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SO(TC) does not need an order

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

proven

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    Natural Language Statement

    Theorem

    A decision problem is SO(TC) definable if and only if it is order-free SO(TC) definable, so PSPACE is the class of the order-free SO(TC) definable problems. A walk can guess its order: one more binary relation variable of the state holds a candidate order, the source sentence checks that it is linear, every step keeps it unchanged, and the three sentences read it in place of the order symbol. The logics of the classes below, deterministic or clausal, have no relation variable to guess an order with.

    Concept map
    22 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 mem_PSPACE_iff_sotcDefinableFree proven

    2 sotcDefinable_iff_free 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: SO(TC) does not need an order
    24type: theorem
    25---
    26A decision problem is SO(TC) definable if and only if it is order-free
    27SO(TC) definable, so PSPACE is the class of the order-free SO(TC) definable
    28problems. A walk can guess its order: one more binary relation variable of
    29the state holds a candidate order, the source sentence checks that it is
    30linear, every step keeps it unchanged, and the three sentences read it in
    31place of the order symbol. The logics of the classes below, deterministic
    32or clausal, have no relation variable to guess an order with.
    33-/
    34
    35namespace Lax134656.TransitiveClosureWithoutOrder
    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/-- SO(TC) definability and its order-free form coincide. -/
    48axiom sotcDefinable_iff_free : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    49 SOTCDefinable P ↔ SOTCDefinableFree P
    50
    51/-- PSPACE is order-free SO(TC) definability. -/
    52axiom mem_PSPACE_iff_sotcDefinableFree :
    53 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    54 PSPACE.Mem P ↔ SOTCDefinableFree P
    55
    56end Lax134656.TransitiveClosureWithoutOrder
    57
    Show ProofShow Proof
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