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CSE544

Wednesday, March 29, 2006

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Theory

• Relational databases invented by a theoretician (Codd)

• Fundamental principle: separate the WHAT from the HOW - data independence

• WHAT: First Order Logic (FO)• HOW: Relational algebra (RA)

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Given:

• A vocabulary: R1, …, Rk

• An arity, ar(Ri), for each i=1,…,k

• An infinite supply of variables x1, x2, x3, …

• Constants: c1, c2, c3, ...

FO Syntax

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• Terms (t) and FO formulas () are:

FO Syntax

t ::= x | c

::= R(t1, ..., tar(R)) | ti = tj

| ’ | ’ | ’

| x. | x.

t ::= x | c

::= R(t1, ..., tar(R)) | ti = tj

| ’ | ’ | ’

| x. | x.

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FO Examples

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Most interesting case:Vocabulary = one binary relation R (encodes a graph)

1 2

2 1

2 3

1 4

3 4

R=

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FO Sentences

• Does there exists a loop in the graph ?

• Are there paths of length >2 ?

• Is there a “sink” node ?

x.R(x,x) x.R(x,x)

x.y.z.u.(R(x,y) R(y,z) R(z,u)) x.y.z.u.(R(x,y) R(y,z) R(z,u))

x.y.R(x,y) x.y.R(x,y)

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Semantics

• Given a vocabulary R1, …, Rk

• A model is D = (D, R1D, …, Rk

D)

– D = a set, called domain, or universe

– RiD D D ... D, (ar(Ri) times)

i = 1,...,k

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Semantics

• Given:– A model D = (D, R1

D, ..., RkD)

– A formula

– A substitution s : {x1, x2, ...} D

• We define next the relation:

meaning “D satisfies with s”

D |= [s]D |= [s]

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Semantics

D |= (R(t1, ..., tn)) [s]D |= (R(t1, ..., tn)) [s]

D |= (t= t’) [s]D |= (t= t’) [s]

If (s(t1), ..., s(tn)) RD

If s(t) = s(t’)

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Semantics

D |= ( ’) [s]D |= ( ’) [s]

D |= ( ’) [s]D |= ( ’) [s]

D |= (’) [s]D |= (’) [s]

If D |= ()[s] and D |= (’) [s]

If D |= ()[s] or D |= (’) [s]

If not D |= ()[s]

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First Order Logic: Semantics

D |= (x.) [s]D |= (x.) [s]

D |= (x.) [s]D |= (x.) [s]

If for all s’ s.t. s(y) = s’(y) for allvariables y other than x, D |= ()[s’]

If for some s’ s.t. s(y) = s’(y) for allvariables y other than x, D |= ()[s’]

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FO and Databases

• FO:a sentence is true in D if D |=

• Databases:a formula with free variables x1, ..., xn defines the query:

(D) = {(s(x1), ..., s(xn)) | D |= [s]}

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FO Queries

• Find all nodes connected by a path of length 2:

• Find all nodes without outgoing edges:

(x,y) u.(R(x,u) R(u,y))(x,y) u.(R(x,u) R(u,y))

(x) u.(R(u,x) y.R(x,y)(x) u.(R(u,x) y.R(x,y)

These are open formulas

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In Class

• Retrieve all nodes with at least two children

• A node x is more important than y if every child of y is also a child of x. Retrieve all ‘most important nodes’ in the graph

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FO in Databases

FO Databases

Vocabulary:

R1, ..., Rn

Database schema:

R1, ..., Rn

Model:D = (D, R1

D, …, RkD)

Database instance:D = (D, R1

D, …, RkD)

Sentences are

true or falseFormulas compute

queries

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FO Semantics

• In FO we express WHAT we want

• Sometimes it’s even unclear HOW to get it

• See accompanying slides on FO semantics– They explain HOW to get it, but it’s impractical

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Relational Algebra

• An algebra over relations

• Five operators:, -, , ,

• Meaning:R1 R2 = set union

R1 - R2 = set difference

R1 R2 = cartesian product

c(R) = subset of tuples satisfying condition c

a(R) = projection on the attributes in a

R1 R2 = set union

R1 - R2 = set difference

R1 R2 = cartesian product

c(R) = subset of tuples satisfying condition c

a(R) = projection on the attributes in a

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FO RA

(x) R(x,x)(x) R(x,x) 1(1=2(R))1(1=2(R))

(x,y) z.u.(R(x,z) R(z,u) R(u,y))

(x,y) z.u.(R(x,z) R(z,u) R(u,y)) 16(2=34=5 (R R R))16(2=34=5 (R R R))

(x) y.R(x,y)(x) y.R(x,y) ?

16 ((R join2=1 R) join4=1 R))16 ((R join2=1 R) join4=1 R))

WHAT HOW

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FO v.s. RA

Theorem. Every query in RA can be expressed in FO

Proof

This shows how to go from HOW to WHATnot very interesting

What about the converse ?

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The Drinkers/Beers Example

• Vocabulary:

• Find all drinkers that frequent some bar that serve some beer that they like:

Likes(drinker,beer), Serves(bar,beer), Frequents(drinker,bar)Likes(drinker,beer), Serves(bar,beer), Frequents(drinker,bar)

(d) ba. be.(F(d,ba) L(d,be))(d) ba. be.(F(d,ba) L(d,be))

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Lots of Fun Examples (in class)

• Find drinkers that frequent some bar that serves only beer they like

• Find drinkers that frequent only bars that serve some beer they like

• Find drinkers that frequent only bars that serve only beer they like

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Unsafe FO Queries

• Find all nodes that are not in the graph:

what’s wrong ?

x)z.R(z,y)y.R(x,q(x) ¬∃∧¬∃=

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Unsafe FO Queries

• Find all nodes that are connected to “everything”:

what’s wrong ?

y)y.R(x,q(x) ∀=

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Unsafe FO Queries

• Find all pairs of employees or offices:

what’s wrong ?

• We don’t want such queries !

Office(y)Emp(x)y)q(x, ∨=

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Safe Queries

A model D = (D, R1D, …, Rk

D)

• In FO:– both D and R1

D, …, RkD may be infinite

• In databases:– D may infinite (int, string, etc)– R1

D, …, RkD are always finite

– We call this a finite model

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Safe Queries

is a finite query if for every finite model D, (D) is finite

is safe, or domain independent, if for every two models D, D’ having the same relations:

D = (D, R1D, …, Rk

D), D’ = (D’, R1D, …, Rk

D)

we have (D) = (D’)

• If is safe then it is also finite (why ?)

• Note: book has different but equivalent definition

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Safe Queries

• Definition. Given D = (D, R1D, …, Rk

D), the active domain is Da = the set of all constants in R1

D, …, RkD

• Example. Given a graph D = (D, R)Da = { x | y.R(x,y) z.R(z,x)}

• Property. If a query is safe, it suffices to range quantifiers only over the active domain (why ?)

• Hence we can compute safe queries

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Safe Queries

• The safe relational calculus consists only of safe queries. However:

• Theorem It is undecidable if a given a FO query is safe.

• Need to write only safe queries, but how do we know how which queries are safe ?

• Work around: write them in an obviously safe way– Range restricted queries - formally defined in [AHU]

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FO v.s. RA

Theorem. Every safe query in FO can be expressed in RA

Proof

From WHAT to HOWthis is really interesting and motivated the relational model

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Limited Expressive Power

• Vocabulary: binary relation R

• The following queries cannot be expressed in FO:

• Transitive closure: x.y. there exists x1, ..., xn s.t.

R(x,x1) R(x1,x2) ... R(xn-1,xn) R(xn,y)

• Parity: the number of edges in R is even