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logic-foundations / Finite model theory; descriptive complexity

Lichter and Schweitzer's question: does witnessed symmetric choice strictly increase the expressive power of CPT with counting?

Theorem 1.1: over a vocabulary of eight binary relations there is a CPT+WSC sentence with a single WSC operator that returns a Boolean value on every finite structure and whose true-model class is not defined by any CPT sentence, even with counting and unbounded set rank; so CPT+WSC is strictly more expressive than CPT in Lichter and Schweitzer's sense. Not shown: whether CPT+WSC captures polynomial time, or the effect of nested WSC operators, both of which the paper leaves open.

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logic-foundationsSep 24, 2026Significance 18/100Registry: lean checked

Lichter and Schweitzer's question: does witnessed symmetric choice strictly increase the expressive power of CPT with counting?

Prior state unknown→proved

Theorem 1.1: over a vocabulary of eight binary relations there is a CPT+WSC sentence with a single WSC operator that returns a Boolean value on every finite structure and whose true-model class is not defined by any CPT sentence, even with counting and unbounded set rank; so CPT+WSC is strictly more expressive than CPT in Lichter and Schweitzer's sense. Not shown: whether CPT+WSC captures polynomial time, or the eff…

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Theorem 1.1: over a vocabulary of eight binary relations there is a CPT+WSC sentence with a single WSC operator that returns a Boolean value on every finite structure and whose true-model class is not defined by any CPT sentence, even with counting and unbounded set rank; so CPT+WSC is strictly more expressive than CPT in Lichter and Schweitzer's sense. Not shown: whether CPT+WSC captures polynomial time, or the effect of nested WSC operators, both of which the paper leaves open.

Theorem 1.1: over a vocabulary of eight binary relations there is a CPT+WSC sentence with a single WSC operator that returns a Boolean value on every finite structure and whose true-model class is not defined by any CPT sentence, even with counting and unbounded set rank; so CPT+WSC is strictly more expressive than CPT in Lichter and Schweitzer's sense. Not shown: whether CPT+WSC captures polynomial time, or the effect of nested WSC operators, both of which the paper leaves open.

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