{ "records": [ { "id": "lax-3", "state": "draft", "environment": "v4.30.0", "title": "First-Order Model Checking on Nowhere Dense Graph Classes in Almost Linear Time", "concepts": [ { "id": "Lax3.ColoredGraphs", "title": "Colored graphs and walk distance", "type": "definition" }, { "id": "Lax3.DistFO", "title": "Distance logic and distance rank", "type": "definition" }, { "id": "Lax3.FirstOrder", "title": "First-order logic on graphs", "type": "definition" }, { "id": "Lax3.Locality", "title": "Locality theorem for distance logic", "type": "theorem" }, { "id": "Lax3.ModelChecking", "title": "First-order model checking on nowhere dense classes in almost linear time", "type": "theorem" }, { "id": "Lax3.NeighborhoodCoverBound", "title": "Neighborhood covers of weak coloring degree", "type": "theorem" }, { "id": "Lax3.NeighborhoodCovers", "title": "Sparse neighborhood covers", "type": "definition" }, { "id": "Lax3.NormalForm", "title": "Normal form for distance logic", "type": "theorem" }, { "id": "Lax3.NowhereDenseSplitter", "title": "Splitter wins on nowhere dense classes", "type": "theorem" }, { "id": "Lax3.OrderedNeighborhoodCover", "title": "Neighborhood covers from vertex orderings", "type": "theorem" }, { "id": "Lax3.ScatterSentences", "title": "Scatter sentences", "type": "definition" }, { "id": "Lax3.SplitterGame", "title": "The isolation splitter game", "type": "definition" } ], "proofs": [ "Lax3Proofs.Assembly.locality", "Lax3Proofs.Assembly.normalForm", "Lax3Proofs.CoverConstruction.exists_neighborhoodCover_degree_wcol", "Lax3Proofs.CoverConstruction.isNeighborhoodCover_wreach", "Lax3Proofs.ModelChecking.exists_almostLinearTime_program_modelChecking", "Lax3Proofs.SplitterWin.splitterWins_of_nowhereDense" ] }, { "id": "lax-5", "state": "registered", "environment": "v4.30.0", "title": "Almost Linear Neighborhood Complexity of Monadically Dependent Graph Classes", "concepts": [ { "id": "Lax5.AdlerAdler", "title": "Nowhere dense classes are monadically dependent", "type": "theorem" }, { "id": "Lax5.AlmostLinearNC", "title": "Monadically dependent classes have almost linear neighborhood complexity", "type": "theorem" }, { "id": "Lax5.GraphClasses", "title": "Weakly sparse graph classes", "type": "definition" }, { "id": "Lax5.GraphTransductions", "title": "Graph transductions", "type": "definition" }, { "id": "Lax5.MonadicDependence", "title": "Monadic dependence", "type": "definition" }, { "id": "Lax5.Transductions", "title": "First-order transductions", "type": "definition" }, { "id": "Lax5.WeaklySparseDependent", "title": "Weakly sparse monadically dependent classes are nowhere dense", "type": "theorem" } ], "proofs": [ "Lax5Proofs.AdlerAdler.monadicallyDependent_of_nowhereDense", "Lax5Proofs.Corollary6a.nowhereDense_of_weaklySparse_of_monadicallyDependent", "Lax5Proofs.Theorem2.hasAlmostLinearNC_of_monadicallyDependent" ] }, { "id": "lax-9", "state": "draft", "environment": "v4.30.0", "title": "χ-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs", "concepts": [ { "id": "Lax9.BoundedMergeWidthChiBounded", "title": "χ-Boundedness of Bounded Merge-Width Classes", "type": "theorem" }, { "id": "Lax9.BoundedMergeWidthLinearNeighborhoodComplexity", "title": "Linear Neighbourhood Complexity of Bounded Merge-Width Classes", "type": "theorem" }, { "id": "Lax9.ChiBoundedness", "title": "χ-Boundedness", "type": "definition" }, { "id": "Lax9.MergeWidth", "title": "Merge-Width", "type": "definition" }, { "id": "Lax9.NeighborhoodComplexity", "title": "Linear Neighbourhood Complexity", "type": "definition" } ], "proofs": [ "Lax9Proofs.chiBounded_of_boundedMergeWidth", "Lax9Proofs.linearNeighborhoodComplexity_of_boundedMergeWidth" ] }, { "id": "lax-10", "state": "draft", "environment": "v4.30.0", "title": "Graphs without a 3-Connected Subgraph are 4-Colourable", "concepts": [ { "id": "Lax10.Fragile", "title": "Fragile and m-fragile graphs", "type": "definition" }, { "id": "Lax10.FragileFourColorable", "title": "Graphs without a 3-connected subgraph are 4-colorable", "type": "theorem" }, { "id": "Lax10.Theorem2", "title": "Theorem 2", "type": "theorem" }, { "id": "Lax10.ThreeConnectedAndColorable", "title": "3-connected graphs and 4-colorability", "type": "definition" } ], "proofs": [ "Lax10Proofs.Main.fragile_four_colorable", "Lax10Proofs.Main.theorem2" ] }, { "id": "lax-11", "state": "draft", "environment": "v4.30.0", "title": "Algorithmic Experiments on a Random Access Machine", "concepts": [ { "id": "Lax11.CliqueExpr", "title": "k-expressions, and the graphs of cliquewidth at most k", "type": "definition" }, { "id": "Lax11.ConnectedComponents", "title": "Connected components in linear time", "type": "theorem" }, { "id": "Lax11.Courcelle", "title": "Courcelle's theorem on a word RAM", "type": "theorem" }, { "id": "Lax11.GraphEncoding", "title": "Compressed sparse row encoding of a graph", "type": "definition" }, { "id": "Lax11.InstanceEncoding", "title": "Encoding a graph together with a k-expression", "type": "definition" }, { "id": "Lax11.Mso", "title": "Monadic second-order logic on graphs", "type": "definition" }, { "id": "Lax11.VertexCover", "title": "A graph with a parameter appended", "type": "definition" } ], "proofs": [ "Lax11Proofs.CCMain.exists_linearTime_program_ccLabels", "Lax11Proofs.Courcelle.exists_linearTime_program_modelChecking" ] }, { "id": "lax-12", "state": "registered", "environment": "v4.30.0", "title": "Sparsity Lectures: Nowhere Denseness, Quasi-Wideness, and Generalized Coloring Numbers", "concepts": [ { "id": "Lax12.Admissibility", "title": "Admissibility", "type": "definition" }, { "id": "Lax12.AdmissibilityBound", "title": "Admissibility is bounded by topological shallow-minor density", "type": "theorem" }, { "id": "Lax12.ColoringNumbers", "title": "Generalized coloring numbers", "type": "definition" }, { "id": "Lax12.GraphClasses", "title": "Graph classes", "type": "definition" }, { "id": "Lax12.NeighborhoodComplexity", "title": "Neighborhood complexity", "type": "definition" }, { "id": "Lax12.NowhereDenseClasses", "title": "Nowhere dense graph classes", "type": "definition" }, { "id": "Lax12.NowhereDenseDensity", "title": "Nowhere dense classes have subpolynomial shallow-minor density", "type": "theorem" }, { "id": "Lax12.NowhereDenseNC", "title": "Nowhere dense classes have almost linear neighborhood complexity", "type": "theorem" }, { "id": "Lax12.NowhereDenseUQW", "title": "Nowhere dense classes are uniformly quasi-wide", "type": "theorem" }, { "id": "Lax12.NowhereDenseWcol", "title": "Nowhere dense classes have subpolynomial weak coloring numbers", "type": "theorem" }, { "id": "Lax12.ShallowMinorDensity", "title": "Edge density of shallow minors", "type": "definition" }, { "id": "Lax12.ShallowTopologicalMinors", "title": "Shallow topological minors", "type": "definition" }, { "id": "Lax12.StrongColoringBound", "title": "Strong coloring numbers are bounded by admissibility", "type": "theorem" }, { "id": "Lax12.UniformQuasiWideness", "title": "Uniform quasi-wideness", "type": "definition" }, { "id": "Lax12.WeakColoringBound", "title": "Weak coloring numbers are bounded by strong coloring numbers", "type": "theorem" } ], "proofs": [ "Lax12Proofs.AdmissibilityBound.adm_le_of_hasTopologicalDensityAtMost", "Lax12Proofs.NowhereDenseDensity.hasSubpolynomialDensity_of_nowhereDense", "Lax12Proofs.NowhereDenseNC.hasAlmostLinearNC_of_nowhereDense", "Lax12Proofs.NowhereDenseUQW.uniformlyQuasiWide_of_nowhereDense", "Lax12Proofs.NowhereDenseWcol.hasSubpolynomialWcol_of_nowhereDense", "Lax12Proofs.StrongColoringBound.scol_le_of_adm", "Lax12Proofs.WeakColoringBound.wcol_le_of_scol" ] }, { "id": "lax-13", "state": "registered", "environment": "v4.30.0", "title": "The Word RAM", "concepts": [ { "id": "Lax13.Ram", "title": "The word RAM", "type": "definition" }, { "id": "Lax13.RamComputes", "title": "Computing a function within a time bound", "type": "definition" } ], "proofs": [] }, { "id": "lax-14", "state": "registered", "environment": "v4.30.0", "title": "Finite Ramsey Theorems for Pairs and Tuples", "concepts": [ { "id": "Lax14.MulticolorRamsey", "title": "Ramsey's theorem for colourings of pairs", "type": "theorem" }, { "id": "Lax14.OrderTypes", "title": "Order type of a tuple", "type": "definition" }, { "id": "Lax14.Ramsey", "title": "Ramsey's theorem", "type": "theorem" }, { "id": "Lax14.TupleRamsey", "title": "Ramsey's theorem for tuples", "type": "theorem" } ], "proofs": [ "Lax14Proofs.MulticolorRamsey.exists_monochromatic_set", "Lax14Proofs.Ramsey.exists_clique_or_indepSet", "Lax14Proofs.TupleRamsey.exists_orderType_homogeneous" ] }, { "id": "lax-17", "state": "draft", "environment": "v4.30.0", "title": "Exponent 8 ($\\times$ Polylogarithmic) Bound for the Grid-Minor Theorem", "concepts": [ { "id": "Lax17.Crossbar", "title": "Crossbars and pseudo-grids", "type": "definition" }, { "id": "Lax17.CrossbarOrPseudoGrid", "title": "Crossbar-or-pseudo-grid