Compositional Methods Alex Rabinovich Department of Computer - - PowerPoint PPT Presentation

Compositional Methods

Alex Rabinovich Department of Computer Science Tel-Aviv University

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Composition Theorem s

Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts.

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Composition Theorem s

Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. Reduce verification of

• n FO structure

to verification of

• ☎
• n

. . .

• ✞
• n

.

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Composition Theorem s

Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. Reduce verification of

• n FO structure

to verification of

• ☎
• n

. . .

• ✞
• n

. Composition theorems (1)Provide a very powerful technique for decidability/definability.

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Composition Theorem s

Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. Reduce verification of

• n FO structure

to verification of

• ☎
• n

. . .

• ✞
• n

. Composition theorems (1)Provide a very powerful technique for decidability/definability. (2) Among few fundamental theorems which hold in finite models

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Landmark papers

Mostowski 1952 - Direct products and power

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Landmark papers

Mostowski 1952 - Direct products and power Feferman - Vaught 1959 -generalized product

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Landmark papers

Mostowski 1952 - Direct products and power Feferman - Vaught 1959 -generalized product Shelah 1975 -generalized sum.

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Plan

Generalized Product (Feferman Vaught 1959). Generalized Sum (Shelah 1975, Rabinovich 1997). Applications to decidability and definability. Recursively (inductively?) Defined Types.

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The Aim of this talk:

• 1. explain the definitions and
• 2. state the composition theorems.

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Generalized Product - Two examples

Example 1 Cartesian Product. Let

(

) be a family of structures for

.

• Cartesian product.

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Generalized Product - Two examples

Example 1 Cartesian Product. Let

(

) be a family of structures for

.

• Cartesian product.

Universe: all functions

with domain

such that

.

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Generalized Product - Two examples

Example 1 Cartesian Product. Let

(

) be a family of structures for

.

• Cartesian product.

Universe: all functions

with domain

such that

. Interpretation of the relations: e.g.

binary symbol,

iff

for every

.

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Generalized Product - Two examples

Example 2 Ordinal Product.

a linear order and

(

) a family of linear orders.

is a structure for

.

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Generalized Product - Two examples

Example 2 Ordinal Product.

a linear order and

(

) a family of linear orders.

is a structure for

. Universe: like before, all functions

with domain

such that

.

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Generalized Product - Two examples

Example 2 Ordinal Product.

a linear order and

(

) a family of linear orders.

is a structure for

. Universe: like before, all functions

with domain

such that

. Interpretation of

(lexicographical order):

iff there is

such that

and

for all

.

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Generalized Product

Ingredients:

a structure for

;

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Generalized Product

Ingredients:

a structure for

; A family

(

• f structures for

;

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Generalized Product

Ingredients:

a structure for

; A family

(

• f structures for

; A structure

for

to be defined.

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Generalized Product

Ingredients:

a structure for

; A family

(

• f structures for

; A structure

for

to be defined. Universe of the product:

.

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Generalized Product

Ingredients:

a structure for

; A family

(

• f structures for

; A structure

for

to be defined. Universe of the product:

. The interpretation of the relational symbols in

will be defined by “Conditions”.

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Conditions

• ary Condition - Syntax

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Conditions

• ary Condition - Syntax

1.

formulas in

.

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Conditions

• ary Condition - Syntax

1.

formulas in

. 2.

a formula in the second-order monadic language for

.

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Conditions

• ary Condition - Syntax

1.

formulas in

. 2.

a formula in the second-order monadic language for

.

• ary Condition - Semantics

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Conditions

• ary Condition - Syntax

1.

formulas in

. 2.

a formula in the second-order monadic language for

.

• ary Condition - Semantics

We have to define when

satisfies the condition.

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Conditions

• ary Condition - Syntax

1.

formulas in

. 2.

a formula in the second-order monadic language for

.

• ary Condition - Semantics

We have to define when

satisfies the condition. Let

. . .

.

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Conditions

• ary Condition - Syntax

1.

formulas in

. 2.

a formula in the second-order monadic language for

.

• ary Condition - Semantics

We have to define when

satisfies the condition. Let

. . .

. Def.

satisfies the condition iff

.

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Two Examples

Conditions for Cartesian Product. E.g.

binary relation 1.

is defined as

. 2.

is

. The condition for ordinal product (

). 1.

is

and

is

2.

is

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Rules for Generalized Product

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Rules for Generalized Product

Syntax.

