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Translations between SO-Krom and FO(TC)

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

proven

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    Natural Language Statement

    Theorem

    If a problem PP is SO-Krom definable, then its complement PcP^c is FO(TC) definable: a Krom program instantiated on a structure is a 2-CNF, which is unsatisfiable exactly when the goal clause fires or some literal reaches its negation and back in the implication graph, a reachability condition. Conversely, if PP is FO(TC) definable, then PcP^c is SO-Krom definable: the program guesses a set of nodes containing the targets and closed under predecessors, and rejects when the set contains a source.

    Concept map
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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 sigmaSOKromDefinable_compl_of_tcDefinable proven

    2 tcDefinable_compl_of_sigmaSOKromDefinable 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: Translations between SO-Krom and FO(TC)
    24type: theorem
    25---
    26If a problem PP is SO-Krom definable, then its complement PcP^c is FO(TC)
    27definable: a Krom program instantiated on a structure is a 2-CNF, which is
    28unsatisfiable exactly when the goal clause fires or some literal reaches its
    29negation and back in the implication graph, a reachability condition.
    30Conversely, if PP is FO(TC) definable, then PcP^c is SO-Krom definable:
    31the program guesses a set of nodes containing the targets and closed under
    32predecessors, and rejects when the set contains a source.
    33-/
    34
    35namespace Lax485149.KromAndTransitiveClosure
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Sat
    40open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    41open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    42open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    43open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    44
    45/-- The complement of an SO-Krom definable problem is FO(TC) definable. -/
    46axiom tcDefinable_compl_of_sigmaSOKromDefinable :
    47 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    48 SigmaSOKromDefinable P → TCDefinable (DecisionProblem.compl P)
    49
    50/-- The complement of an FO(TC) definable problem is SO-Krom definable. -/
    51axiom sigmaSOKromDefinable_compl_of_tcDefinable :
    52 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    53 TCDefinable P → SigmaSOKromDefinable (DecisionProblem.compl P)
    54
    55end Lax485149.KromAndTransitiveClosure
    56
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