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PTIME ⊆ NP

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

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    Natural Language Statement

    Theorem

    Every problem of PTIME is in NP: every SO-Horn definable problem is definable in existential second-order logic. The guards of a Horn program being over the ordered expansion of the vocabulary, the inclusion is not read off the syntax; every problem of PTIME reduces to HORN-SAT, which is in NP, and NP is closed under ordered first-order reductions.

    Concept map
    24 concepts
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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: PTIME ⊆ NP
    27type: theorem
    28---
    29Every problem of PTIME is in NP: every SO-Horn definable problem is
    30definable in existential second-order logic. The guards of a Horn program
    31being over the ordered expansion of the vocabulary, the inclusion is not
    32read off the syntax; every problem of PTIME reduces to HORN-SAT, which is in
    33NP, and NP is closed under ordered first-order reductions.
    34-/
    35
    36namespace Lax535992.PTIMESubsetNP
    37
    38open FirstOrder FirstOrder.Language
    39open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    40open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    41open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    42open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    43open Lax485149.ClassNL Lax485149.ClassL
    44open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    45open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    46open Lax535992.ClassPTIME
    47
    48/-- PTIME is contained in NP. -/
    49axiom PTIME_subset_NP : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    50 PTIME.Mem P → NP.Mem P
    51
    52end Lax535992.PTIMESubsetNP
    53
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