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**Toon Calders t.calders@tue.nl**

Deductive databases Toon Calders

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**Important Information**

This material was developed by Dr. Toon Calders and prof. Dr. Jan Paredaens at University of Technology, Eindhoven. The original document can be accessed at the following address: All changes and adaptations are my own responsability, Rogelio Dávila Pérez

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**Motivation: Deductive DB**

Motivation is two-fold: add deductive capabilities to databases; the database contains: facts (extensional relations) rules to generate derived facts (intensional relations) Database is knowledge base Extend the querying datalog allows for recursion

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**Motivation: Deductive DB**

Datalog as engine of deductive databases similarities with Prolog has facts and rules rules define -possibly recursive- views Semantics not always clear safety negation recursion

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

Relational algebra = safe Datalog with negation and without recursion Optimization techniques Conclusions

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**Syntax of Datalog Datalog query/program: Rules**

facts traditional relational tables rules define intensional views Rules if-then rules can contain recursion can contain negations Semantics of program can be ambiguous

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Syntax of Datalog Example father(X,Y) :- person(X,m), parent(X,Y). grandson(X,Y) :- parent(Y,Z), parent(Z,X), person(X,m). hbrothers(X,Y) :- person(X,m), person(Y,m), parent(Z,X), parent(Z,Y). person parent Name Sex Parent Child ingrid f ingrid toon alex m alex toon rita f rita an jan m jan an toon m an bernd an f an mattijs bernd m toon bernd mattijs m toon mattijs

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**Syntax of Datalog Variables: X, Y Constants: m, f, rita, …**

Positive literal: p(t1,…,tn) p is the name of a relation (EDB or IDB) t1, …, tn constants or variables Negative literal: not p(t1, …, tn) Rule: h :- l1, …, ln h positive literal, l1, …, ln literals

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**Syntax of Datalog Variables: X, Y Constants: m, f, rita, …**

Positive literal: p(t1,…,tn) p is the name of a relation (EDB or IDB) t1, …, tn constants or variables Negative literal: not p(t1, …, tn) Rule: h :- l1, …, ln h positive literal, l1, …, ln literals In Datalog: Classic logical negation ( In contrast to Prolog’s negation by failure )

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**Syntax of Datalog Rules can be recursive**

Arithmetic operations considered as special predicates A<B : smaller(A,B) A+B=C : plus(A,B,C)

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

non-recursive recursive datalog aggregation Relational algebra = safe Datalog with negation and without recursion Optimization techniques Conclusions

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**Semantics of Non-Recursive Datalog Programs**

Ground instantiation of a rule h :- l1, …, ln : replace every variable in the rule by a constant Example: father(X,Y) :- person(X,m), parent(X,Y) instantiation: father(toon,an) :- person(toon,m), parent(toon,an).

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**Semantics of Non-Recursive Datalog Programs**

Let I be a set of facts The body of a rule instantiation R’ is satisfied by I if: every positive literal in the body of R’ is in I no negative literal in the body of R’ is in I Example: person(toon,m), parent(toon,an) not satisfied by the facts given before

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**Semantics of Non-Recursive Datalog Programs**

Let I be a set of facts R is a rule h :- l1, …, ln Infer(R,I) = { h’ : h’ :- l1’, …, ln’ is a ground instantiation of R l1’ … ln’ is satisfied by I } R={R1, …, Rn} Infer(R,I) = Infer(R1,I) … Infer(Rn,I)

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**Semantics of Non-Recursive Datalog Programs**

A rule h :- l1, …, ln is in layer 1: l1, …, ln only involve extensional predicates A rule h :- l1, …, ln is in layer i for all 0<j<i, it is not in layer j l1, …, ln only involve predicates that are extensional and in the layers 1, …, i-1

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**Semantics of Non-Recursive Datalog Programs**

Let I0 be the facts in a datalog program Let R1 be the rules at layer 1 … Let Rn be the rules at layer n I1 = I0 Infer(R1, I0) I2 = I1 Infer(R2, I1) In = In-1 Infer(Rn, In-1)

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**Semantics of Non-Recursive Datalog Programs**

Example: person parent Name Sex Parent Child ingrid f ingrid toon alex m alex toon rita f rita an jan m jan an toon m an bernd an f an mattijs bernd m toon bernd mattijs m toon mattijs father(X,Y) :- person(X,m), parent(X,Y). grandfather(X,Y) :- father(X,Y), parent(Y,Z). hbrothers(X,Y) :- person(X,m), person(Y,m), parent(Z,X), parent(Z,Y).