dichotomy", "type": "theorem" }, { "id": "Lax17.CrossbarStitching", "title": "Crossbar stitching", "type": "theorem" }, { "id": "Lax17.CutMatchingTheorem", "title": "Cut-matching expansion theorem", "type": "theorem" }, { "id": "Lax17.Degree", "title": "Degree bounds", "type": "definition" }, { "id": "Lax17.EdgeMenger", "title": "Edge-Menger theorem", "type": "theorem" }, { "id": "Lax17.ExpanderGrid", "title": "Grid minors from separator expansion", "type": "theorem" }, { "id": "Lax17.Expansion", "title": "Edge expansion and cut-matching transcripts", "type": "definition" }, { "id": "Lax17.ExponentTenCrossbarDichotomy", "title": "Exponent-ten crossbar dichotomy", "type": "theorem" }, { "id": "Lax17.Grid", "title": "Square grid graphs", "type": "definition" }, { "id": "Lax17.GridMinor", "title": "Square grid minors", "type": "definition" }, { "id": "Lax17.HairyPathOfSetsFromTreewidth", "title": "A hairy path-of-sets system from treewidth", "type": "theorem" }, { "id": "Lax17.HindOellermann", "title": "Hind--Oellermann deletion--contraction theorem", "type": "theorem" }, { "id": "Lax17.Linkedness", "title": "Linked terminal sets", "type": "definition" }, { "id": "Lax17.LocalRoutingOrGrid", "title": "Local routing-or-grid alternative", "type": "theorem" }, { "id": "Lax17.LowDegreeWellLinkedCore", "title": "A low-degree well-linked core", "type": "theorem" }, { "id": "Lax17.Mader", "title": "Mader's admissible split-off theorem", "type": "theorem" }, { "id": "Lax17.Minor", "title": "Graph minors", "type": "definition" }, { "id": "Lax17.NodeWellLinkedSetFromTreewidth", "title": "A well-linked set from treewidth", "type": "theorem" }, { "id": "Lax17.ParallelClusterSplitting", "title": "Parallel cluster splitting", "type": "theorem" }, { "id": "Lax17.PathOfSets", "title": "Strong and hairy path-of-sets systems", "type": "definition" }, { "id": "Lax17.Paths", "title": "Paths, linkages, and separators", "type": "definition" }, { "id": "Lax17.PolynomialGridMinor", "title": "Exponent 8 ($\\times$ Polylogarithmic) Bound for the Grid-Minor Theorem", "type": "theorem" }, { "id": "Lax17.SinghLau", "title": "Singh--Lau bounded-degree spanning-tree theorem", "type": "theorem" }, { "id": "Lax17.SmallLinkedSubsets", "title": "Linking small terminal subsets", "type": "theorem" }, { "id": "Lax17.SpanningTreeRounding", "title": "Bounded-degree spanning-tree relaxation", "type": "definition" }, { "id": "Lax17.StrongPathExtraction", "title": "Strong path extraction", "type": "theorem" }, { "id": "Lax17.StrongPathOfSetsContainsGrid", "title": "A grid minor from a strong path-of-sets system", "type": "theorem" }, { "id": "Lax17.StrongPathOfSetsFromTreewidth", "title": "A strong path-of-sets system from treewidth", "type": "theorem" }, { "id": "Lax17.StrongTreeOfSetsConstruction", "title": "Strong tree-of-sets construction", "type": "theorem" }, { "id": "Lax17.TerminalConnectivity", "title": "Terminal element connectivity and split-off operations", "type": "definition" }, { "id": "Lax17.TerminalElementMenger", "title": "Terminal element-Menger theorem", "type": "theorem" }, { "id": "Lax17.TreeOfSets", "title": "Strong tree-of-sets systems", "type": "definition" }, { "id": "Lax17.Treewidth", "title": "Treewidth", "type": "definition" }, { "id": "Lax17.TreewidthMinorMonotonicity", "title": "Treewidth monotonicity under graph minors", "type": "theorem" }, { "id": "Lax17.TreewidthSparsifier", "title": "Degree-three treewidth sparsifier", "type": "theorem" }, { "id": "Lax17.VertexMenger", "title": "Vertex-Menger theorem", "type": "theorem" }, { "id": "Lax17.WellLinkednessBoosting", "title": "Well-linkedness boosting", "type": "theorem" } ], "proofs": [ "Lax17Proofs.Exposed.crossbarOrPseudoGrid", "Lax17Proofs.Exposed.crossbarStitching", "Lax17Proofs.Exposed.degreeThreeTreewidthSparsifier", "Lax17Proofs.Exposed.edgeMenger", "Lax17Proofs.Exposed.expanderContainsGrid", "Lax17Proofs.Exposed.exponentTenCrossbarDichotomy", "Lax17Proofs.Exposed.hairyPathOfSetsFromTreewidth", "Lax17Proofs.Exposed.hindOellermannDeletionContraction", "Lax17Proofs.Exposed.localRoutingOrGrid", "Lax17Proofs.Exposed.logarithmicCutMatchingExpansion", "Lax17Proofs.Exposed.lowDegreeWellLinkedCore", "Lax17Proofs.Exposed.maderAdmissibleSplitOff", "Lax17Proofs.Exposed.nodeWellLinkedSetFromTreewidth", "Lax17Proofs.Exposed.parallelClusterSplitting", "Lax17Proofs.Exposed.singhLauBoundedDegreeSpanningTree", "Lax17Proofs.Exposed.smallLinkedSubsets", "Lax17Proofs.Exposed.strongPathExtraction", "Lax17Proofs.Exposed.strongPathOfSetsContainsGrid", "Lax17Proofs.Exposed.strongPathOfSetsFromTreewidth", "Lax17Proofs.Exposed.strongTreeOfSetsConstruction", "Lax17Proofs.Exposed.terminalElementMenger", "Lax17Proofs.Exposed.treewidth_mono_minor", "Lax17Proofs.Exposed.vertexMenger", "Lax17Proofs.Exposed.wellLinkednessBoosting", "Lax17Proofs.Final.polynomial_grid_minor_eight_polylog" ] }, { "id": "lax-18", "state": "draft", "environment": "v4.30.0", "title": "Szemerédi's Regularity Lemma", "concepts": [ { "id": "Lax18.EdgeDensity", "title": "Edge density between two vertex sets", "type": "definition" }, { "id": "Lax18.EnergyIncrement", "title": "The energy-increment refinement lemma", "type": "theorem" }, { "id": "Lax18.EquitableCleanup", "title": "Equitable cleanup of a bounded partition", "type": "theorem" }, { "id": "Lax18.FiniteGraphPartitions", "title": "Finite graph vertex partitions", "type": "definition" }, { "id": "Lax18.PartitionEnergy", "title": "Refinement and the index of a graph partition", "type": "definition" }, { "id": "Lax18.PartitionEnergyBounds", "title": "Bounds for partition energy", "type": "theorem" }, { "id": "Lax18.PartitionEnergyMonotonicity", "title": "Monotonicity of partition energy under refinement", "type": "theorem" }, { "id": "Lax18.RegularPairs", "title": "Regular pairs", "type": "definition" }, { "id": "Lax18.RegularPartitions", "title": "Regular partitions", "type": "definition" }, { "id": "Lax18.SzemerediRegularityLemma", "title": "Szemerédi's regularity lemma", "type": "theorem" } ], "proofs": [ "Lax18Proofs.energy_increment_refinement", "Lax18Proofs.equitable_cleanup", "Lax18Proofs.partitionEnergy_mem_unitInterval", "Lax18Proofs.partitionEnergy_mono_of_refines", "Lax18Proofs.szemeredi_regularity_lemma" ] }, { "id": "lax-41", "state": "registered", "environment": "v4.30.0", "title": "Constructive Lovász Local Lemma", "concepts": [ { "id": "Lax41.HaeuplerSahaSrinivasanDefinitions", "title": "Definitions for the Haeupler–Saha–Srinivasan distributional theorem", "type": "definition" }, { "id": "Lax41.HaeuplerSahaSrinivasanTheorem22", "title": "Haeupler–Saha–Srinivasan Theorem 2.2", "type": "theorem" }, { "id": "Lax41.MoserTardos", "title": "The Moser–Tardos theorem", "type": "theorem" }, { "id": "Lax41.MoserTardosDefinitions", "title": "Definitions for the Moser–Tardos resampling algorithm", "type": "definition" } ], "proofs": [ "Lax41Proofs.moser_tardos", "Lax41Proofs.theorem_2_2" ] }, { "id": "lax-47", "state": "draft", "environment": "v4.30.0", "title": "Tight Inapproximability of Max Independent Set in Triangle-Free Graphs", "concepts": [ { "id": "Lax47.Complexity", "title": "Executable graph encodings and triangle-free approximation", "type": "definition" }, { "id": "Lax47.Gap", "title": "Finite-Turing promise-gap algorithms for Max Independent Set", "type": "definition" }, { "id": "Lax47.Hastad", "title": "Håstad's inapproximability of Max Independent Set", "type": "definition" }, { "id": "Lax47.Machine", "title": "Polynomial-time computation on the Lax51 finite-Turing model", "type": "definition" }, { "id": "Lax47.Theorem12", "title": "Håstad hardness implies tight inapproximability in triangle-free graphs", "type": "theorem" } ], "proofs": [ "Lax47Proofs.theorem_1_2" ] }, { "id": "lax-48", "state": "registered", "environment": "v4.30.0", "title": "Twin-Width Can Be Exponential in Treewidth", "concepts": [ { "id": "Lax48.ExponentialSeparation", "title": "Twin-width can be exponential in treewidth", "type": "theorem" }, { "id": "Lax48.Treewidth", "title": "Treewidth", "type": "definition" }, { "id": "Lax48.TwinWidth", "title": "Twin-width", "type": "definition" } ], "proofs": [ "Lax48Proofs.Main.exists_treewidth_le_and_two_pow_lt_twinWidth" ] }, { "id": "lax-49", "state": "registered", "environment": "v4.30.0", "title": "Functional Equivalence of Twin-Width and Mixed Minor Number", "concepts": [ { "id": "Lax49.FunctionalEquivalence", "title": "Twin-width and mixed minor number are functionally equivalent", "type": "theorem" }, { "id": "Lax49.GraphParameters", "title": "Graph parameters and functional equivalence", "type": "definition" }, { "id": "Lax49.MixedMinorNumber", "title": "Mixed minor number", "type": "definition" }, { "id": "Lax49.MixedMinorNumberFromTwinWidth", "title": "Mixed minor number is bounded by a function of