Feferman–Vaught theorem explained

Feferman–Vaught theorem^[1] in model theory is a theorem by Solomon Feferman and Robert Lawson Vaught that shows how to reduce, in an algorithmic way, the first-order theory of a product of structures to the first-order theory of elements of the structure.

The theorem is considered as one of the standard results in model theory.^[2] ^[3] ^[4] The theorem extends the previous result of Andrzej Mostowski on direct products of theories.^[5] It generalizes (to formulas with arbitrary quantifiers) the property in universal algebra that equalities (identities) carry over to direct products of algebraic structures (which is a consequence of one direction of Birkhoff's theorem).

Direct product of structures

Consider a first-order logic signature L.The definition of product structures takes a family of L-structures

A_i

for

i\inI

for some index set I and defines the product structure

A=\prod_iA_i

, which is also an L-structure, with all functions and relations defined pointwise.

The definition generalizes the direct product in universal algebra to structures for languages that contain not only function symbols but also relation symbols.

r(a_1,\ldots,a_p)

is a relation symbol with

arguments in L and

a_1,\ldots,a_n\in\Pi_iA_i

are elements of the cartesian product, we define the interpretation of

A\modelsr(a_1,\ldots,a_p)\iff \foralli\inI, A_i\modelsr(a_{1(i),\ldots,a}_p(i))

When

is a functional relation, this definition reduces to the definition of direct product in universal algebra.

Statement of the theorem for direct products

For a first-order formula

\phi(\barx)

in signature L with free variables, and for an interpretation

\bara

of the variables

\barx

, we define the set of indices

for which

\phi(\bara)

holds in

A_i

||\phi(\bara)||=\{i\midA_i\models\phi(\bara(i))\}

Given a first-order formula with free variables

\phi(\barx)

, there is an algorithm to compute its equivalent game normal form, which is a finite disjunction

	k-1
vee
	i=0

\theta(\barx)

of mutually contradictory formulas.

The Feferman–Vaught theorem gives an algorithm that takes a first-order formula

\phi(\barx)

and constructs a formula

\phi^*

that reduces the condition that

\phi(\bara)

holds in the product to the condition that

\phi^*

holds in the interpretation of

k+1

sets of indices:

I,||\theta_0(\bara)||,\ldots,||\theta_k-1(\bara)||

Formula

\phi^*

is thus a formula with

k+1

free set variables, for example, in the first-order theory of Boolean algebra of sets.

Proof idea

Formula

\phi^*

can be constructed following the structure of the starting formula

\phi^*

. When

\phi

is quantifier free then, by definition of direct product above it follows

\begin{array}{rl} A\models\phi(\bara)&\iff\foralli\inI. A_i\models\phi(\bara(i))\\ &\iff\{i\midA_i\models\phi(\bara(i))\}=I\\ &\iff||\phi(\bara)||=I \end{array}

Consequently, we can take

	*(U,X
\phi
	1)

to be the equality

U=X₁

in the language of boolean algebra of sets (equivalently, the field of sets).

Extending the condition to quantified formulas can be viewed as a form of quantifier elimination, where quantification over product elements

\bara

\phi

is reduced to quantification over subsets of

Generalized products

It is often of interest to consider substructure of the direct product structure. If the restriction that defines product elements that belong to the substructure can be expressed as a condition on the sets of index elements, then the results can be generalized.

An example is the substructure of product elements that are constant at all but finitely many indices. Assume that the language L contains a constant symbol

and consider the substructure containing only those product elements

for which the set

\{i\midbf{A}_i\modelsa(i) ≠ c\}

is finite. The theorem then reduces the truth value in such substructure to a formula

\phi^*

in the boolean algebra of sets, where certain sets are restricted to be finite.

One way to define generalized products is to consider thosesubstructures where the sets

||\phi(a)||

belong to some boolean algebra

of sets

X\subseteqI

of indices (a subset of the powerset set algebra

2^I

), and where the product substructure admits gluing.^[6] Here admitting gluing refers to the following closure condition: if

a,b

are two product elements and

X\inB

is the element of the boolean algebra, then so is the element

defined by "gluing"

and

according to

c(i)=\left\{\begin{array}{rl} a(i),&ifi\inX\\ b(i),&ifi\in(I\setminusX) \end{array}\right.

Consequences

Feferman–Vaught theorem implies the decidability of Skolem arithmetic by viewing, via the fundamental theorem of arithmetic, the structure of natural numbers with multiplication as a generalized product (power) of Presburger arithmetic structures.

Given an ultrafilter on the set of indices

, we can define a quotient structure on product elements, leading to the theorem of Jerzy Łoś that can be used to construct hyperreal numbers.

Notes and References

Feferman . S . Vaught . R . The first order properties of products of algebraic systems . . 1959 . 47 . 1 . 57–103.
Book: Wilfrid Hodges
. Hodges . Wilfrid . Wilfrid Hodges. Model theory . 1993 . Cambridge University Press . 0521304423 . Section 9.6: Feferman-Vaught theorem.
Karp . Carol . Carol Karp. S. Feferman and R. L. Vaught. The first order properties of products of algebraic systems. Fundamenta mathematicae, vol, 47 (1959), pp. 57–103. (Review) . . August 1967 . 32 . 2 . 276–276 . 10.2307/2271704.
Book: Monk . J. Donald . Mathematical Logic . . Berlin, New York . Graduate Texts in Mathematics . 978-0-387-90170-1 . 1976 . registration . 23: Generalized Products.
Mostowski . Andrzej . On direct products of theories . Journal of Symbolic Logic . March 1952 . 17 . 1 . 1–31 . 10.2307/2267454.
Book: Hodges . Wilfrid . Model theory . 1993 . Cambridge University Press . 0521304423 . Section 9.6: Feferman-Vaught theorem. 459.