What the name means

QF means quantifier-free. FF means finite fields. A field supports addition, subtraction, multiplication, and division by nonzero elements.

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

For a prime p, arithmetic modulo p forms a field: keep the remainder after division by p. In the field with five elements, four plus two equals one. Multiplication also reduces its result modulo five.

Field arithmetic has no usual less-than ordering. It also differs from machine-word arithmetic modulo a power of two. A solver's finite-field support may cover prime fields only, and accepted syntax depends on the implementation.

Worked example

Solve an equation modulo five

In the field with five elements, find x such that 2 × x = 1.

  1. The available values are 0, 1, 2, 3, and 4, with arithmetic modulo five.
  2. Choose x = 3. Multiplying gives six, whose remainder modulo five is one.
  3. The field equation holds, so three is the multiplicative inverse of two in this field.

The query is sat. Its model uses field arithmetic, not ordinary integer equality.

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

Over ordinary integers, 2 × x = 1 is unsat. Modulo eight it is also unsat, because multiplying by two always gives an even remainder. Modulo eight is not a field.

Use cases

  • Check algebraic constraints in arithmetic circuits used by cryptographic proof systems.
  • Study equations in error-correcting code constructions over a specified finite field.

Modeling caution

Fix the field before interpreting a result. A small prime used for an example does not establish the same claim for another modulus or a production cryptographic field.

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.