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**Semantics of Non-Recursive Datalog Programs**

Valuation: Stratum 0 person parent Name Sex Parent Child ingrid f ingrid toon alex m alex toon rita f rita an jan m jan an toon m an bernd an f an mattijs bernd m toon bernd mattijs m toon mattijs father(X,Y) :- person(X,m), parent(X,Y). grandfather(X,Y) :- father(X,Y), parent(Y,Z). hbrothers(X,Y) :- person(X,m), person(Y,m), parent(Z,X), parent(Z,Y).

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**Semantics of Non-Recursive Datalog Programs**

Valuation: Stratum 1 Stratum 0 father alex toon jan an toon bernd toon mattijs hbrothers bernd mattijs mattijs bernd mattijs mattijs bernd bernd person parent Name Sex Parent Child ingrid f ingrid toon alex m alex toon rita f rita an jan m jan an toon m an bernd an f an mattijs bernd m toon bernd mattijs m toon mattijs father(X,Y) :- person(X,m), parent(X,Y). grandfather(X,Y) :- father(X,Y), parent(Y,Z). hbrothers(X,Y) :- person(X,m), person(Y,m), parent(Z,X), parent(Z,Y).

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**Semantics of Non-Recursive Datalog Programs**

Valuation: Stratum 1 Stratum 2 Stratum 0 father grandfather alex toon jan mattijs jan an jan bernd toon bernd alex mattijs toon mattijs alex bernd hbrothers bernd mattijs mattijs bernd mattijs mattijs bernd bernd person parent Name Sex Parent Child ingrid f ingrid toon alex m alex toon rita f rita an jan m jan an toon m an bernd an f an mattijs bernd m toon bernd mattijs m toon mattijs father(X,Y) :- person(X,m), parent(X,Y). grandfather(X,Y) :- father(X,Y), parent(Y,Z). hbrothers(X,Y) :- person(X,m), person(Y,m), parent(Z,X), parent(Z,Y).

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**Caveat: Correct Negation**

Negation in Datalog Negation in Prolog Prolog negation (Negation by failure): not(p(X)) “is true if we fail to prove p(X)” Datalog negation (Correct negation): not(p(X)) “binds X to a value such that p(X) does not hold.”

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**Caveat: Correct Negation**

Example: father(a,b). person(a). person(b). nfather(X) :- person(X), not( father (X,Y) ), person(Y). Datalog: ? nfather(X) { (a), (b) } Prolog: ? nfather(X) { (b) } ? person(a), not(father(a,a)), person(a) yes

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**Caveat: Correct Negation**

Prolog: Order of the clauses is important nfather(X) :- person(X), not( father (X,Y) ), person(Y). versus nfather(X) :- person(X), person(Y), not( father (X,Y) ). Order of the rules is important Datalog: Order not important More declarative

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**Caveat: Correct Negation**

Difference is not fundamental: Prolog: nfather(X) :- person(X), not( father (X,Y) ). Datalog: nfather(X) :- person(X), not(father_of_someone(X)). father_of_someone(X) :- father (X,Y).

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**Caveat: Correct Negation**

Difference is not fundamental: Many systems that claim to implement Datalog, actually implement negation by failure. Debating on whether or not this is correct is pointless; both perspectives are useful Check on beforehand how an engine implements negation … Throughout the course, in all exercises, in the exam, …, we assume correct negation.

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**Safety A rule can make no sense if variables appear in funny ways**

Examples: S(x) :- R(y) S(x) :- not R(x) S(x) :- R(y), x<y In each of these cases the result is infinite even if the relation R is finite

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Safety Even when not leading to infinite relations, such Datalog Programs can be domain-dependent. Example: s(a,b). s(a,a). r(a). r(b). t(X) :- not(s(X,Y)), r(X). If domain is {a,b}: only t(b) holds. If domain is {a,b,c} not only t(b), but also t(a) holds ( Ground instantiation t(a) :- not(s(a,c)), r(a). )

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**Safety Therefore, we will only consider rules that are safe.**

A rule h :- l1, …, ln is safe if: every variable in the head of the rule: also in a non-arithmetic positive literale in body every variable in a negative literal of the body: also in some positive literal of the body

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**Model-Theoretic Semantics**

A model M of a Datalog program is: An instatiation of all intensional relations in the program That satisfies all rules in the program If the body of a ground instantiation of a rule holds in M, also the head must hold Some models are special …

