Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-12
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FALSE: swapping the accepting states of a DPDA automatically complements its language

Statement

False claim: once a language is recognized by a DPDA, one complements the language simply by declaring the old nonaccepting states to be accepting and the old accepting states to be nonaccepting.

Facts & Assumptions

Given: A DPDA with one state q, bottom marker Z0, no accepting states, and one transition (q,Z0)→ε(q,Z0), so the machine loops forever without reading input.

[A1]

The statement refuted is: swapping the accepting states of a DPDA automatically complements its language.

[L1]

By Deterministic pushdown automata and deterministic context-free languages, a DPDA may have an enabled ε-move as its unique move from a configuration.

[L2]

By Deterministic context-free languages are closed under complement, complementation of DPDA languages requires an explicit construction, not merely a relabeling of states.

Refutation

technique · direct
1.1

The displayed machine recognizes the empty language, because on every marked input it follows the unique ε-loop forever and never reaches an accepting final state.

L1given
2.1

If we merely declare the state q to be accepting, the machine still follows exactly the same infinite ε-loop on every marked input, so it still recognizes the empty language. In particular, it does not recognize the complement Σ∗.

L1step 1.1
3.1

This contradicts [A1]. The complement theorem [L2] needs the endmarker and loop repairs; naive state swapping does not suffice.

A1L2step 2.1∎

Depends on

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources