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Abstract State Machines and Computationally Complete Query Languages

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### Abstract State MachinesandComputationally Complete Query Languages

Andreas Blass, U Michigan

Yuri Gurevich, Microsoft Research & U Michigan

Jan Van den Bussche, U Limburg

Outline

- Databases and queries
- Query languages:
- whilenew, whilenewsets
- ASMs

- Notions of polynomial time
- Comparisons

Relational databases

- Database schema = Finite set S of relation names with associated arities
- DatabaseB over S = Finite structure over S
- Finite domain D of atomic values
- For each R S, a k-ary relation RB onD

Relational databases

- Database schema = Finite set S of relation names with associated arities
- DatabaseB over S = Finite structure over S
- Finite domain D of atomic values
- For each R S, a k-ary relation RB onD

arity associated to R in S

Relational databases

- Database schema = Finite set S of relation names with associated arities
- DatabaseB over S = Finite structure over S
- Finite domain D of atomic values
- For each R S, a k-ary relation RB onD
E.g. Graph:

arity associated to R in S

Relational databases

- Database schema = Finite set S of relation names with associated arities
- DatabaseB over S = Finite structure over S
- Finite domain D of atomic values
- For each R S, a k-ary relation RB onD
E.g. Graph:

arity associated to R in S

1

2

3

4

Relational databases

D

1

2

3

4

E

(1,2)

(2,3)

(2,4)

(3,4)

- Database schema = Finite set S of relation names with associated arities
- DatabaseB over S = Finite structure over S
- Finite domain D of atomic values
- For each R S, a k-ary relation RB onD
E.g. Graph:

arity associated to R in S

1

2

3

4

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

(over a common schema)

(of a common arity)

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

- Q(B) is the answer to the query Q on database B.

(over a common schema)

(of a common arity)

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

- Q(B) is the answer to the query Q on database B.
- Arity 0: {( )} or { } Boolean query

(over a common schema)

(of a common arity)

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

- Q(B) is the answer to the query Q on database B.
- Arity 0: {( )} or { } Boolean query
E.g. On a graph:

- Give all pairs of nodes that are targets of a common source.

(over a common schema)

(of a common arity)

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

- Q(B) is the answer to the query Q on database B.
- Arity 0: {( )} or { } Boolean query
E.g. On a graph:

- Give all pairs of nodes that are targets of a common source.
- Is f(m)=2000?

(over a common schema)

(of a common arity)

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

- Q(B) is the answer to the query Q on database B.
- Arity 0: {( )} or { } Boolean query
E.g. On a graph:

- Give all pairs of nodes that are targets of a common source.
- Is f(m)=2000?

(over a common schema)

(of a common arity)

number of edges in graph

Queries

- General definition of query: a (partial, computable) mapping Q
- from databases
- to relations

- Q(B) is the answer to the query Q on database B.
- Arity 0: {( )} or { } Boolean query
E.g. On a graph:

- Give all pairs of nodes that are targets of a common source.
- Is f(m)=2000?

(over a common schema)

(of a common arity)

arbitrary computable function on N

The consistency criterion

- The answer of a query on a database can depend only on information that is logically contained in that database.
- If his an isomorphism B B, then h is also an isomorphism Q(B) Q(B).

Query languages

- In practice: SQL
- first-order logic + counting, summation, …
E.g. Give all pairs of nodes that are targets of a common source:

- first-order logic + counting, summation, …

Query languages

- In practice: SQL
- first-order logic + counting, summation, …
E.g. Give all pairs of nodes that are targets of a common source:

select E1.target, E2.target

from E E1, E E2

where E1.source = E2.source

- first-order logic + counting, summation, …

Query languages

- In practice: SQL
- first-order logic + counting, summation, …
E.g. Give all pairs of nodes that are targets of a common source:

select E1.target, E2.target

from E E1, E E2

where E1.source = E2.source

(x,y) z(E(z,x) E(z,y))

- first-order logic + counting, summation, …

Expressiveness of first-order logic (FO)

2000

0

if m is even

if m is odd

f(m) =

- Many useful queries are expressible in FO.
- But many others are not:
- Connectivity: Is the graph connected?
- Is f(m)=2000, where

Expressiveness of first-order logic (FO)

2000

0

if m is even

if m is odd

f(m) =

- Many useful queries are expressible in FO.
- But many others are not:
- Connectivity: Is the graph connected?
- Is f(m)=2000, where
- (parity query)

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

relation variable of arity j

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

FO-formula over db relations and relation variables

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

- sequential composition

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

- sequential composition
- while-loops:
while do … od

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

- sequential composition
- while-loops:
while do … od

FO-sentence

Towards a complete language: while

- Make FO basis of a small programming language for working with relations:
- relation variables (typed by fixed arities)
- operations on relations provided by FO
- assignment:
X(x1,…,xj)(x1,…,xj)

- sequential composition
- while-loops:
while do … od

- Chandra & Harel [1982]

Example while-program

- Connectivity query:
Seen(2);

Path(2) E;

whilePath Seen do

Seen Path;

Path Path (x,z) y(Path(x,y) E(y,z));

od.

Example while-program

- Connectivity query:
Seen(2);

Path(2) E;

whilePath Seen do

Seen Path;

Path Path (x,z) y(Path(x,y) E(y,z));

od.

- Parity query:

Example while-program

- Connectivity query:
Seen(2);

Path(2) E;

whilePath Seen do

Seen Path;

Path Path (x,z) y(Path(x,y) E(y,z));

od.