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**Model-Theoretic Semantics**

father(a,b). person(X) :- father(X,Y). person(Y) :- father(X,Y). M1: father person a b a b M2: father person b a b a a

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**Model-Theoretic Semantics**

A model is minimal if « we cannot remove tuples » M1: father person a b a b M2: father person b a b a a Minimal Not Minimal

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**Model-Theoretic Semantics**

For non-recursive, safe datalog programs semantics is well defined The model = all facts that can be derived from the program Closed-World Assumption: if a fact cannot be derived from the database, then it is not true Is a minimal model

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**Model-Theoretic Semantics**

Minimal model is, however, not necessarily unique Example: r(a). t(X) :- r(X), not s(X). minimal models: M M2 r s t r s t a a a a

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

non-recursive recursive datalog aggregation Relational algebra = safe Datalog with negation and without recursion Optimization techniques Conclusions

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**Semantics of Recursive Datalog Programs**

g(a,b). g(b,c). g(a,d). reach(X,X) :- g(X,Y). reach(Y,Y) :- g(X,Y) reach(X,Y) :- g(X,Y). reach(X,Z) :- reach(X,Y), reach(Y,Z). Fixpoint of a set of rules R, starting with set of facts I: repeat Old_I := I I := I infer(R,I) until I = Old_I Always termination (inflationary fixpoint)

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**Semantics of Recursive Datalog Programs**

g(a,b). g(b,c). g(a,d). reach(X,X) :- g(X,Y). reach(Y,Y) :- g(X,Y) reach(X,Y) :- g(X,Y). reach(X,Z) :- reach(X,Y), reach(Y,Z). Step 0: reach = { } Step 1: reach = { (a,a), (b,b), (c,c), (d,d), (a,b), (b,c), (a,d) } Step 2: reach = {(a,a), (b,b), (c,c), (d,d), (a,b), (b,c), (a,d), (a,c) } Step 3: reach = {(a,a), (b,b), (c,c), (d,d), (a,b), (b,c), (a,d), (a,c) } STOP

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**Semantics of Recursive Datalog Programs**

Datalog without negation: Always a unique minimal model. Semantics of recursive datalog with negation is less clear. Example: T(a). R(X) :- T(X), not S(X). S(X) :- T(X), not R(X). What about R(a)? S(a)?

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**Semantics of Recursive Datalog Programs**

For some classes of Datalog queries with negation still a natural semantics can be defied Important class: stratified programs T depends on S if some rule with T in the head contains S or (recursively) some predicate that depends on S, in the body. Stratified program: If T depends on (not S), then S cannot depend on T or (not T).

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**Semantics of Recursive Datalog Programs**

The program: T(a). R(X) :- T(X), not S(X). S(X) :- T(X), not R(X). is not stratified; R depends negatively on S S depends negatively on R R T S

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**Semantics of Recursive Datalog Programs**

g(a,b). g(b,c). g(a,d). reach(X,X) :- g(X,Y). reach(Y,Y) :- g(X,Y). reach(X,Y) :- g(X,Y). reach(X,Z) :- reach(X,Y), reach(Y,Z). node(X) :- g(X,Y). node(Y) :- g(X,Y). unreach(X,Y) :- node(X), node(Y), not reach(X,Y). g reach node unreach

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**Semantics of Recursive Datalog Programs**

If a program is stratified, the tables in the program can be partitioned into strata: Stratum 0: All database tables. Stratum I: Tables defined in terms of tables in Stratum I and lower strata. If T depends on (not S), S is in lower stratum than T.

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**Semantics of Recursive Datalog Programs**

g(a,b). g(b,c). g(a,d). reach(X,X) :- g(X,Y). reach(Y,Y) :- g(X,Y). reach(X,Y) :- g(X,Y). reach(X,Z) :- reach(X,Y), reach(Y,Z). node(X) :- g(X,Y). node(Y) :- g(X,Y). unreach(X,Y) :- node(X), node(Y), not reach(X,Y). g reach node unreach 1 2

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**Semantics of Recursive Datalog Programs**

Semantics of a stratified program given by: First, compute the least fixpoint of all tables in Stratum 1. (Stratum 0 tables are fixed.) Then, compute the least fixpoint of tables in Stratum 2; then the lfp of tables in Stratum 3, and so on, stratum-by-stratum.

