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EXPTIME and EXPSPACE as PTIME and PSPACE read exponentially

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

proven

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    Natural Language Statement

    Theorem

    EXPTIME is PTIME one exponential up and EXPSPACE is PSPACE one exponential up: these are the capture theorems FO(≤, LFP) = PTIME and FO(≤, PFP) = PSPACE read on the expanded universe. The order that the expansion's sentences read can be removed from both definitions, the expansion guessing it into its block.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 EXPSPACE_eq_PSPACE_exp proven

    2 EXPTIME_eq_PTIME_exp proven

    3 mem_EXPSPACE_iff_sopfpDefinableFree proven

    4 mem_EXPTIME_iff_solfpDefinableFree 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: EXPTIME and EXPSPACE as PTIME and PSPACE read exponentially
    19type: theorem
    20---
    21EXPTIME is PTIME one exponential up and EXPSPACE is PSPACE one exponential
    22up: these are the capture theorems FO(≤, LFP) = PTIME and FO(≤, PFP) =
    23PSPACE read on the expanded universe. The order that the expansion's
    24sentences read can be removed from both definitions, the expansion guessing
    25it into its block.
    26-/
    27
    28namespace Lax480241.ExponentialCaptures
    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/-- EXPTIME is PTIME one exponential up. -/
    38axiom EXPTIME_eq_PTIME_exp :
    39 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    40 EXPTIME.Mem P ↔ (expClass PTIME).Mem P
    41
    42/-- EXPSPACE is PSPACE one exponential up. -/
    43axiom EXPSPACE_eq_PSPACE_exp :
    44 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    45 EXPSPACE.Mem P ↔ (expClass PSPACE).Mem P
    46
    47/-- EXPTIME without the order. -/
    48axiom mem_EXPTIME_iff_solfpDefinableFree :
    49 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    50 EXPTIME.Mem P ↔ SOLFPDefinableFree P
    51
    52/-- EXPSPACE without the order. -/
    53axiom mem_EXPSPACE_iff_sopfpDefinableFree :
    54 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    55 EXPSPACE.Mem P ↔ SOPFPDefinableFree P
    56
    57end Lax480241.ExponentialCaptures
    58
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