twin-width", "type": "theorem" }, { "id": "Lax49.TwinWidthFromMixedMinorNumber", "title": "Twin-width is bounded by a function of mixed minor number", "type": "theorem" } ], "proofs": [ "Lax49Proofs.Main.twin_width_functionally_equivalent_mixed_minor_number", "Lax49Proofs.MixedMinorNumberFromTwinWidth.exists_mixedMinorNumber_bound_of_twinWidth", "Lax49Proofs.TwinWidthFromMixedMinorNumber.exists_twinWidth_bound_of_mixedMinorNumber" ] }, { "id": "lax-51", "state": "draft", "environment": "v4.30.0", "title": "Computability and polynomial-time equivalence of Turing machines and word RAMs", "concepts": [ { "id": "Lax51.BinaryWordEncoding", "title": "Binary encoding of finite words", "type": "definition" }, { "id": "Lax51.RamPolytime", "title": "Polynomial-time computation by a word RAM", "type": "definition" }, { "id": "Lax51.RamToTuringGenericTime", "title": "Generic-time simulation of word RAMs by Turing machines", "type": "theorem" }, { "id": "Lax51.TuringPolytime", "title": "Polynomial-time computation by a Turing machine", "type": "definition" }, { "id": "Lax51.TuringRamEquivalence", "title": "Equivalence of Turing machines and word RAMs", "type": "theorem" }, { "id": "Lax51.TuringRamPolytimeEquivalence", "title": "Polynomial-time equivalence of Turing machines and word RAMs", "type": "theorem" }, { "id": "Lax51.TuringToRamGenericTime", "title": "Generic-time simulation of Turing machines by word RAMs", "type": "theorem" } ], "proofs": [ "Lax51Proofs.GenericTimeSimulation.ramInTime_to_turingInPolynomialOverhead", "Lax51Proofs.GenericTimeSimulation.turingWithInputTime_to_ramInPolynomialOverhead", "Lax51Proofs.TuringRamEquivalence.ramComputable_iff_computable", "Lax51Proofs.TuringRamPolytimeEquivalence.ramPolytime_iff_turingPolytime" ] }, { "id": "lax-52", "state": "draft", "environment": "v4.30.0", "title": "MSO-automata", "concepts": [ { "id": "Lax52.MSOAutomataEquivalence", "title": "Büchi-Elgot-Trakhtenbrot theorem for finite words", "type": "theorem" }, { "id": "Lax52.MSOSemantics", "title": "Semantics of monadic second-order logic", "type": "definition" }, { "id": "Lax52.MSOSyntax", "title": "Monadic second-order syntax", "type": "definition" }, { "id": "Lax52.MSOToNFA", "title": "MSO-definable word languages are NFA-recognizable", "type": "theorem" }, { "id": "Lax52.NFARecognizable", "title": "Recognition by a finite nondeterministic automaton", "type": "definition" }, { "id": "Lax52.NFAToMSO", "title": "Finite automata are MSO-definable", "type": "theorem" }, { "id": "Lax52.WordStructure", "title": "Words as finite relational structures", "type": "definition" } ], "proofs": [ "Lax52Proofs.existsUnique_wordStructure_proof", "Lax52Proofs.mso_definable_is_nfaRecognizable_proof", "Lax52Proofs.nfa_definable_by_mso_proof", "Lax52Proofs.nfaRecognizable_iff_msoDefinable_proof" ] }, { "id": "lax-53", "state": "draft", "environment": "v4.30.0", "title": "MSO and tree automata on finite ranked trees", "concepts": [ { "id": "Lax53.AutomatonLinearTime", "title": "Word-RAM evaluation of tree automata", "type": "theorem" }, { "id": "Lax53.Determinization", "title": "Determinization of tree automata", "type": "theorem" }, { "id": "Lax53.MSOLinearTime", "title": "Uniform word-RAM model checking for intrinsic MSO sentences", "type": "definition and theorem" }, { "id": "Lax53.MSOTreeAutomataEquivalence", "title": "Thatcher–Wright–Doner theorem for finite ranked trees", "type": "theorem" }, { "id": "Lax53.RankedTree", "title": "Finite ranked trees", "type": "definition" }, { "id": "Lax53.StructuralRepresentations", "title": "Certified structural representations for ranked-tree model checking", "type": "definition and theorem" }, { "id": "Lax53.TreeAutomaton", "title": "Bottom-up tree automata", "type": "definition" }, { "id": "Lax53.TreeModelCheckingEncoding", "title": "Distinguished structural inputs for tree-automaton evaluation", "type": "definition and theorem" }, { "id": "Lax53.TreeStructure", "title": "Ranked trees as finite relational structures", "type": "definition" }, { "id": "Lax53.ValueTranslations", "title": "Value-level translations between MSO and tree automata", "type": "theorem" } ], "proofs": [ "Lax53Proofs.AutomatonLinearTime.exists_fixed_automatonAcceptance_proof", "Lax53Proofs.AutomatonLinearTime.exists_uniform_automatonAcceptance_proof", "Lax53Proofs.Determinization.exists_deterministic_equivalent_proof", "Lax53Proofs.FixedSentenceModelChecking.exists_fixed_sentence_modelChecking_proof", "Lax53Proofs.IntrinsicUniformModelChecking.exists_uniform_msoModelChecking_proof", "Lax53Proofs.MSOLinearTime.modelCheckingInput_length_proof", "Lax53Proofs.MSOToTreeAutomata.mso_definable_is_recognizable_proof", "Lax53Proofs.MSOTreeAutomataEquivalence.recognizable_iff_msoDefinable_proof", "Lax53Proofs.StructuralRepresentations.formula_structural_proof", "Lax53Proofs.StructuralRepresentations.sentence_lawful_proof", "Lax53Proofs.StructuralRepresentations.tree_lawful_proof", "Lax53Proofs.StructuralRepresentations.tree_structural_proof", "Lax53Proofs.TreeAutomataToMSO.automaton_definable_by_mso_proof", "Lax53Proofs.TreeModelCheckingEncoding.automatonInput_length_proof", "Lax53Proofs.ValueTranslations.language_equivalence_proof" ] }, { "id": "lax-54", "state": "draft", "environment": "v4.30.0", "title": "Erdős–Hajnal for graphs with no 5-hole", "concepts": [ { "id": "Lax54.AveragingLemma", "title": "Sparse graph thinning lemma", "type": "theorem" }, { "id": "Lax54.BipartiteCombLemma", "title": "Bipartite comb lemma", "type": "theorem" }, { "id": "Lax54.CriticalCombInput", "title": "Quantitative critical-comb consequence", "type": "theorem" }, { "id": "Lax54.ErdosHajnalC5", "title": "Erdős–Hajnal theorem for the five-cycle", "type": "theorem" }, { "id": "Lax54.GraphDefinitions", "title": "Finite graph notions for the five-cycle Erdős–Hajnal theorem", "type": "definition" }, { "id": "Lax54.KeyCombLemma", "title": "Stable hubbed comb in a critical graph", "type": "theorem" }, { "id": "Lax54.MaximumDegreeReduction", "title": "Maximum-degree form of Rödl's theorem", "type": "theorem" }, { "id": "Lax54.RodlTheorem", "title": "Rödl's theorem for induced-subgraph-free graphs", "type": "theorem" } ], "proofs": [ "Lax54Proofs.bipartite_comb_lemma", "Lax54Proofs.CriticalCombInput.exists_critical_comb_parameters", "Lax54Proofs.erdos_hajnal_C5", "Lax54Proofs.key_comb_lemma", "Lax54Proofs.maximum_degree_reduction", "Lax54Proofs.RodlTheorem.rodl_theorem", "Lax54Proofs.sparse_graph_thinning" ] }, { "id": "lax-56", "state": "draft", "environment": "v4.30.0", "title": "Large Finite Point Sets Have 4 Collinear Points or a 6-Clique", "concepts": [ { "id": "Lax56.ConvexLayers", "title": "Convex layers and minimal polygons", "type": "definition" }, { "id": "Lax56.Geometry", "title": "Point visibility in the real plane", "type": "definition" }, { "id": "Lax56.HujterKisfaludiBak", "title": "The unoptimized empty-hexagon bound", "type": "definition" }, { "id": "Lax56.MainTheorem", "title": "Large finite point sets have four collinear points or a visible six-clique", "type": "theorem" }, { "id": "Lax56.ValtrFourLayer", "title": "Valtr's four-layer lemma", "type": "theorem" }, { "id": "Lax56.VertexRemovalStability", "title": "Quantitative vertex-removal Erdős--Simonovits stability for K₆", "type": "theorem" } ], "proofs": [ "Lax56Proofs.EmptyHexagon.exists_emptyConvexHexagon", "Lax56Proofs.MainTheorem.large_point_set_four_collinear_or_visible_six", "Lax56Proofs.ValtrFourLayer.exists_emptyHexagon_of_four_layers", "Lax56Proofs.VertexRemovalStability.exists_fiveColorable_delete" ] }, { "id": "lax-57", "state": "draft", "environment": "v4.30.0", "title": "Erdős–Hajnal for the five-vertex path", "concepts": [ { "id": "Lax57.AnticomponentBlockade", "title": "Anticomponent or complete blockade", "type": "theorem" }, { "id": "Lax57.BlockadeThinning", "title": "Simultaneous thinning of a semisparse blockade", "type": "theorem" }, { "id": "Lax57.ErdosHajnalP5", "title": "Erdős–Hajnal theorem for the five-vertex path", "type": "theorem" }, { "id": "Lax57.GraphDefinitions", "title": "Finite graph notions for the five-vertex path theorem", "type": "definition" }, { "id": "Lax57.HouseDichotomy", "title": "Restricted set or uniform blockade in a house-free graph", "type": "theorem" }, { "id": "Lax57.PreparedHouseBlockade", "title": "Prepared sparse-or-complete house blockades", "type": "theorem" }, { "id": "Lax57.SemisparseBlockade", "title": "Polynomial semisparse blockades for the house", "type": "theorem" }, { "id": "Lax57.SparseHouseAcceleration", "title": "Sparse-house acceleration", "type": "theorem" }, { "id": "Lax57.SparseHouseTools", "title": "Anticomplete pairs in sparse $P_5$-free graphs", "type": "theorem" }, { "id": "Lax57.SparseHouseTrichotomy", "title": "The sparse-house trichotomy", "type": "theorem" } ], "proofs": [ "Lax57Proofs.anticomponent_or_complete_blockade", "Lax57Proofs.erdos_hajnal_P5", "Lax57Proofs.house_dichotomy", "Lax57Proofs.prepared_house_blockade", "Lax57Proofs.semisparse_blockade_thinning", "Lax57Proofs.semisparse_house_blockade", "Lax57Proofs.sparse_house_acceleration", "Lax57Proofs.sparse_house_trichotomy", "Lax57Proofs.sparse_P5_anticomplete_pair" ] }, { "id": "lax-58", "state": "draft", "environment": "v4.30.0", "title": "Certified structural representations of finite data", "concepts": [ { "id": "Lax58.BitPolynomialTime", "title": "Polynomial time with polynomial word length", "type": "definition" }, { "id": "Lax58.CertifiedDerivation", "title": "Field-encoding agreement (infrastructure)", "type": "infrastructure" }, { "id": "Lax58.CertifiedDerivationElab", "title": "Certified encoding derivation (elaboration infrastructure)", "type": "elaboration infrastructure" }, { "id": "Lax58.FormulaExample", "title": "Example — structural encoding of propositional formulas", "type": "definition" }, { "id": "Lax58.RamComplexity", "title": "Time and word-capacity bounds on the word-RAM", "type": "definition" }, { "id": "Lax58.RamComplexityElab", "title": "Automatic RAM encoding selection (elaboration infrastructure)", "type": "elaboration infrastructure" }, { "id": "Lax58.RamComplexityExample", "title": "Examples — separate resource bounds and polynomial time", "type": "definition" }, { "id": "Lax58.RamPolynomialComparison", "title": "Two polynomial-time conventions, and their separation", "type": "definition and theorem" }, { "id": "Lax58.StructuralCombinators", "title": "Certified structural presentation combinators", "type": "definition and theorem" }, { "id": "Lax58.StructuralDerivation", "title": "Unrestricted structural folds (elaboration infrastructure)", "type": "elaboration infrastructure" }, { "id": "Lax58.StructuralPresentation", "title": "Structural presentations of finite data", "type": "definition" }, { "id": "Lax58.WordArena", "title": "Distinguished immutable word arenas", "type": "definition and theorem" } ], "proofs": [ "Lax58Proofs.RamExponentialExample.exponentialLength_bitPolynomialTime", "Lax58Proofs.RamPolynomialComparison.bitPolynomialTime_iff_ramPolytime", "Lax58Proofs.RamPolynomialSeparation.exponentialLength_not_polynomialTime", "Lax58Proofs.StructuralCombinators.constructor_eq_iff_proof", "Lax58Proofs.StructuralCombinators.structural_combinator_size_laws", "Lax58Proofs.StructuralCombinators.structural_combinators_lawful", "Lax58Proofs.WordArena.encode_fits_proof", "Lax58Proofs.WordArena.encode_memoryWords_proof", "Lax58Proofs.WordArena.encode_represents_proof", "Lax58Proofs.WordArena.encode_toInput_length_proof", "Lax58Proofs.WordArena.encode_totalWords_proof", "Lax58Proofs.WordArena.encodeRaw_dense_proof", "Lax58Proofs.WordArena.encodeRaw_fits_proof", "Lax58Proofs.WordArena.encodeRaw_memoryWords_proof", "Lax58Proofs.WordArena.encodeRaw_represents_proof", "Lax58Proofs.WordArena.encodeRaw_toInput_length_proof", "Lax58Proofs.WordArena.encodeRaw_totalWords_proof", "Lax58Proofs.WordArena.encodeRaw_wellFormed_proof" ] }, { "id": "lax-59", "state": "draft", "environment": "v4.30.0", "title": "Nagura's Theorem and the Interesting Numbers", "concepts": [ { "id": "Lax59.InterestingNumbers", "title": "Classification of Interesting Numbers", "type": "theorem" }, { "id": "Lax59.Nagura", "title": "Nagura's Prime Interval Theorem", "type": "theorem" } ], "proofs": [ "Lax59Proofs.InterestingNumbers.interesting_iff", "Lax59Proofs.nagura_bound" ] }, { "id": "lax-62", "state": "draft", "environment": "v4.30.0", "title": "A Refinement Framework for the Word RAM", "concepts": [], "proofs": [] }, { "id": "lax-67", "state": "draft", "environment": "v4.30.0", "title": "The Word RAM", "supersedes": "lax-13", "concepts": [ { "id": "Lax67.Ram", "title": "The word RAM", "type": "definition" }, { "id": "Lax67.RamComputes", "title": "Computing a function within a time bound", "type": "definition" } ], "proofs": [] }, { "id": "lax-68", "state": "draft", "environment": "v4.30.0", "title": "Planar Graph Classes", "concepts": [ { "id": "Lax68.GraphMinors", "title": "Graph minors", "type": "definition" }, { "id": "Lax68.GraphTopologicalMinors", "title": "Topological graph minors", "type": "definition" }, { "id": "Lax68.GridPlanar", "title": "Grids are planar", "type": "theorem" }, { "id": "Lax68.GridsAndWalls", "title": "Grids and walls", "type": "definition" }, { "id": "Lax68.HalinGraphs", "title": "Halin graphs", "type": "definition" }, { "id": "Lax68.HalinPlanar", "title": "Halin graphs are planar", "type": "theorem" }, { "id": "Lax68.KuratowskiPlanarity", "title": "Kuratowski's theorem in straight-line form", "type": "theorem" }, { "id": "Lax68.LadderGrid", "title": "Ladders are grids", "type": "theorem" }, { "id": "Lax68.LadderOuterplanar", "title": "Ladders are outerplanar", "type": "theorem" }, { "id": "Lax68.LadderPlanar", "title": "Ladders are planar", "type": "theorem" }, { "id": "Lax68.Ladders", "title": "Ladders", "type": "definition" }, { "id": "Lax68.LadderSeriesParallel", "title": "Ladders are series-parallel", "type": "theorem" }, { "id": "Lax68.MaximalOuterplanar", "title": "Maximal outerplanar graphs", "type": "definition" }, { "id": "Lax68.MaximalOuterplanarOuterplanar", "title": "Maximal outerplanar graphs are outerplanar", "type": "theorem" }, { "id": "Lax68.MaximalOuterplanarPlanar", "title": "Maximal outerplanar graphs are planar", "type": "theorem" }, { "id": "Lax68.Outerplanar", "title": "Outerplanar graphs", "type": "definition" }, { "id": "Lax68.OuterplanarPlanar", "title": "Outerplanar graphs are planar", "type": "theorem" }, { "id": "Lax68.PathOuterplanar", "title": "Paths are outerplanar", "type": "theorem" }, { "id": "Lax68.PathPlanar", "title": "Paths are planar", "type": "theorem" }, { "id": "Lax68.Paths", "title": "Paths", "type": "definition" }, { "id": "Lax68.PathTree", "title": "Paths are trees", "type": "theorem" }, { "id": "Lax68.Planar", "title": "Planar graphs", "type": "definition" }, { "id": "Lax68.PlanarExcludedMinors", "title": "Wagner's theorem", "type": "theorem" }, { "id": "Lax68.SeriesParallel", "title": "Series-parallel graphs", "type": "definition" }, { "id": "Lax68.SeriesParallelPlanar", "title": "Series-parallel graphs are planar", "type": "theorem" }, { "id": "Lax68.StarOuterplanar", "title": "Stars are outerplanar", "type": "theorem" }, { "id": "Lax68.StarPlanar", "title": "Stars are planar", "type": "theorem" }, { "id": "Lax68.Stars", "title": "Stars", "type": "definition" }, { "id": "Lax68.StarTree", "title": "Stars are trees", "type": "theorem" }, { "id": "Lax68.StraightLineDrawings", "title": "Straight-line graph drawings", "type": "definition" }, { "id": "Lax68.TreeOuterplanar", "title": "Trees are outerplanar", "type": "theorem" }, { "id": "Lax68.TreePlanar", "title": "Trees are planar", "type": "theorem" }, { "id": "Lax68.Trees", "title": "Trees", "type": "definition" }, { "id": "Lax68.TriangleMaximalOuterplanar", "title": "Triangles are maximal outerplanar", "type": "theorem" }, { "id": "Lax68.TriangleOuterplanar", "title": "Triangles are outerplanar", "type": "theorem" }, { "id": "Lax68.TrianglePlanar", "title": "Triangles are planar", "type": "theorem" }, { "id": "Lax68.Triangles", "title": "Triangles", "type": "definition" }, { "id": "Lax68.TriangulationPlanar", "title": "Triangulations of planar graphs are planar", "type": "theorem" }, { "id": "Lax68.Triangulations", "title": "Triangulations", "type": "definition" }, { "id": "Lax68.WagnerObstructionBridge", "title": "The Kuratowski-Wagner obstruction bridge", "type": "theorem" }, { "id": "Lax68.WallPlanar", "title": "Walls are planar", "type": "theorem" }, { "id": "Lax68.WheelHalin", "title": "Wheels are Halin graphs", "type": "theorem" }, { "id": "Lax68.WheelPlanar", "title": "Wheels are planar", "type": "theorem" }, { "id": "Lax68.Wheels", "title": "Wheels", "type": "definition" } ], "proofs": [ "Lax68Proofs.halin_planar", "Lax68Proofs.ladder_grid", "Lax68Proofs.ladder_planar", "Lax68Proofs.maximalOuterplanar_outerplanar", "Lax68Proofs.maximalOuterplanar_planar", "Lax68Proofs.outerplanar_planar", "Lax68Proofs.path_outerplanar", "Lax68Proofs.path_planar", "Lax68Proofs.planar_iff_excludedMinors", "Lax68Proofs.star_outerplanar", "Lax68Proofs.star_planar", "Lax68Proofs.tree_planar", "Lax68Proofs.triangle_outerplanar", "Lax68Proofs.triangle_planar", "Lax68Proofs.triangulationOf_planar", "Lax68Proofs.wheel_planar" ] }, { "id": "lax-132576", "state": "draft", "environment": "v4.30.0", "title": "Transducers, Part B: Rational Functions", "concepts": [ { "id": "Lax132576.BimachineOfRational", "title": "Rational functions are computed by bimachines", "type": "theorem" }, { "id": "Lax132576.Bimachines", "title": "Bimachines", "type": "definition" }, { "id": "Lax132576.EpsilonEliminationExtended", "title": "Elimination of ε-transitions, with extended