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Inclusions between the polynomial and exponential classes

Lax480241.ExponentialInclusions · concepts/Lax480241/ExponentialInclusions.lean · lax-480241

proven

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    Natural Language Statement

    Theorem

    PTIME ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE, with NP ⊆ NEXPTIME, PSPACE ⊆ EXPSPACE, PH ⊆ EXPTIME, and NLexp⁡^{\exp} ⊆ EXPTIME. The inclusions one exponential up are those below carried by the exponential of classes; PSPACE ⊆ EXPTIME reads the configurations of a space-bounded computation as the points of an expansion.

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

    This concept declares 9 statements. Each proof establishes one of them relative to its assumptions.

    1 EXPTIME_subset_EXPSPACE proven

    2 EXPTIME_subset_NEXPTIME proven

    3 NEXPTIME_subset_EXPSPACE proven

    4 NL_exp_subset_EXPTIME proven

    5 NP_subset_NEXPTIME proven

    6 PH_subset_EXPTIME proven

    7 PSPACE_subset_EXPSPACE proven

    8 PSPACE_subset_EXPTIME proven

    9 PTIME_subset_EXPTIME proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Classes
    3import Lax904597.Machines
    4import Lax485149.Problems
    5import Lax485149.Complement
    6import Lax485149.ClassNL
    7import Lax535992.ClassPTIME
    8import Lax564036.Hierarchy
    9import Lax564036.AlternatingMachines
    10import Lax134656.ClassPSPACE
    11import Lax480241.Expansions
    12import Lax480241.SecondOrderFixedPoints
    13import Lax480241.AlternatingSpace
    14import Lax480241.ExponentialClasses
    15
    16/-!
    17---
    18title: Inclusions between the polynomial and exponential classes
    19type: theorem
    20---
    21PTIME ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE, with NP ⊆ NEXPTIME, PSPACE ⊆
    22EXPSPACE, PH ⊆ EXPTIME, and NLexp⁡^{\exp} ⊆ EXPTIME. The inclusions one
    23exponential up are those below carried by the exponential of classes; PSPACE
    24⊆ EXPTIME reads the configurations of a space-bounded computation as the
    25points of an expansion.
    26-/
    27
    28namespace Lax480241.ExponentialInclusions
    29
    30open FirstOrder FirstOrder.Language
    31open Lax904597.Problems Lax904597.Classes Lax904597.Machines Lax485149.Problems
    32 Lax485149.Complement Lax485149.ClassNL
    33open Lax535992.ClassPTIME Lax564036.Hierarchy Lax564036.AlternatingMachines Lax134656.ClassPSPACE
    34open Lax480241.Expansions Lax480241.SecondOrderFixedPoints Lax480241.AlternatingSpace
    35 Lax480241.ExponentialClasses
    36
    37/-- PSPACE ⊆ EXPTIME. -/
    38axiom PSPACE_subset_EXPTIME :
    39 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    40 PSPACE.Mem P → EXPTIME.Mem P
    41
    42/-- EXPTIME ⊆ NEXPTIME. -/
    43axiom EXPTIME_subset_NEXPTIME :
    44 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    45 EXPTIME.Mem P → NEXPTIME.Mem P
    46
    47/-- NEXPTIME ⊆ EXPSPACE. -/
    48axiom NEXPTIME_subset_EXPSPACE :
    49 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    50 NEXPTIME.Mem P → EXPSPACE.Mem P
    51
    52/-- EXPTIME ⊆ EXPSPACE. -/
    53axiom EXPTIME_subset_EXPSPACE :
    54 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    55 EXPTIME.Mem P → EXPSPACE.Mem P
    56
    57/-- PTIME ⊆ EXPTIME. -/
    58axiom PTIME_subset_EXPTIME :
    59 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    60 PTIME.Mem P → EXPTIME.Mem P
    61
    62/-- NP ⊆ NEXPTIME. -/
    63axiom NP_subset_NEXPTIME :
    64 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    65 NP.Mem P → NEXPTIME.Mem P
    66
    67/-- PSPACE ⊆ EXPSPACE. -/
    68axiom PSPACE_subset_EXPSPACE :
    69 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    70 PSPACE.Mem P → EXPSPACE.Mem P
    71
    72/-- PH ⊆ EXPTIME. -/
    73axiom PH_subset_EXPTIME :
    74 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    75 PH.Mem P → EXPTIME.Mem P
    76
    77/-- NL one exponential up is in EXPTIME. -/
    78axiom NL_exp_subset_EXPTIME :
    79 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    80 (expClass NL).Mem P → EXPTIME.Mem P
    81
    82end Lax480241.ExponentialInclusions
    83
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