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The Immerman–Szelepcsényi theorem: FO(TC) is closed under complement

Lax485149.ImmermanSzelepcsenyi · concepts/Lax485149/ImmermanSzelepcsenyi.lean · lax-485149

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    Natural Language Statement

    Theorem

    If a decision problem PP is FO(TC) definable, then so is its complement PcP^c. This is the Immerman–Szelepcsényi theorem in logical form: the absence of a path between source and target nodes of a definable graph on tuples of an ordered structure is witnessed by a walk that counts, inductively on the distance, the nodes reachable from the sources.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax485149.SecondOrderAtoms
    10import Lax485149.KromFragment
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.FirstOrderDefinability
    14import Lax485149.HeadAutomata
    15import Lax485149.Reachability
    16import Lax485149.DeterministicReachability
    17import Lax485149.TwoSat
    18import Lax485149.ClassNL
    19import Lax485149.ClassL
    20
    21/-!
    22---
    23title: The Immerman–Szelepcsényi theorem: FO(TC) is closed under complement
    24type: theorem
    25---
    26If a decision problem PP is FO(TC) definable, then so is its complement
    27PcP^c. This is the Immerman–Szelepcsényi theorem in logical form: the
    28absence of a path between source and target nodes of a definable graph on
    29tuples of an ordered structure is witnessed by a walk that counts,
    30inductively on the distance, the nodes reachable from the sources.
    31-/
    32
    33namespace Lax485149.ImmermanSzelepcsenyi
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Sat
    38open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    39open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    40open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    41open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    42
    43/-- The complement of an FO(TC) definable problem is FO(TC) definable. -/
    44axiom tcDefinable_compl : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    45 TCDefinable P → TCDefinable (DecisionProblem.compl P)
    46
    47end Lax485149.ImmermanSzelepcsenyi
    48
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