, where

assigns to every

• ary symbol in

a finite set of

• ary conditions.

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Rules for Generalized Product

Syntax.

, where

assigns to every

• ary symbol in

a finite set of

• ary conditions.
• Semantics. Given

and

define

as follows:

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Rules for Generalized Product

Syntax.

, where

assigns to every

• ary symbol in

a finite set of

• ary conditions.
• Semantics. Given

and

define

as follows: The universe

.

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Rules for Generalized Product

Syntax.

, where

assigns to every

• ary symbol in

a finite set of

• ary conditions.
• Semantics. Given

and

define

as follows: The universe

. The interpretation of

consists of all the tuples that satisfy at least one condition assigned by

to

.

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Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

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Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

The full statement of the theorem is:

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Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

The full statement of the theorem is: Theorem Let

be a rule and let

• ✂

be a formula in

. Then there exists a finite set

• f conditions such that for every

and every family

the relation definable by

• in

is the same as the relation definable by

in

.

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Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

The full statement of the theorem is: Theorem Let

be a rule and let

• ✂

be a formula in

. Then there exists a finite set

• f conditions such that for every

and every family

the relation definable by

• in

is the same as the relation definable by

in

. Moreover,

is computable from

and

• .

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Consequences

• Corollary. if

then

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Consequences

• Corollary. if

then

• Def. Generalized Power

.

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Consequences

• Corollary. if

then

• Def. Generalized Power

.

• Corollary. Theory of

is recursive in the theory of

and the monadic theory of

.

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Example

Direct Power (Mostowski 1952). The number of such examples could be multiplied indefinitely. It must be said, however, that only exceptionally does one come to a really interesting example

• f a power theory

Decidability of

with multiplication

. Reduce to

with + (Presburger Arithmetic) Represent a number as the product of prime factors

• ❍

• ❍

Hence

• ❑

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Example

Direct Power (Mostowski 1952). The number of such examples could be multiplied indefinitely. It must be said, however, that only exceptionally does one come to a really interesting example

• f a power theory

Decidability of

with multiplication

. Reduce to

with + (Presburger Arithmetic) Represent a number as the product of prime factors

• ❍

• ❍

Hence

• ❑

The structure

can be represented as the (weak) power of

• ver the

index structure

.

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Example

Direct Power (Mostowski 1952). The number of such examples could be multiplied indefinitely. It must be said, however, that only exceptionally does one come to a really interesting example

• f a power theory

Decidability of

with multiplication

. Reduce to

with + (Presburger Arithmetic) Represent a number as the product of prime factors

• ❍

• ❍

Hence

• ❑

The structure

can be represented as the (weak) power of

• ver the

index structure

. Hence

is decidable

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Generalized Sum - Two examples

Example 1 Disjoint Union.

a family of (disjoint) structures for the summand language

. Universe.

has the universe

. Interpretation of a relation

iff there is

such that

and

.

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Generalized Sum - examples

Example 2 Lexicographical Sum

• a linearly ordered set.

a family of linearly ordered sets.

is defined as follows: Universe;

Interpretation of <:

if either 1.

for some

and

• r

2.

,

and

.

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Generalized Sum

Ingredients:

a structure for

;

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Generalized Sum

Ingredients:

a structure for

; A family

(

• f structures for

;

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Generalized Sum

Ingredients:

a structure for

; A family

(

• f structures for

; A structure

for

to be defined.

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Generalized Sum

Ingredients:

a structure for

; A family

(

• f structures for

; A structure

for

to be defined. Universe of the sum:

(

are assumed to be disjoint).

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Generalized Sum

Ingredients:

a structure for

; A family

(

• f structures for

; A structure

for

to be defined. Universe of the sum:

(

are assumed to be disjoint). The interpretation of symbols in

will be defined by “Conditions”.

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Conditions for binary relations

For

we have to specify when

holds. Two type of conditions:

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Conditions for binary relations

For

we have to specify when

holds. Two type of conditions: case 1

and

are in the same component (say

). In this case a condition consists of

and

. The pair

satisfies the condition if

and

.

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Conditions for binary relations

For

we have to specify when

holds. Two type of conditions: case 1

and

are in the same component (say

). In this case a condition consists of

and

. The pair

satisfies the condition if

and

. case 2

and

are in the distinct summands (say in

and

). In this case a condition consists of

and

. The pair

satisfies the condition if

and

.