transitions", "type": "theorem" }, { "id": "Lax132576.EpsilonEliminationFinite", "title": "Elimination of ε-transitions for finitely valued relations", "type": "theorem" }, { "id": "Lax132576.EpsilonFreeAutomata", "title": "Extended transitions and the elimination of ε-transitions", "type": "definition" }, { "id": "Lax132576.HomomorphismComplement", "title": "The complement of a homomorphism is rational", "type": "theorem" }, { "id": "Lax132576.LabelledAutomata", "title": "Automata with labelled transitions", "type": "definition" }, { "id": "Lax132576.LeftDistance", "title": "Left distance and bounded variation", "type": "definition" }, { "id": "Lax132576.LengthPreservingDecidable", "title": "Deciding length preservation of a rational function", "type": "theorem" }, { "id": "Lax132576.LengthPreservingNormalForm", "title": "Length preserving rational functions have length preserving automata", "type": "theorem" }, { "id": "Lax132576.LengthPreservingTyping", "title": "Length preservation through a typing of the states", "type": "theorem" }, { "id": "Lax132576.MealyDecidable", "title": "Deciding whether a rational function is a Mealy machine", "type": "theorem" }, { "id": "Lax132576.MealyMachineIndependent", "title": "Machine-independent characterisation of Mealy machines", "type": "theorem" }, { "id": "Lax132576.PrimeRationalFunctions", "title": "The prime rational functions", "type": "definition" }, { "id": "Lax132576.PrimesOfRational", "title": "Rational functions decompose into prime rational functions", "type": "theorem" }, { "id": "Lax132576.RationalComposition", "title": "Rational relations are closed under composition", "type": "theorem" }, { "id": "Lax132576.RationalContinuity", "title": "Rational relations are continuous", "type": "theorem" }, { "id": "Lax132576.RationalEquivalenceDecidable", "title": "Decidable equivalence of rational functions", "type": "theorem" }, { "id": "Lax132576.RationalEquivalenceUndecidable", "title": "Equivalence of rational relations is undecidable", "type": "theorem" }, { "id": "Lax132576.RationalFunctions", "title": "Rational functions", "type": "definition" }, { "id": "Lax132576.RationalMachineIndependent", "title": "Machine-independent characterisation of rational functions", "type": "theorem" }, { "id": "Lax132576.RationalMealyCharacterisation", "title": "Which rational functions are Mealy machines", "type": "theorem" }, { "id": "Lax132576.RationalOfBimachine", "title": "Bimachines compute rational functions", "type": "theorem" }, { "id": "Lax132576.RationalOfPrimes", "title": "Compositions of prime rational functions are rational", "type": "theorem" }, { "id": "Lax132576.RationalOfWeightedPrecomposition", "title": "Functions that weighted automata can be pre-composed with are rational", "type": "theorem" }, { "id": "Lax132576.RationalPrimes", "title": "The rational functions are the compositions of prime rational functions", "type": "theorem" }, { "id": "Lax132576.RationalRelations", "title": "Nondeterministic automata with output and rational relations", "type": "definition" }, { "id": "Lax132576.RationalUnambiguousBimachine", "title": "Eilenberg's theorem: rational functions, unambiguous automata and bimachines", "type": "theorem" }, { "id": "Lax132576.RationalViaWeighted", "title": "Rational functions characterised by weighted automata", "type": "theorem" }, { "id": "Lax132576.SequentialCharacterisation", "title": "Machine-independent characterisation of sequential functions", "type": "theorem" }, { "id": "Lax132576.SequentialTransducers", "title": "Sequential transducers", "type": "definition" }, { "id": "Lax132576.StringHomomorphisms", "title": "String homomorphisms", "type": "definition" }, { "id": "Lax132576.SubsequentialCharacterisation", "title": "Machine-independent characterisation of subsequential functions", "type": "theorem" }, { "id": "Lax132576.SubsequentialTransducers", "title": "Subsequential transducers", "type": "definition" }, { "id": "Lax132576.TransducerCodes", "title": "Codes of automata with output, and decidability under a promise", "type": "definition" }, { "id": "Lax132576.UnambiguousOfRational", "title": "Rational functions are computed by unambiguous automata", "type": "theorem" }, { "id": "Lax132576.Uniformisation", "title": "Uniformisation of total rational relations", "type": "theorem" }, { "id": "Lax132576.WeightedAutomata", "title": "Weighted automata", "type": "definition" }, { "id": "Lax132576.WeightedCodes", "title": "Codes of weighted automata over the rationals", "type": "definition" }, { "id": "Lax132576.WeightedEquivalenceDecidable", "title": "Decidable equivalence of weighted automata over the rationals", "type": "theorem" }, { "id": "Lax132576.WeightedPrecomposition", "title": "Weighted automata are closed under pre-composition with rational functions", "type": "theorem" }, { "id": "Lax132576.WeightedZeronessDecidable", "title": "Decidable zeroness of weighted automata over the rationals", "type": "theorem" } ], "proofs": [ "Lax132576Proofs.Results.compClosure_primeRational_of_isRationalFun", "Lax132576Proofs.Results.decidable_codeRel_eq", "Lax132576Proofs.Results.decidable_isMealy", "Lax132576Proofs.Results.decidable_lengthPreserving", "Lax132576Proofs.Results.decidable_wcodeEval_eq", "Lax132576Proofs.Results.decidable_wcodeEval_eq_zero", "Lax132576Proofs.Results.exists_extended_epsilonFree", "Lax132576Proofs.Results.exists_isUnambiguousRel_le", "Lax132576Proofs.Results.exists_nfao_epsilonFree", "Lax132576Proofs.Results.exists_nfao_length_eq", "Lax132576Proofs.Results.isBimachine_of_isRationalFun", "Lax132576Proofs.Results.isMealy_iff", "Lax132576Proofs.Results.isMealy_iff_of_isRationalFun", "Lax132576Proofs.Results.isRationalFun_iff", "Lax132576Proofs.Results.isRationalFun_iff_compClosure_primeRational", "Lax132576Proofs.Results.isRationalFun_iff_weighted_precomp", "Lax132576Proofs.Results.isRationalFun_of_compClosure_primeRational", "Lax132576Proofs.Results.isRationalFun_of_isBimachine", "Lax132576Proofs.Results.isRationalFun_of_weighted_precomp", "Lax132576Proofs.Results.isRationalRel_comp", "Lax132576Proofs.Results.isRationalRel_ne_homOf", "Lax132576Proofs.Results.isSequential_iff", "Lax132576Proofs.Results.isSubsequential_iff", "Lax132576Proofs.Results.isUnambiguousRel_of_isRationalFun", "Lax132576Proofs.Results.isWeighted_comp_of_isRationalFun", "Lax132576Proofs.Results.lengthPreserving_iff_typing", "Lax132576Proofs.Results.not_computablePred_codeRel_eq", "Lax132576Proofs.Results.relContinuous_of_isRationalRel", "Lax132576Proofs.Results.tfae_rational_unambiguous_bimachine" ] }, { "id": "lax-157538", "state": "draft", "environment": "v4.30.0", "title": "Transducers", "concepts": [], "proofs": [] }, { "id": "lax-194892", "state": "draft", "environment": "v4.30.0", "title": "Transducers, Part D: Polyregular Functions", "concepts": [ { "id": "Lax194892.BalancedRunReachability", "title": "Balanced runs between configurations are regular", "type": "theorem" }, { "id": "Lax194892.ChildConfigurationGraphs", "title": "Children of a configuration and child configuration graphs", "type": "definition" }, { "id": "Lax194892.ChildGraphOfConfiguration", "title": "A for-transducer produces the child configuration graph", "type": "theorem" }, { "id": "Lax194892.ChildrenOfChildGraph", "title": "A for-transducer reads the children off a child configuration graph", "type": "theorem" }, { "id": "Lax194892.ChildrenOfConfiguration", "title": "A for-transducer produces the children of a configuration", "type": "theorem" }, { "id": "Lax194892.ForComposition", "title": "For-transducers are closed under composition", "type": "theorem" }, { "id": "Lax194892.ForIffPolyregular", "title": "For-transducers compute exactly the polyregular functions", "type": "theorem" }, { "id": "Lax194892.ForOfPebble", "title": "Pebble transducers are computed by for-transducers", "type": "theorem" }, { "id": "Lax194892.ForOfPolyregular", "title": "Polyregular functions are computed by for-transducers", "type": "theorem" }, { "id": "Lax194892.ForTransducers", "title": "For-transducers", "type": "definition" }, { "id": "Lax194892.MarkedSquaring", "title": "Marked squaring", "type": "definition" }, { "id": "Lax194892.PebbleConfigurationEncoding", "title": "String representations of pebble configurations, and balanced runs", "type": "definition" }, { "id": "Lax194892.PebbleContinuity", "title": "Pebble transducers are continuous", "type": "theorem" }, { "id": "Lax194892.PebbleIffFor", "title": "Pebble transducers and for-transducers compute the same functions", "type": "theorem" }, { "id": "Lax194892.PebbleOfFor", "title": "For-transducers are computed by pebble transducers", "type": "theorem" }, { "id": "Lax194892.PebbleReachability", "title": "Reachability between configurations of a pebble transducer is regular", "type": "theorem" }, { "id": "Lax194892.PebbleTransducers", "title": "Pebble