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L is acceptance by deterministic two-way multihead automata

Lax485149.LByAutomata · concepts/Lax485149/LByAutomata.lean · lax-485149

proven

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    Natural Language Statement

    Theorem

    A decision problem PP over a relational vocabulary LL is in L if and only if there are a number kk and a deterministic two-way kk-head automaton over LL that accepts, for every nonempty finite LL-structure AA and every linear order on AA, exactly when AA is a yes-instance of PP. The same holds for FO(DTC) definability.

    Concept map
    20 concepts
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    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 dtcDefinable_iff_automaton proven

    2 mem_LOGSPACE_iff_automaton proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax485149.SecondOrderAtoms
    10import Lax485149.KromFragment
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.FirstOrderDefinability
    14import Lax485149.HeadAutomata
    15import Lax485149.Reachability
    16import Lax485149.DeterministicReachability
    17import Lax485149.TwoSat
    18import Lax485149.ClassNL
    19import Lax485149.ClassL
    20
    21/-!
    22---
    23title: L is acceptance by deterministic two-way multihead automata
    24type: theorem
    25---
    26A decision problem PP over a relational vocabulary LL is in L if and only
    27if there are a number kk and a deterministic two-way kk-head automaton
    28over LL that accepts, for every nonempty finite LL-structure AA and every
    29linear order on AA, exactly when AA is a yes-instance of PP. The same
    30holds for FO(DTC) definability.
    31-/
    32
    33namespace Lax485149.LByAutomata
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Sat
    38open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    39open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    40open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    41open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    42
    43/-- L is acceptance by a deterministic two-way multihead automaton. -/
    44axiom mem_LOGSPACE_iff_automaton : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    45 LOGSPACE.Mem P ↔ ∃ (k : ℕ) (M : HeadAutomaton L k), M.IsDeterministic ∧
    46 ∀ (A : Type) [L.Structure A] [LinearOrder A] [Finite A] [Nonempty A], P A ↔ M.Accepts A
    47
    48/-- FO(DTC) definability is acceptance by a deterministic two-way multihead
    49automaton. -/
    50axiom dtcDefinable_iff_automaton : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    51 DTCDefinable P ↔ ∃ (k : ℕ) (M : HeadAutomaton L k), M.IsDeterministic ∧
    52 ∀ (A : Type) [L.Structure A] [LinearOrder A] [Finite A] [Nonempty A], P A ↔ M.Accepts A
    53
    54end Lax485149.LByAutomata
    55
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