
part), (derek, frank), {(bert, derek), (bert, pat),(alice, frank), (bert, frank)}
Step 3: D4 = .
The advantage of the semi-naive method is that at each step a differential termDi is used in each equation instead of the whole Ri. In this way, the timecomplexity of a computation is decreased drastically.
Jan. 2012 Yangjun Chen ACS-3902 53
Deductive Databases
•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.
Jan. 2012 Yangjun Chen ACS-3902 54
Deductive Databases
•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.
Jan. 2012 Yangjun Chen ACS-3902 55
Deductive Databases
•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.
Jan. 2012 Yangjun Chen ACS-3902 56
Deductive Databases
•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 A1, …, An
and Ai is positive, then l(Ai) l(B) for all i, 1 i n. If Ai is negative, then l(B) < l(Ai) for all i, 1 i n.
Jan. 2012 Yangjun Chen ACS-3902 57
Deductive Databases
•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 A1, …, An
and Ai is positive, then l(Ai) l(B) for all i, 1 i n. If Ai is negative, then l(B) < l(Ai) for all i, 1 i n.
Jan. 2012 Yangjun Chen ACS-3902 58
Deductive Databases
•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.
Jan. 2012 Yangjun Chen ACS-3902 59
Deductive Databases
•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: