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SUCCINCT-REACH is PSPACE-complete

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

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    Natural Language Statement

    Theorem

    SUCCINCT-REACH is PSPACE-complete under first-order reductions. It is SO(TC) definable, a state of the system being an assignment of a unary relation variable; and every SO(TC) definable problem reduces to it, the three sentences of a specification becoming the three groups of clauses of a transition system, as the first-order kernel of a definition becomes a CNF formula in the Cook–Levin theorem.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    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: SUCCINCT-REACH is PSPACE-complete
    24type: theorem
    25---
    26SUCCINCT-REACH is PSPACE-complete under first-order reductions. It is
    27SO(TC) definable, a state of the system being an assignment of a unary
    28relation variable; and every SO(TC) definable problem reduces to it, the
    29three sentences of a specification becoming the three groups of clauses of
    30a transition system, as the first-order kernel of a definition becomes a
    31CNF formula in the Cook–Levin theorem.
    32-/
    33
    34namespace Lax134656.SuccinctReachPSPACEComplete
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Machines
    39open Lax485149.Problems Lax485149.Complement
    40open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    41open Lax564036.Hierarchy
    42open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    43open Lax134656.PartialFixedPoint
    44open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    45
    46/-- The problem is PSPACE-complete. -/
    47axiom succinctReach_PSPACE_complete : PSPACE.Complete SUCCINCTREACH
    48
    49end Lax134656.SuccinctReachPSPACEComplete
    50
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