transducers", "type": "definition" }, { "id": "Lax194892.PolyregularContinuity", "title": "Polyregular functions are continuous", "type": "theorem" }, { "id": "Lax194892.PolyregularFunctions", "title": "Polyregular functions", "type": "definition" }, { "id": "Lax194892.PolyregularOfFor", "title": "For-transducers compute polyregular functions", "type": "theorem" }, { "id": "Lax194892.PrenexNormalForm", "title": "Every for-transducer has a prenex form", "type": "theorem" } ], "proofs": [ "Lax194892Proofs.Results.continuous_of_isPebbleTransducer", "Lax194892Proofs.Results.continuous_of_isPolyregular", "Lax194892Proofs.Results.exists_forTransducer_cgOut", "Lax194892Proofs.Results.exists_forTransducer_childGraph", "Lax194892Proofs.Results.exists_forTransducer_children", "Lax194892Proofs.Results.exists_prenexForm", "Lax194892Proofs.Results.exists_regular_balancedLang", "Lax194892Proofs.Results.exists_regular_reachLang", "Lax194892Proofs.Results.isForTransducer_comp", "Lax194892Proofs.Results.isForTransducer_of_isPebbleTransducer", "Lax194892Proofs.Results.isForTransducer_of_isPolyregular", "Lax194892Proofs.Results.isPebbleTransducer_iff_isForTransducer", "Lax194892Proofs.Results.isPebbleTransducer_of_isForTransducer", "Lax194892Proofs.Results.isPolyregular_iff_isForTransducer", "Lax194892Proofs.Results.isPolyregular_of_isForTransducer" ] }, { "id": "lax-195003", "state": "draft", "environment": "v4.30.0", "title": "Near-Linear Time Computation of Welzl Orders on Graphs with Linear Neighborhood Complexity", "concepts": [ { "id": "Lax195003.WelzlOrders", "title": "Welzl orders", "type": "definition" }, { "id": "Lax195003.WelzlOrdersComputation", "title": "Near-linear-time computation of Welzl orders", "type": "theorem" }, { "id": "Lax195003.WelzlOrdersInGraphs", "title": "Welzl orders in graphs", "type": "definition" }, { "id": "Lax195003.WelzlOrdersNeighborhoodComplexity", "title": "Neighborhood complexity", "type": "definition" }, { "id": "Lax195003.WelzlOrdersNeighborhoodSetSystem", "title": "Neighborhood set systems of graphs", "type": "definition" }, { "id": "Lax195003.WordRamRandomness", "title": "Randomized computation on the word RAM", "type": "definition" } ], "proofs": [] }, { "id": "lax-214022", "state": "draft", "environment": "v4.30.0", "title": "Welzl Orders of Cographs", "concepts": [ { "id": "Lax214022.Cographs", "title": "Cographs", "type": "definition" }, { "id": "Lax214022.CographWelzlLowerBound", "title": "Cographs can require logarithmic Welzl orders", "type": "theorem" }, { "id": "Lax214022.CographWelzlUpperBound", "title": "Cographs have logarithmic Welzl orders", "type": "theorem" } ], "proofs": [ "Lax214022Proofs.CographWelzlLowerBound.exists_cograph_requiring_crossingNumber_at_least", "Lax214022Proofs.CographWelzlUpperBound.exists_welzlOrder_crossingNumber_le_four_clog_add_one" ] }, { "id": "lax-242665", "state": "draft", "environment": "v4.30.0", "title": "An Introduction to Lax", "concepts": [ { "id": "Lax242665.BertrandPostulate", "title": "Bertrand's postulate", "type": "lemma" }, { "id": "Lax242665.InfinitelyManyPrimes", "title": "There are infinitely many primes", "type": "theorem" }, { "id": "Lax242665.OddPrimeBetween", "title": "An odd prime between n and 2n", "type": "theorem" }, { "id": "Lax242665.OddPrimes", "title": "Every prime other than 2 is odd", "type": "lemma" }, { "id": "Lax242665.Primes", "title": "Prime numbers", "type": "definition" } ], "proofs": [ "Lax242665Proofs.InfinitelyManyPrimes.exists_prime_gt", "Lax242665Proofs.OddPrimeBetween.exists_odd_prime_between", "Lax242665Proofs.OddPrimes.odd_of_prime" ] }, { "id": "lax-251941", "state": "draft", "environment": "v4.30.0", "title": "Undecidability of the Post Correspondence Problem", "concepts": [ { "id": "Lax251941.Acceptance", "title": "The acceptance problem for machines", "type": "definition" }, { "id": "Lax251941.AcceptanceUndecidable", "title": "The acceptance problem is undecidable", "type": "theorem" }, { "id": "Lax251941.PostCorrespondence", "title": "The Post correspondence problem", "type": "definition" }, { "id": "Lax251941.PostCorrespondenceIndexUndecidable", "title": "The Post correspondence problem in index form is undecidable", "type": "theorem" }, { "id": "Lax251941.PostCorrespondenceReduction", "title": "The acceptance problem reduces to the Post correspondence problem", "type": "theorem" }, { "id": "Lax251941.PostCorrespondenceUndecidable", "title": "The Post correspondence problem is undecidable", "type": "theorem" }, { "id": "Lax251941.TapeAcceptanceUndecidable", "title": "The acceptance problem for Turing machines is undecidable", "type": "theorem" }, { "id": "Lax251941.TuringCompleteness", "title": "Every partial recursive function is computed by a Turing machine", "type": "theorem" }, { "id": "Lax251941.TuringMachines", "title": "Single-tape Turing machines as string rewriting", "type": "definition" } ], "proofs": [ "Lax251941Proofs.Results.computablePred_accepts_of_hasMatch", "Lax251941Proofs.Results.exists_tm_accepts_iff_dom", "Lax251941Proofs.Results.not_computablePred_accepts", "Lax251941Proofs.Results.not_computablePred_hasMatch", "Lax251941Proofs.Results.not_computablePred_solvable", "Lax251941Proofs.Results.not_turingDecidable_ATM" ] }, { "id": "lax-307052", "state": "draft", "environment": "v4.30.0", "title": "Savitch's Theorem", "concepts": [ { "id": "Lax307052.Acceptance", "title": "Correctness of the configuration search", "type": "lemma" }, { "id": "Lax307052.BoundedConfigurations", "title": "Bounded machine configurations", "type": "definition" }, { "id": "Lax307052.ConfigCount", "title": "Number of bounded configurations", "type": "lemma" }, { "id": "Lax307052.ConfigurationGraph", "title": "The bounded configuration graph", "type": "definition" }, { "id": "Lax307052.FiniteSearch", "title": "Deciding finite reachability", "type": "lemma" }, { "id": "Lax307052.PolyBounds", "title": "Constructing polynomial space bounds", "type": "lemma" }, { "id": "Lax307052.PolynomialSpace", "title": "PSPACE equals NPSPACE", "type": "theorem" }, { "id": "Lax307052.Reachability", "title": "Bounded reachability", "type": "definition" }, { "id": "Lax307052.Recursion", "title": "Correctness of recursive reachability", "type": "lemma" }, { "id": "Lax307052.Runs", "title": "Runs and configuration reachability", "type": "lemma" }, { "id": "Lax307052.Savitch", "title": "Savitch's theorem", "type": "theorem" }, { "id": "Lax307052.SearchBounds", "title": "Quadratic space for the recursion stack", "type": "lemma" }, { "id": "Lax307052.SearchMachine", "title": "Space bound for the search machine", "type": "lemma" }, { "id": "Lax307052.ShortPaths", "title": "Short paths in a finite graph", "type": "lemma" }, { "id": "Lax307052.SpaceConstructibility", "title": "Space-constructible bounds", "type": "definition" }, { "id": "Lax307052.SplitWalk", "title": "Splitting a bounded walk", "type": "lemma" } ], "proofs": [ "Lax307052Proofs.configuration_acceptance", "Lax307052Proofs.configuration_count", "Lax307052Proofs.finite_reachability", "Lax307052Proofs.path_splitting", "Lax307052Proofs.polynomial_constructible", "Lax307052Proofs.polynomial_space", "Lax307052Proofs.quadratic_stack", "Lax307052Proofs.recursive_reachability", "Lax307052Proofs.savitch", "Lax307052Proofs.search_acceptance", "Lax307052Proofs.search_machine", "Lax307052Proofs.short_paths" ] }, { "id": "lax-314295", "state": "draft", "environment": "v4.30.0", "title": "Transducers, Part C: Regular Functions in Terms of Logic", "concepts": [ { "id": "Lax314295.AperiodicBimachineOfFORelabelling", "title": "First-order relabellings are computed by aperiodic bimachines", "type": "theorem" }, { "id": "Lax314295.AperiodicOfFO", "title": "First-order definable languages are aperiodic", "type": "theorem" }, { "id": "Lax314295.BuchiTheorem", "title": "Büchi's theorem: regular languages are the MSO-definable ones", "type": "theorem" }, { "id": "Lax314295.FOIffAperiodic", "title": "First-order definable languages are exactly the aperiodic ones", "type": "theorem" }, { "id": "Lax314295.FOOfAperiodic", "title": "Aperiodic automata recognise first-order definable languages", "type": "theorem" }, { "id": "Lax314295.FORelabellingIffAperiodicBimachine", "title": "First-order relabellings are exactly the aperiodic bimachines", "type": "theorem" }, { "id": "Lax314295.FORelabellingOfAperiodicBimachine", "title": "Aperiodic bimachines compute first-order relabellings", "type": "theorem" }, { "id": "Lax314295.KTypes", "title": "The k-type of a string", "type": "definition" }, { "id": "Lax314295.KTypesAperiodicity", "title": "k-types are aperiodic", "type": "theorem" }, { "id": "Lax314295.KTypesCongruence", "title": "k-types are a congruence for concatenation", "type": "theorem" }, { "id": "Lax314295.KTypesFOEquivalence", "title": "k-types capture first-order sentences of quantifier rank k", "type": "theorem" }, { "id": "Lax314295.KTypesRefinement", "title": "k-types