What the name means

QF means quantifier-free. UFFF joins UF, uninterpreted functions, with FF, finite fields.

Quantifier-free formulas have no explicit "for every" or "there exists" binders. They can still contain unknown values that the solver must find.

A finite field supplies the arithmetic laws. An uninterpreted function has a declared input and output type but no given formula. Equal inputs require equal outputs, including equalities discovered through field arithmetic.

The function is otherwise arbitrary. A name such as hash adds no collision resistance or cryptographic assumption. Solver support for this combined fragment must be checked separately from support for field equations alone.

Worked example

Apply a function to equal field elements

Use the field with five elements and an abstract function tag from that field to itself. Compare tag(4 + 2) with tag(1).

  1. In this field, 4 + 2 reduces to 1.
  2. Both tag calls therefore receive the same field element.
  3. Ask whether their outputs differ. Function consistency requires the outputs to be equal.

The query is unsat. The field equation forces equality of the abstract function results.

This is an authored mathematical example, not captured Axeyum output. Here, sat means the constraints can all hold; unsat means they cannot.

A contrasting case

The condition tag(0) = tag(1) is sat. A constant function, which gives the same output for every input, meets that constraint.

Use cases

  • Check consistency among field calculations and abstract helper operations in an arithmetic-circuit model.
  • Explore which conclusions follow from field laws before specifying the formulas of every component.

Modeling caution

An uninterpreted function does not model a concrete cryptographic primitive's security. A sat result can exploit behaviors the real primitive does not have.

Axeyum coverage

The examples explain the logic's meaning. They do not establish complete Axeyum support for this division. Acceptance, solving, and evidence coverage depend on the input and the solver route.

This division appears in the site's retained regression scoreboard. Read the individual slices and their source records for the tested scope.

A solver may return unknown if it cannot settle a query. That result establishes neither sat nor unsat. Parsing a declared logic name alone does not prove support for every formula in it.

Engine support documentation at the site's recorded snapshot, 03ae4a3c9, 2026-10-05.

Definitions and references

Definitions consulted September 6, 2026. The worked examples and use-case explanations are authored for this guide.