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GAME is PTIME-complete

Lax535992.GamePTIMEComplete · concepts/Lax535992/GamePTIMEComplete.lean · lax-535992

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    Natural Language Statement

    Theorem

    GAME, alternating reachability, is PTIME-complete under first-order reductions. It is FO(LFP) definable, the winning positions being a least fixed point, hence in PTIME. Hardness reads unit propagation as a game, which the existential player wins exactly when the Horn formula is unsatisfiable: the complement of HORN-SAT reduces to GAME, and PTIME is closed under complement.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax904597.Machines
    8import Lax485149.Problems
    9import Lax485149.Complement
    10import Lax485149.SecondOrderAtoms
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.ClassNL
    14import Lax485149.ClassL
    15import Lax535992.HornFragment
    16import Lax535992.LeastFixedPoint
    17import Lax535992.InflationaryFixedPoint
    18import Lax535992.HornSat
    19import Lax535992.CircuitValue
    20import Lax535992.Game
    21import Lax535992.DeterministicMachines
    22import Lax535992.ClassPTIME
    23
    24/-!
    25---
    26title: GAME is PTIME-complete
    27type: theorem
    28---
    29GAME, alternating reachability, is PTIME-complete under first-order
    30reductions. It is FO(LFP) definable, the winning positions being a least
    31fixed point, hence in PTIME. Hardness reads unit propagation as a game,
    32which the existential player wins exactly when the Horn formula is
    33unsatisfiable: the complement of HORN-SAT reduces to GAME, and PTIME is
    34closed under complement.
    35-/
    36
    37namespace Lax535992.GamePTIMEComplete
    38
    39open FirstOrder FirstOrder.Language
    40open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    41open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    42open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    43open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    44open Lax485149.ClassNL Lax485149.ClassL
    45open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    46open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    47open Lax535992.ClassPTIME
    48
    49/-- GAME is PTIME-complete. -/
    50axiom game_PTIME_complete : PTIME.Complete GAME
    51
    52end Lax535992.GamePTIMEComplete
    53
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