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NL ⊆ NP

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

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    Natural Language Statement

    Theorem

    Every problem of NL is in NP: every SO-Krom definable problem is definable in existential second-order logic. The inclusion is not syntactic, the guards of a Krom program being over the ordered expansion of the vocabulary while a Σ1\Sigma_1 definition is order-free. It goes through complete problems: every problem of NL reduces to 2SAT, which a Horn program defines, and every problem a Horn program defines reduces to HORN-SAT, which is in NP.

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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: NL ⊆ NP
    24type: theorem
    25---
    26Every problem of NL is in NP: every SO-Krom definable problem is definable
    27in existential second-order logic. The inclusion is not syntactic, the
    28guards of a Krom program being over the ordered expansion of the vocabulary
    29while a Σ1\Sigma_1 definition is order-free. It goes through complete
    30problems: every problem of NL reduces to 2SAT, which a Horn program
    31defines, and every problem a Horn program defines reduces to HORN-SAT,
    32which is in NP.
    33-/
    34
    35namespace Lax485149.NLSubsetNP
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Sat
    40open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    41open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    42open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    43open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    44
    45/-- NL is contained in NP. -/
    46axiom NL_subset_NP : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    47 NL.Mem P → NP.Mem P
    48
    49end Lax485149.NLSubsetNP
    50
    Show Proof
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