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**Semantics of Recursive Datalog Programs**

Fixpoint of a set of rules R, starting with set of facts I: repeat Old_I := I I := I infer(R,I) until I = Old_I Fixpoint within one stratum always terminates Due to monotonicity within the strata Only positive dependence between tables in stratum l. Due to finite program, number of strata is finite as well

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**Semantics of Recursive Datalog Programs**

Stratum 0: g(a,b). g(b,c). g(a,d). Stratum 1: node(a), node(b), node(c), node(d), reach(a,a), reach(b,b), reach(c,c), reach(d,d), reach(a,b), reach(b,c), … Stratum 2: unreach(b,a), unreach(c,a), …

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

non-recursive recursive datalog aggregation Relational algebra = safe Datalog with negation and without recursion Optimization techniques Conclusions

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Aggregate Operators Degree(X, SUM(<Y>)) :- g(X,Y). The < … > notation in the head indicates grouping; the remaining arguments (X, in this example) are the GROUP BY fields. In order to apply such a rule, must have all of relation g available. Stratification with respect to use of < … > is similar to negation.

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**Aggregate Operators bi(X,Y) :- g(X,Y). g bi(Y,X) :- g(X,Y).**

Degree(X, SUM(<Y>)) :- bi(X,Y). bi degree

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**Aggregate Operators bi(X,Y) :- g(X,Y). g bi(Y,X) :- g(X,Y).**

Degree(X, SUM(<Y>)) :- bi(X,Y). bi degree 1 2

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**Aggregate Operators bi(X,Y) :- g(X,Y). g bi(Y,X) :- g(X,Y).**

Degree(X, SUM(<Y>)) :- bi(X,Y). bi degree Compute stratum by stratum Assume strata 1 k fixed when computing k+1 1 2

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**Aggregate Operators r(a,b). r(a,c). s(a,d).**

t(X,SUM(<Y>)) :- r(X,Y). r(X,Y) :- t(X,Z), Z=2, s(X,Y).

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**Aggregate Operators r(a,b). r(a,c). s(a,d). t**

t(X,SUM(<Y>)) :- r(X,Y). r(X,Y) :- t(X,Z), Z=2, s(X,Y). r s

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**Aggregate Operators r(a,b). r(a,c). s(a,d). t**

t(X,SUM(<Y>)) :- r(X,Y). r(X,Y) :- t(X,Z), Z=2, s(X,Y). r s “a is aggregating over a moving target” Step 1: t(a,2) is added Step 2: r(a,d) is added Step 3: t(a,3) added, t(a,2) no longer true … hence r(a,d) should not have been added

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

Relational algebra = Safe Datalog with negation and without recursion Optimization techniques Conclusions

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**RA = Non-Recursive Datalog**

Every operator of RA can be simulated by non-recursive datalog Project on the first attribute of the ternary relation r: query (A) :–r(A, B, C). Cartesian product of relations r1 and r2. query (X1, X2, ..., Xn, Y1, Y1, Y2, ..., Ym ) :– r1 (X1, X2, ..., Xn ), r2 (Y1, Y2, ..., Ym ). Union of relations r1 and r2. query (X1, X2, ..., Xn ) :–r1 (X1, X2, ..., Xn ), query (X1, X2, ..., Xn ) :–r2 (X1, X2, ..., Xn ), Set difference of r1 and r2. query (X1, X2, ..., Xn ) :–r1(X1, X2, ..., Xn ), not r2 (X1, X2, ..., Xn )

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**RA = Non-Recursive Datalog**

Every operator of RA can be simulated by non-recursive datalog Result of our construction is always safe, and equivalent for stratified semantics $1=$3 (($1R) x R) query1(A) :- R(A,B). query2(A,B,C) :- query1(A), R(B,C). result(A,B,A) :- query2(A,B,A)

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**RA = Non-Recursive Datalog**

Every rule can be expressed by one RA expression: Translate every atom separately Negation/arithmetic: use “complement” construction Essential: safety Combine atoms with Cartesian product Do the joins with a selection Project on the relevant attributes Strata determine the order of evaluation Because of no recursion: every rule only executed once.

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**RA = Non-Recursive Datalog**

sister(X,Y) :- person(X,f), parent(Z,X), parent(Z,Y), not(X=Y). person(X,f): $2=‘f’ Person parent(Z,X) and parent(Z,Y): Parent not(X=Y): complement construction X comes from parent(Z,X) $2 Parent Y from parent(Z,Y) $2 Parent not(X=Y) $1$2 ($2 Parent x $2 Parent)

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**RA = Non-Recursive Datalog**

sister(X,Y) :- person(X,f), parent(Z,X), parent(Z,Y), not(X=Y). $1,$6 $1=$4, $3=$5, $1=$7, $6=$8 ($2=‘f’ Person x Parent x Parent x $1$2 ($2 Parent x $2 Parent))

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**RA = Non-Recursive Datalog**

Hence, the following two are equivalent in expressive power: Safe Datalog with negation, without recursion or aggregation, under the stratified semantics Relational Algebra Every rule separately can be expressed by a relational algebra expression Makes it very suitable for implementation on top of a relational database

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

Relational algebra = Datalog with negation and without recursion Optimization techniques Conclusions

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**Evaluation of Datalog Programs**

Running example: root(r). child(r,a). child(r,b). child(a,c). child(a,d). child(c,e). child(d,f). child(b,h). sg(X,Y) :- root(X),root(Y). sg(X,Y) :- child(X,U), sg(U,V), child(Y,V). r a b c d h e f

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**Evaluation of Datalog Programs: Issues**

Repeated inferences: recursive rules are repeatedly applied in the naïve way; same inferences in several iterations. Unnecessary inferences: if we just want to find sg of a particular node, say e, computing the fixpoint of the sg program and then selecting tuples with e in the first column is wasteful, in that we compute many irrelevant facts.

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**Evaluation of Datalog Programs**

Running example: Query: ? sg(e,X) (r, r) (a,a), (b,b), (a,b), (b,a) (c,c), (c,d), (c,h), (d,c), (d,d), … (e,e), (f,f), (e,f), (f,e) r a b c d h e f

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**Avoiding Repeated Inferences**

Seminaive Fixpoint Evaluation: Avoid repeated inferences: at least one of the body facts generated in the most recent iteration. For each recursive table P, use a table delta_P. Rewrite the program to use the delta tables. A second evaluation of the rule: r(X,Y) :- s(X), t(Y), u(X,Z), v(Y,Z). only gives new tuples (X,Y) for ground instantiations in which at least one of the atoms is new.

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**Avoiding Unnecessary Inferences**

Still, in the running example: many unnecessary deductions when query is: ? sg(e,X) Compare with top-down as in Prolog only facts that are connected to the ultimate goal are being considered

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**The “Prolog Way” ? sg(e,X). try: root(e) FAIL try: child(e,U) U=c**

sg(X,Y) :- root(X),root(Y). sg(X,Y) :- child(X,U), sg(U,V), child(Y,V). ? sg(e,X). try: root(e) FAIL try: child(e,U) U=c try: sg(c,V) try: child(c,U) U=a … r a b c d h e f

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**The “Prolog Way” ? sg(e,X). try: root(e) FAIL try: child(e,U) U=c**

sg(X,Y) :- root(X),root(Y). sg(X,Y) :- child(X,U), sg(U,V), child(Y,V). ? sg(e,X). try: root(e) FAIL try: child(e,U) U=c try: sg(c,V) try: child(c,U) U=a … r a b c d h e f

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**“Magic Sets” Idea We want to do something similar for Datalog**

Idea: Define a “filter” table: computes all relevant values, restricts the computation of sg(e,X). sg(X,Y) :- m(X), root(X), root(Y). sg(X,Y) :- m(X), child(X,U), sg(U,V),child(Y,V). m(X) :- m(Y), child(Y,X). m(e).

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Magic Sets It is always possible to do this in such a way that bottom-up becomes as efficient as top-down! Different proposals exist in literature how to introduce the magic filters

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**Optimization Techniques**

Many other techniques exist as well Standard relational indexing techniques (partly) materializing intensional relations on beforehand Trade-off memory query time performance (See also the OLAP-part for a similar technique) Different representations for relations BDD (Stanford)

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**Outline Syntax of the Datalog language Semantics of a Datalog program**

Relational algebra = Datalog with negation and without recursion Optimization techniques Conclusions

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**Conclusions Datalog adds deductive capabilities to databases**

extensional relations intensional relations Datalog without recursion, with negation safety requirement stratification equal in power to relational algebra Closed World Assumption

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**Conclusions Datalog without Negation**

Always a unique minimal model Datalog with negation and recursion semantics not always clear stratified negation Evaluation of datalog queries: without negation = RA-optimization with recursion: semi-naive recursion magic sets

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**Conclusions Very nice idea, but …**

Deductive databases did not make it as a database paradigm Yet, many ideas survived recursion in SQL … And others may re-surface in future. Increasing need for adding meta-information in databases

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