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L ⊆ NL ⊆ PTIME

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

proven

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    Natural Language Statement

    Theorem

    Every problem of NL is in PTIME, and hence so is every problem of L. A Krom program is not a Horn program, Krom clauses having possibly two positive literals, so the inclusion is not syntactic: every problem of NL reduces to 2SAT, which a Horn program defines by deriving reachability in the implication graph and rejecting when a variable reaches its negation and back.

    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 LOGSPACE_subset_PTIME proven

    2 NL_subset_PTIME 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: L ⊆ NL ⊆ PTIME
    27type: theorem
    28---
    29Every problem of NL is in PTIME, and hence so is every problem of L. A Krom
    30program is not a Horn program, Krom clauses having possibly two positive
    31literals, so the inclusion is not syntactic: every problem of NL reduces to
    322SAT, which a Horn program defines by deriving reachability in the
    33implication graph and rejecting when a variable reaches its negation and
    34back.
    35-/
    36
    37namespace Lax535992.NLSubsetPTIME
    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/-- NL is contained in PTIME. -/
    50axiom NL_subset_PTIME : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    51 NL.Mem P → PTIME.Mem P
    52
    53/-- L is contained in PTIME. -/
    54axiom LOGSPACE_subset_PTIME : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    55 LOGSPACE.Mem P → PTIME.Mem P
    56
    57end Lax535992.NLSubsetPTIME
    58
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