Stratification (mathematics)

Stratification has several usages in mathematics.

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, we say that a set of clauses of the form ${\displaystyle Q_{1}\wedge \dots \wedge Q_{n}\wedge \neg Q_{n+1}\wedge \dots \wedge \neg Q_{n+m}\rightarrow P}$ is stratified if and only if there is a stratification assignment S that fulfills the following conditions:

The notion of stratified negation leads to a very effective operational semantics for stratified programs in terms of the stratified least fixpoint, that is obtained by iteratively applying the fixpoint operator to each stratum of the program, from the lowest one up. Stratification is not 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).

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