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PTIME by deterministic Turing machines

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

proven

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    Natural Language Statement

    Theorem

    Deterministic machine acceptance is PTIME-complete under first-order reductions, and a decision problem is in PTIME if and only if it reduces to deterministic machine acceptance by an ordered first-order reduction: the class defined by the Horn fragment is polynomial time on deterministic Turing machines, the reduction supplying the machine, its input and its polynomial bound. Membership is an FO(LFP) definition of the unique run; hardness builds, inside the instance, the machine that runs unit propagation on a Horn formula.

    Concept map
    24 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 dtmAccept_PTIME_complete proven

    2 mem_PTIME_iff_le_dtmAccept proven

    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: PTIME by deterministic Turing machines
    27type: theorem
    28---
    29Deterministic machine acceptance is PTIME-complete under first-order
    30reductions, and a decision problem is in PTIME if and only if it reduces to
    31deterministic machine acceptance by an ordered first-order reduction: the
    32class defined by the Horn fragment is polynomial time on deterministic
    33Turing machines, the reduction supplying the machine, its input and its
    34polynomial bound. Membership is an FO(LFP) definition of the unique run;
    35hardness builds, inside the instance, the machine that runs unit
    36propagation on a Horn formula.
    37-/
    38
    39namespace Lax535992.DeterministicMachinePTIMEComplete
    40
    41open FirstOrder FirstOrder.Language
    42open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    43open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    44open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    45open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    46open Lax485149.ClassNL Lax485149.ClassL
    47open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    48open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    49open Lax535992.ClassPTIME
    50
    51/-- Deterministic machine acceptance is PTIME-complete. -/
    52axiom dtmAccept_PTIME_complete : PTIME.Complete DTMAccept
    53
    54/-- PTIME is reducibility to deterministic machine acceptance. -/
    55axiom mem_PTIME_iff_le_dtmAccept : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    56 PTIME.Mem P ↔ Nonempty (OrderedFOReduction P DTMAccept)
    57
    58end Lax535992.DeterministicMachinePTIMEComplete
    59
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