- Parity query: Not!

A complete language: whilenew

- S. Abiteboul & V. Vianu [1988]
- Allow introduction of new domain elements in the computation.
- New operator:

A complete language: whilenew

- S. Abiteboul & V. Vianu [1988]
- Allow introduction of new domain elements in the computation.
- New operator: new

A complete language: whilenew

X

R

a b a

c d b

f g c

a b

c d

f g

- S. Abiteboul & V. Vianu [1988]
- Allow introduction of new domain elements in the computation.
- New operator: new
X(3) newR(2)

A complete language: whilenew

X

R

a b a

c d b

f g c

a b

c d

f g

- S. Abiteboul & V. Vianu [1988]
- Allow introduction of new domain elements in the computation.
- New operator: new
X(3) newR(2)

- Every partial computable query can be programmed in whilenew.

Parity in whilenew

- Easy to check parity of a set Sequipped with a successor relation:
Even(0) true;

Visit(1) first element of S ;

whileVisit do

Even Even;

Visit succ(x)Visit(x)

od.

Parity in whilenew

- Make a set S of new elements, one for each edge:
S0 newE;

S 3(S0);

Parity in whilenew

- Compute a successor relation on S:
Impossible!

Parity in whilenew

- Compute the tree T of all m! successor relations, where m = |S|:
T new ;

Seen ;

Extend r,xTr Sx;

whileExtend do

X newExtend;

T T p3X; succ succ p1,3X;

Seen Seen n,x nXn,x,n

xx Seenn,x;

Extend n,xn p3X Sx Seen(n,x

od.

We can’t do better!

- whilenew-PSPACE: class of whilenew-programs running in polynomial space.
Theorem: [Abiteboul–Vianu 1991] The parity query cannot be done in whilenew-PSPACE.

- Intuition: In whilenew you cannot make arbitrary choices (recall consistency criterion)
- Instead of choosing one successor relation, we must work with them all.

- whilenew-PTIME: class of whilenew-PSPACE-programs running in polynomial time.

BGS

- Blass, Gurevich, Shelah [1996]:
- How can we formalize algorithms that never have to make arbitrary choices?
- What can such algorithms still do in polynomial time?

- Instantiation of ASMs for expressing database queries.

BGS ASMs

- Universe: HF(D)
- every x D is in HF(D);
- every finite set of elements of HF(D) is itself in HF(D).

- Infinite, but at any point only finitely many sets are “active”.
- Set-theoretic static functions:
- pairing
- bounded set-construction

- forall do (parallel ASMs)

Connectivity with a BGS-ASM

ifMode0then

forallxDdoFrontierxx enddo,

Mode 1

endif,

ifMode= 1 then

forallxDdo

Reached(x) := Reached(x) Frontier(x),

Frontier(x) := {y D z Frontier(x):

E(z,y) y Reached(x) Frontier(x)}

enddo,

Halt := {Frontier(x)x D} =

endif.

BGS-PTIME

- BGS-PTIME: class of BGS-ASMs
- running for at most polynomially many steps
- constructing at most polynomially many sets

- “Choiceless polynomial time”

BGS-PTIME versus whilenew-PTIME?

- Structure In:
- There is a PTIME BGS-program that outputs:

• • •

• • •

n

2n

BGS-PTIME versus whilenew-PTIME?

- Structure In:
- There is a PTIME BGS-program that outputs:
- true on every In with neven;

• • •

• • •

n

2n

BGS-PTIME versus whilenew-PTIME?

- Structure In:
- There is a PTIME BGS-program that outputs:
- true on every In with neven;
- falseodd.

• • •

• • •

n

2n

BGS-PTIME versus whilenew-PTIME?

- Structure In:
- There is a PTIME BGS-program that outputs:
- true on every In with neven;
- falseodd.

- (Just construct all red subsets of even size.)

• • •

• • •

n

2n

BGS-PTIME versus whilenew-PTIME?

- Structure In:
- There is a PTIME BGS-program that outputs:
- true on every In with neven;
- falseodd.

- (Just construct all red subsets of even size.)
Theorem: There is no such PSPACE whilenew-program (let alone PTIME).

• • •

• • •

n

2n

Sets versus lists

- BGS programs can construct sets.
- whilenew programs can only construct lists.
- operator new works tuple- ( list-) based.

- Lists are ordered; sets can be unordered.
- If you want to simulate something unordered by something ordered, you have to work with all orders.
- (Recall parity in whilenew.)

- BGS-PTIME strictly encompasses whilenew-PTIME.

The language whilenewsets

- Theory of object-based query languages, studied late 80s – early 90s.
- Operator new from whilenew is really tuple-new.
- We need also a set-new!
- Language whilenewsets

Equivalence results

- whilenewsets and BGS can simulate each other.
- Simulation:
- linear step overhead
- polynomial space overhead

- BGS-PTIME =whilenewsets-PTIME whilenew-PTIME

Concluding remarks

- ASMs and query languages are quite related, and share the common concern of computation on the “logical” level.
- Purely mathematically,
- basic parallel ASMs
- whilenew
are essentially the same thing.

- ASMs clearly win from query languages in flexibility, appeal to practitioners, developed philosophy, and people like Yuri and Egon.
- whilenew never “escaped” database theory!
- Challenge: the Web (querying XML, WWW, …)

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