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SO-Horn = FO(LFP)

Lax535992.HornIsLeastFixedPoint · concepts/Lax535992/HornIsLeastFixedPoint.lean · lax-535992

proven

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    Natural Language Statement

    Theorem

    A decision problem is SO-Horn definable if and only if it is FO(LFP) definable. A Horn program is a rule system and its goal clauses an output sentence, which gives one direction. Conversely, an FO(LFP) definition is compiled into a Horn program that derives the complement of the fixed point stage by stage along the order and evaluates the output sentence clause by clause. This is the equivalence of the two logics on ordered structures, due to Grädel.

    Concept map
    24 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 lfpDefinable_sigmaSOHornDefinable proven

    2 sigmaSOHornDefinable_lfpDefinable 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 Lax904597.Machines
    8import Lax485149.Problems
    9import Lax485149.Complement
    10import Lax485149.SecondOrderAtoms
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.ClassNL
    14import Lax485149.ClassL
    15import Lax535992.HornFragment
    16import Lax535992.LeastFixedPoint
    17import Lax535992.InflationaryFixedPoint
    18import Lax535992.HornSat
    19import Lax535992.CircuitValue
    20import Lax535992.Game
    21import Lax535992.DeterministicMachines
    22import Lax535992.ClassPTIME
    23
    24/-!
    25---
    26title: SO-Horn = FO(LFP)
    27type: theorem
    28---
    29A decision problem is SO-Horn definable if and only if it is FO(LFP)
    30definable. A Horn program is a rule system and its goal clauses an output
    31sentence, which gives one direction. Conversely, an FO(LFP) definition is
    32compiled into a Horn program that derives the complement of the fixed point
    33stage by stage along the order and evaluates the output sentence clause by
    34clause. This is the equivalence of the two logics on ordered structures, due
    35to Grädel.
    36-/
    37
    38namespace Lax535992.HornIsLeastFixedPoint
    39
    40open FirstOrder FirstOrder.Language
    41open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    42open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    43open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    44open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    45open Lax485149.ClassNL Lax485149.ClassL
    46open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    47open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    48open Lax535992.ClassPTIME
    49
    50/-- Every SO-Horn definable problem is FO(LFP) definable. -/
    51axiom sigmaSOHornDefinable_lfpDefinable :
    52 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    53 SigmaSOHornDefinable P → LFPDefinable P
    54
    55/-- Every FO(LFP) definable problem is SO-Horn definable. -/
    56axiom lfpDefinable_sigmaSOHornDefinable :
    57 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    58 LFPDefinable P → SigmaSOHornDefinable P
    59
    60end Lax535992.HornIsLeastFixedPoint
    61
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