refine each other", "type": "theorem" }, { "id": "Lax314295.LogicPrecomputation", "title": "Precomputing the answers of MSO formulas by a rational function", "type": "theorem" }, { "id": "Lax314295.MSOAnnotationRegular", "title": "The correctly annotated strings of an MSO relabelling form a regular language", "type": "theorem" }, { "id": "Lax314295.MSODefinableOfRegular", "title": "Regular languages are MSO-definable", "type": "theorem" }, { "id": "Lax314295.MSOFreeVariables", "title": "Formulas with free variables define regular languages of annotated strings", "type": "theorem" }, { "id": "Lax314295.MSOLogic", "title": "Monadic second-order logic on strings", "type": "definition" }, { "id": "Lax314295.MSORelabellings", "title": "MSO relabellings", "type": "definition" }, { "id": "Lax314295.MSOTransductionIffRegular", "title": "MSO transductions define exactly the regular functions", "type": "theorem" }, { "id": "Lax314295.MSOTransductionOfRegular", "title": "Regular functions are MSO transductions", "type": "theorem" }, { "id": "Lax314295.MSOTransductions", "title": "String-to-string MSO transductions", "type": "definition" }, { "id": "Lax314295.RationalIffRelabelling", "title": "Rational functions are exactly the MSO relabellings", "type": "theorem" }, { "id": "Lax314295.RationalOfRelabelling", "title": "MSO relabellings are rational", "type": "theorem" }, { "id": "Lax314295.RegularOfMSODefinable", "title": "MSO-definable languages are regular", "type": "theorem" }, { "id": "Lax314295.RegularOfMSOTransduction", "title": "MSO transductions define regular functions", "type": "theorem" }, { "id": "Lax314295.RelabellingOfRational", "title": "Rational functions are MSO relabellings", "type": "theorem" } ], "proofs": [ "Lax314295Proofs.Results.exists_aperiodic_dfa_of_foDefinable", "Lax314295Proofs.Results.exists_rational_precomputation", "Lax314295Proofs.Results.exists_tp_npow_eq", "Lax314295Proofs.Results.foDefinable_iff_aperiodic_dfa", "Lax314295Proofs.Results.foDefinable_of_aperiodic_dfa", "Lax314295Proofs.Results.isAperiodicBimachine_of_isFORelabelling", "Lax314295Proofs.Results.isFORelabelling_iff_isAperiodicBimachine", "Lax314295Proofs.Results.isFORelabelling_of_isAperiodicBimachine", "Lax314295Proofs.Results.isMSORelabelling_of_isRationalFun", "Lax314295Proofs.Results.isMSOTransduction_iff_isRegularFun", "Lax314295Proofs.Results.isMSOTransduction_of_isRegularFun", "Lax314295Proofs.Results.isRationalFun_iff_isMSORelabelling", "Lax314295Proofs.Results.isRationalFun_of_isMSORelabelling", "Lax314295Proofs.Results.isRegular_annotated", "Lax314295Proofs.Results.isRegular_annotation", "Lax314295Proofs.Results.isRegular_iff_msoDefinable", "Lax314295Proofs.Results.isRegular_of_msoDefinable", "Lax314295Proofs.Results.isRegularFun_of_isMSOTransduction", "Lax314295Proofs.Results.msoDefinable_of_isRegular", "Lax314295Proofs.Results.tp_append_congr", "Lax314295Proofs.Results.tp_eq_iff_fo_equiv", "Lax314295Proofs.Results.tp_eq_of_tp_succ_eq" ] }, { "id": "lax-434930", "state": "draft", "environment": "v4.30.0", "title": "Classical Complexity Classes", "concepts": [ { "id": "Lax434930.BasicProperties", "title": "Elementary properties of the complexity classes", "type": "theorem" }, { "id": "Lax434930.Certificates", "title": "Binary encoding of an input and a certificate", "type": "definition" }, { "id": "Lax434930.ComplementClasses", "title": "The complexity classes coNL and coNP", "type": "definition" }, { "id": "Lax434930.ExponentialTime", "title": "The complexity class EXPTIME", "type": "definition" }, { "id": "Lax434930.LogarithmicSpace", "title": "The complexity class L", "type": "definition" }, { "id": "Lax434930.NondeterministicLogarithmicSpace", "title": "The complexity class NL", "type": "definition" }, { "id": "Lax434930.NondeterministicPolynomialSpace", "title": "The complexity class NPSPACE", "type": "definition" }, { "id": "Lax434930.NondeterministicPolynomialTime", "title": "The complexity class NP", "type": "definition" }, { "id": "Lax434930.PolynomialSpace", "title": "The complexity class PSPACE", "type": "definition" }, { "id": "Lax434930.PolynomialTime", "title": "The complexity class P", "type": "definition" }, { "id": "Lax434930.SpaceBounds", "title": "Deterministic and nondeterministic space bounds", "type": "definition" }, { "id": "Lax434930.SpaceMachines", "title": "Finite Turing machines with a read-only input tape", "type": "definition" } ], "proofs": [ "Lax434930Proofs.BasicProperties.co_co", "Lax434930Proofs.BasicProperties.L_subset_NL", "Lax434930Proofs.BasicProperties.L_subset_PSPACE", "Lax434930Proofs.BasicProperties.mem_coNP_iff", "Lax434930Proofs.BasicProperties.NL_subset_NPSPACE", "Lax434930Proofs.BasicProperties.P_subset_EXPTIME", "Lax434930Proofs.BasicProperties.PSPACE_subset_NPSPACE", "Lax434930Proofs.Certificates.pair_injective", "Lax434930Proofs.Certificates.pair_length", "Lax434930Proofs.Certificates.unpair_pair" ] }, { "id": "lax-485480", "state": "draft", "environment": "v4.30.0", "title": "Welzl Trees of Cographs", "concepts": [ { "id": "Lax485480.CographWelzlTreeLowerBound", "title": "Cographs can require logarithmic Welzl trees", "type": "theorem" }, { "id": "Lax485480.CographWelzlTreeUpperBound", "title": "Cographs have logarithmic Welzl trees", "type": "theorem" }, { "id": "Lax485480.WelzlTrees", "title": "Welzl trees", "type": "definition" } ], "proofs": [ "Lax485480Proofs.CographWelzlTreeLowerBound.exists_cograph_requiring_treeCrossingNumber_at_least", "Lax485480Proofs.CographWelzlTreeUpperBound.exists_welzlTree_crossingNumber_le_four_clog_add_one" ] }, { "id": "lax-489179", "state": "draft", "environment": "v4.30.0", "title": "ETH, SETH, Weighted APSP, and 3SUM", "concepts": [ { "id": "Lax489179.Algorithms", "title": "Deterministic and bounded-error algorithms", "type": "definition" }, { "id": "Lax489179.APSPHypothesis", "title": "The weighted APSP complexity assumption", "type": "definition" }, { "id": "Lax489179.DeterministicSemantics", "title": "Exact correctness of deterministic computations", "type": "lemma" }, { "id": "Lax489179.DistanceProperties", "title": "Uniqueness and diagonal entries of shortest-path distances", "type": "lemma" }, { "id": "Lax489179.EncodingProperties", "title": "Integer encodings and distinct 3-SUM entries", "type": "lemma" }, { "id": "Lax489179.ETH", "title": "The Exponential Time Hypothesis (ETH)", "type": "definition" }, { "id": "Lax489179.HypothesisProperties", "title": "Negations and algorithm conventions for the four hypotheses", "type": "lemma" }, { "id": "Lax489179.IntegerEncoding", "title": "Encoding signed integers and missing distances", "type": "definition" }, { "id": "Lax489179.Satisfiability", "title": "Bounded-width CNF satisfiability and its encoding", "type": "definition" }, { "id": "Lax489179.SATTime", "title": "Exponential running time for bounded-width SAT", "type": "definition" }, { "id": "Lax489179.SETH", "title": "The Strong Exponential Time Hypothesis (SETH)", "type": "definition" }, { "id": "Lax489179.ThreeSUM", "title": "The integer 3-SUM problem", "type": "definition" }, { "id": "Lax489179.ThreeSUMHypothesis", "title": "The 3-SUM Hypothesis", "type": "definition" }, { "id": "Lax489179.TimeProperties", "title": "Monotonicity of running-time bounds", "type": "lemma" }, { "id": "Lax489179.TuringMachine", "title": "Finite multitape Turing machines", "type": "definition" }, { "id": "Lax489179.WeightedAPSP", "title": "Weighted all-pairs shortest paths", "type": "definition" }, { "id": "Lax489179.WordPrograms", "title": "Word-RAM programs with optional fair coins", "type": "definition" }, { "id": "Lax489179.WordTime", "title": "Polynomial running time on logarithmic words", "type": "definition" } ], "proofs": [ "Lax489179Proofs.decode_encode_int", "Lax489179Proofs.decode_encode_option", "Lax489179Proofs.diagonal_zero", "Lax489179Proofs.distance_unique", "Lax489179Proofs.encode_int_injective", "Lax489179Proofs.encode_option_injective", "Lax489179Proofs.matrix_length", "Lax489179Proofs.not_apsp", "Lax489179Proofs.not_eth", "Lax489179Proofs.not_seth", "Lax489179Proofs.not_three_sum", "Lax489179Proofs.randomized_apsp", "Lax489179Proofs.randomized_eth", "Lax489179Proofs.randomized_seth", "Lax489179Proofs.randomized_three_sum", "Lax489179Proofs.sat_exponent_mono", "Lax489179Proofs.sat_randomized", "Lax489179Proofs.sat_width_mono", "Lax489179Proofs.three_entries", "Lax489179Proofs.turing_exact", "Lax489179Proofs.word_exact", "Lax489179Proofs.word_exponent_mono", "Lax489179Proofs.word_randomized" ] }, { "id": "lax-554803", "state": "draft", "environment": "v4.30.0", "title": "Polynomial Time and Closure under Complement", "concepts": [ { "id": "Lax554803.ComplementClosure", "title": "P is closed under complement", "type": "theorem" }, { "id": "Lax554803.FiniteStackEquivalence", "title": "Finite stack alphabets