Stratification (mathematics) Totally Explained

Everything about Stratification Mathematics totally explained

Stratification has several usages in mathematics.

In mathematical logic

In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, for Horn clause theories, we say that such a theory is stratified if and only if there is a stratification assignment S that fulfills the following conditions:

If a predicate P is positively derived from a predicate Q, then the stratification number of P must be greater than or equal to the stratification number of Q, in short $S(P) geq S(Q)$ .
If a predicate P is derived from a negated predicate Q, then the stratification number of P must be greater than the stratification number of Q, in short $S(P) > S(Q)$ .

The notion of stratified negation leads to a very effective operational semantics for stratified programs in terms of the stratified least fixpoint, that's obtained by iteratively applying the fixpoint operator to each stratum of the program, from the lowest one up. Stratification isn't only useful for guaranteeing unique interpretation of Horn clause theories. It has also been used by W.V. Quine (1937) to address Russell's paradox, which undermined Frege's central work Grundgesetze der Arithmetik (1902).

In set theory

In New Foundations (NF) and related set theories, a formula

phi

in the language of first-order logic with equality and membership is said to be stratified if and only if there's a function

sigma

which sends each variable appearing in

phi

(considered as an item of syntax) to a natural number (this works equally well if all integers are used) in such a way that any atomic formula

x in y

appearing in

phi

satisfies

sigma(x)+1 = sigma(y)

and any atomic formula

x = y

appearing in

phi

satisfies

sigma(x) = sigma(y)

.
It turns out that it's sufficient to require that these conditions be satisfied only when both variables in an atomic formula are bound in the set abstract

under consideration. A set abstract satisfying this weaker condition is said to be weakly stratified.
The stratification of New Foundations generalizes readily to languages with more predicates and with term constructions. Each primitive predicate needs to have specified required displacements between values of

sigma

at its (bound) arguments in a (weakly) stratified formula. In a language with term constructions, terms themselves need to be assigned values under

sigma

, with fixed displacements from the values of each of their (bound) arguments in a (weakly) stratified formula. Defined term constructions are neatly handled by (possibly merely implicitly) using the theory of descriptions: a term

(iota x.phi)

(the x such that

phi

) must be assigned the same value under

sigma

as the variable x.
A formula is stratified if and only if it's possible to assign types to all variables appearing in the formula in such a way that it'll make sense in a version TST of the theory of types described in the New Foundations article, and this is probably the best way to understand the stratification of New Foundations in practice.
The notion of stratification can be extended to the lambda calculus; this is found in papers of Randall Holmes.

In singularity theory

In singularity theory, there's a different meaning, of a decomposition of a topological space X into disjoint subsets (so that stratification is to spaces what partition is to sets). This isn't a useful notion when unrestricted; but when the various strata are defined by some recognisable set of conditions (for example being locally closed), and fit together manageably, this idea is often applied in geometry. Hassler Whitney defined formal conditions for stratification. See topologically stratified space.

In statistics

See stratified sampling.

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