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PH ⊆ PSPACE

Lax134656.HierarchyInPSPACE · concepts/Lax134656/HierarchyInPSPACE.lean · lax-134656

proven

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    Natural Language Statement

    Theorem

    NP, polynomial time, every level Σkp\Sigma_k^p and Πkp\Pi_k^p of the polynomial hierarchy, and hence PH, are contained in PSPACE. An existential block of second-order quantifiers is a walk that guesses its state and takes no step, and a walk can guess a block into its own state and never touch it again, so an existential block in front of an SO(TC) condition is again one; universal blocks are handled by complementing twice, PSPACE being closed under complement.

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

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 NP_subset_PSPACE proven

    2 PH_subset_PSPACE proven

    3 piP_subset_PSPACE proven

    4 PTIME_subset_PSPACE proven

    5 sigmaP_subset_PSPACE proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Machines
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax564036.Hierarchy
    13import Lax134656.SecondOrderTransitiveClosure
    14import Lax134656.OrderFreeTransitiveClosure
    15import Lax134656.PartialFixedPoint
    16import Lax134656.Qsat
    17import Lax134656.SuccinctReach
    18import Lax134656.SpaceBoundedMachines
    19import Lax134656.ClassPSPACE
    20
    21/-!
    22---
    23title: PH ⊆ PSPACE
    24type: theorem
    25---
    26NP, polynomial time, every level Σkp\Sigma_k^p and Πkp\Pi_k^p of the
    27polynomial hierarchy, and hence PH, are contained in PSPACE. An existential
    28block of second-order quantifiers is a walk that guesses its state and
    29takes no step, and a walk can guess a block into its own state and never
    30touch it again, so an existential block in front of an SO(TC) condition is
    31again one; universal blocks are handled by complementing twice, PSPACE being
    32closed under complement.
    33-/
    34
    35namespace Lax134656.HierarchyInPSPACE
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Machines
    40open Lax485149.Problems Lax485149.Complement
    41open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    42open Lax564036.Hierarchy
    43open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    44open Lax134656.PartialFixedPoint
    45open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    46
    47/-- NP is contained in PSPACE. -/
    48axiom NP_subset_PSPACE : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    49 NP.Mem P → PSPACE.Mem P
    50
    51/-- PTIME is contained in PSPACE. -/
    52axiom PTIME_subset_PSPACE : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    53 PTIME.Mem P → PSPACE.Mem P
    54
    55/-- Every `Σ` level is contained in PSPACE. -/
    56axiom sigmaP_subset_PSPACE :
    57 ∀ (k : ℕ) {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    58 (SigmaP k).Mem P → PSPACE.Mem P
    59
    60/-- Every `Π` level is contained in PSPACE. -/
    61axiom piP_subset_PSPACE : ∀ (k : ℕ) {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    62 (PiP k).Mem P → PSPACE.Mem P
    63
    64/-- PH is contained in PSPACE. -/
    65axiom PH_subset_PSPACE : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    66 PH.Mem P → PSPACE.Mem P
    67
    68end Lax134656.HierarchyInPSPACE
    69
    Show ProofShow ProofShow ProofShow ProofShow Proof
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