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General

• ary Conditions

For

define an equivalence

• n

.

iff

and

are in the same summand.

specifies the distribution of

among the summands. For every equivalence

we will define

• conditions.

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• Condition

Let

be an equivalence relation on

. Let

be

• equivalence classes. Let

be

where

are all elements of

. (

are defined similarly.)

• Condition -Syntax

1.

formulas from

.

• 2. Sentences

from

3.

a formula from

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• condition - Semantics

, . . . ,

satisfy the

condition iff

• 1. They agree with

, i.e., there are

s.t

iff

. 2.

for

. 3.

, where

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Rules Generalized Sum

Syntax.

, where

assigns to every

• ary symbol in

a finite set of

• ary conditions.
• Semantics. Given

and

define

as follows: The universe

. The interpretation of

consists of all the tuples that satisfy at least one condition assigned by

to

.

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Example - Tree Sum

Let

be a tree. Let

(

) be a family of trees. The sum of

• ver

is a tree defined as follows: The universe:

. The interpretation of

:

if either

and

are in the the summand

and

• r

is a root of

,

is in

and

.

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Example - Tree Sum

Let

be a tree. Let

(

) be a family of trees. The sum of

• ver

is a tree defined as follows: The universe:

. The interpretation of

:

if either

and

are in the the summand

and

• r

is a root of

,

is in

and

. Formal conditions:

One condition for the first case: The formulas:

is defined to be

and

is TRUE One condition for the second case:

is

is TRUE

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Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

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Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

The full statement of the theorem is as follows:

– p.24/30

Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

The full statement of the theorem is as follows: Theorem Let

be a rule and let

• ✂

be a formula in

. Then there exists a finite set

• f conditions such that for every

and every family

the relation definable by

• in

is the same as the relation definable by

in

.

– p.24/30

Composition Theorem

The composition theorem expresses a fundamental property of the first-order language: if every atomic relation in

is expressed by a finite set of conditions, then for every formula

• in

the relation definable by

• is also definable by a finite set of conditions.

The full statement of the theorem is as follows: Theorem Let

be a rule and let

• ✂

be a formula in

. Then there exists a finite set

• f conditions such that for every

and every family

the relation definable by

• in

is the same as the relation definable by

in

. Moreover,

is computable from

and

• .

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The

formulas

The structure of formulas in

faithfully reflects the structure of

• .

If

is quantifier free, then (1) qr of all

formulas is bounded by qr of

• .

(2) if

• has

variables then

formulas have

variables. (3) if

contains only positive formulas (

,

,

), then in

there are only positive formulas. (4) the alternation rank of all

formulas is bounded by the alternation rank of

• .

(5) etc.

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Simple Applications To Definability

Two important Theorem For

• f qr

and

• iff

• Theorem For
• f qr

and

• iff

• Many theorems in Descriptive Complexity can be reduced

in a modular way to these theorems by the compositional method.

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Modular reduction - Example

For

• f qr

• 1. if

• iff

• 2. if

• iff

• ✁

• iff

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Recursively defined Structures

. So

is a solution of the equation

It is the least solution. The composition theorem implies that

is computable from the FO theories of the components (here structures

and

). The rationals satisfy the equation:

Moreover

is the least solution of

The composition theorem implies that

is computable from the FO theories of the components (here structures

and

).

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I found composition theorems for a variety of recursive equations constructed from sum and product. The solution is taken as the least solution.

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Further Problems

What should be a general recursive definition for relational structures?

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Further Problems

What should be a general recursive definition for relational structures? How to treat the greatest solution for recursive definitions.

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Further Problems

What should be a general recursive definition for relational structures? How to treat the greatest solution for recursive definitions. Interesting Examples of recursively defined structures.

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Further Problems

What should be a general recursive definition for relational structures? How to treat the greatest solution for recursive definitions. Interesting Examples of recursively defined structures. Possible lack of examples: maybe recursive definitions can be reduced to the generalized products and sums?

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Further Problems

What should be a general recursive definition for relational structures? How to treat the greatest solution for recursive definitions. Interesting Examples of recursively defined structures. Possible lack of examples: maybe recursive definitions can be reduced to the generalized products and sums? THANK YOU

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