suffice", "type": "theorem" }, { "id": "Lax554803.MachineModels", "title": "Finite-stack and elementary single-tape definitions of P", "type": "definition" }, { "id": "Lax554803.ModelEquivalence", "title": "Single-tape characterization of P", "type": "theorem" }, { "id": "Lax554803.PolynomialTime", "title": "The complexity class P", "type": "definition" }, { "id": "Lax554803.SingleTapeComplement", "title": "Single-tape P is closed under complement", "type": "theorem" } ], "proofs": [ "Lax554803Proofs.closed_under_complement", "Lax554803Proofs.ModelEquivalence.finiteStackP_eq_P", "Lax554803Proofs.ModelEquivalence.singleTape_closed_under_complement", "Lax554803Proofs.ModelEquivalence.singleTapeP_eq_P" ] }, { "id": "lax-678846", "state": "draft", "environment": "v4.30.0", "title": "Fagin’s theorem", "concepts": [ { "id": "Lax678846.ExistentialSecondOrder", "title": "Existential second-order logic", "type": "definition and theorem" }, { "id": "Lax678846.Fagin", "title": "Fagin’s theorem", "type": "theorem" }, { "id": "Lax678846.FiniteStructures", "title": "Finite relational structures and their properties", "type": "definition" }, { "id": "Lax678846.NondeterministicPolynomialTime", "title": "NP through polynomially bounded certificates", "type": "definition" } ], "proofs": [ "Lax678846Proofs.Fagin.capturesNP", "Lax678846Proofs.Fagin.definableInNP", "Lax678846Proofs.Fagin.npDefinable", "Lax678846Proofs.Isomorphism.satisfiesInvariant" ] }, { "id": "lax-709149", "state": "draft", "environment": "v4.30.0", "title": "Transducers, Part C: Regular Functions in Terms of Combinators", "concepts": [ { "id": "Lax709149.RegularOfTerm", "title": "Regular terms define regular functions", "type": "theorem" }, { "id": "Lax709149.RegularTerms", "title": "Regular terms", "type": "definition" }, { "id": "Lax709149.RegularUnderRepresentation", "title": "Regular functions on types under string representation", "type": "definition" }, { "id": "Lax709149.Types", "title": "Types and their string representation", "type": "definition" } ], "proofs": [ "Lax709149Proofs.Results.isRegularUnderRepr_of_isRegularTermFun" ] }, { "id": "lax-765601", "state": "draft", "environment": "v4.30.0", "title": "Transducers, Part A: Mealy Machines", "concepts": [ { "id": "Lax765601.Aperiodicity", "title": "Aperiodic string-to-string functions", "type": "definition" }, { "id": "Lax765601.AperiodicityMinimalMachine", "title": "Aperiodicity through the state transformations of the minimal machine", "type": "theorem" }, { "id": "Lax765601.AperiodicMealy", "title": "Aperiodic Mealy machines are exactly the compositions of flip-flops", "type": "theorem" }, { "id": "Lax765601.AperiodicOfFlipFlops", "title": "Compositions of flip-flop machines are aperiodic", "type": "theorem" }, { "id": "Lax765601.AperiodicPumping", "title": "Aperiodicity as a pumping property", "type": "theorem" }, { "id": "Lax765601.CompositionClosure", "title": "Closure of a family of functions under composition", "type": "definition" }, { "id": "Lax765601.Continuity", "title": "Continuous string-to-string functions", "type": "definition" }, { "id": "Lax765601.Derivatives", "title": "Derivatives of a string-to-string function", "type": "definition" }, { "id": "Lax765601.ElementaryProperties", "title": "Prefix preservation and length preservation", "type": "definition" }, { "id": "Lax765601.FlipFlopsOfAperiodic", "title": "Aperiodic Mealy machines are compositions of flip-flops", "type": "theorem" }, { "id": "Lax765601.KrohnRhodes", "title": "The Krohn–Rhodes decomposition theorem", "type": "theorem" }, { "id": "Lax765601.MapLifting", "title": "Map lifting", "type": "definition" }, { "id": "Lax765601.MapLiftingDecomposition", "title": "Map lifting preserves decompositions into primes", "type": "theorem" }, { "id": "Lax765601.MealyComposition", "title": "Mealy machines are closed under composition", "type": "theorem" }, { "id": "Lax765601.MealyContinuity", "title": "Mealy machines are continuous", "type": "theorem" }, { "id": "Lax765601.MealyDerivatives", "title": "Myhill–Nerode for Mealy machines", "type": "theorem" }, { "id": "Lax765601.MealyEquivalenceBound", "title": "Decidable equivalence of Mealy machines", "type": "theorem" }, { "id": "Lax765601.MealyMachine", "title": "Mealy machines", "type": "definition" }, { "id": "Lax765601.PrimeMealyMachines", "title": "Prime Mealy machines: reversible and flip-flop", "type": "definition" }, { "id": "Lax765601.ReversibleComposition", "title": "Reversible Mealy machines are closed under composition", "type": "theorem" }, { "id": "Lax765601.StateTransformationDecomposition", "title": "The state transformation transducer is a composition of primes", "type": "theorem" }, { "id": "Lax765601.StateTransformations", "title": "State transformations of a pre-automaton", "type": "definition" } ], "proofs": [ "Lax765601Proofs.Results.aperiodic_iff_compClosure_flipFlop", "Lax765601Proofs.Results.aperiodic_iff_pumping", "Lax765601Proofs.Results.aperiodic_iff_transAperiodic", "Lax765601Proofs.Results.aperiodic_of_compClosure_flipFlop", "Lax765601Proofs.Results.compClosure_flipFlop_of_aperiodic", "Lax765601Proofs.Results.compClosure_mapLift", "Lax765601Proofs.Results.compClosure_primeMealy_of_isMealy", "Lax765601Proofs.Results.compClosure_stateTransTransducer", "Lax765601Proofs.Results.continuous_of_isMealy", "Lax765601Proofs.Results.eval_eq_iff_short", "Lax765601Proofs.Results.isMealy_comp", "Lax765601Proofs.Results.isMealy_iff_derivatives", "Lax765601Proofs.Results.isReversibleMealy_comp" ] }, { "id": "lax-916827", "state": "draft", "environment": "v4.30.0", "title": "Transducers, Part C: Regular Functions, Two-Way Transducers and Streaming String Transducers", "concepts": [ { "id": "Lax916827.ConfigurationGraphOutput", "title": "Checking the output of a configuration graph against a regular language", "type": "theorem" }, { "id": "Lax916827.ConfigurationGraphRational", "title": "Computing the reachable configuration graph is rational", "type": "theorem" }, { "id": "Lax916827.ConfigurationGraphs", "title": "The string representation of the reachable configuration graph", "type": "definition" }, { "id": "Lax916827.DuplicationContinuous", "title": "String duplication is continuous", "type": "theorem" }, { "id": "Lax916827.MapLiftingContinuity", "title": "The map lifting of a continuous function is continuous", "type": "theorem" }, { "id": "Lax916827.RegularComposition", "title": "Regular functions are closed under composition", "type": "theorem" }, { "id": "Lax916827.RegularConcatenation", "title": "Regular functions are closed under concatenation", "type": "theorem" }, { "id": "Lax916827.RegularConditional", "title": "Regular functions are closed under conditionals over regular languages", "type": "theorem" }, { "id": "Lax916827.RegularContinuity", "title": "Regular functions are continuous", "type": "theorem" }, { "id": "Lax916827.RegularEquivalenceDecidable", "title": "Decidable equivalence of regular functions", "type": "theorem" }, { "id": "Lax916827.RegularFunctions", "title": "Regular functions", "type": "definition" }, { "id": "Lax916827.RegularMapLifting", "title": "Regular functions are closed under map lifting", "type": "theorem" }, { "id": "Lax916827.RegularOfSST", "title": "Every streaming string transducer computes a regular function", "type": "theorem" }, { "id": "Lax916827.RegularOfTwoWay", "title": "Every two-way transducer computes a regular function", "type": "theorem" }, { "id": "Lax916827.RegularSum", "title": "The sum of two regular functions on disjoint alphabets", "type": "theorem" }, { "id": "Lax916827.ReversalContinuous", "title": "String reversal is continuous", "type": "theorem" }, { "id": "Lax916827.SnakeGraphs", "title": "Snake graphs and their outputs", "type": "definition" }, { "id": "Lax916827.SnakeLemma", "title": "The snake lemma: the output of a snake graph is a regular function", "type": "theorem" }, { "id": "Lax916827.SSTIffRegular", "title": "Streaming string transducers compute exactly the regular functions", "type": "theorem" }, { "id": "Lax916827.SSTOfRegular", "title": "Every regular function is computed by a streaming string transducer", "type": "theorem" }, { "id": "Lax916827.StreamingStringTransducers", "title": "Streaming string transducers", "type": "definition" }, { "id": "Lax916827.TwoWayCodes", "title": "Codes of two-way transducers", "type": "definition" }, { "id": "Lax916827.TwoWayComposition", "title": "Two-way transducers are closed under composition", "type": "theorem" }, { "id": "Lax916827.TwoWayContinuity", "title": "Two-way transducers are continuous", "type": "theorem" }, { "id": "Lax916827.TwoWayIffRegular", "title": "Two-way transducers compute exactly the regular functions", "type": "theorem" }, { "id": "Lax916827.TwoWayMealyPrecomposition", "title": "Two-way transducers are closed under pre-composition with Mealy machines", "type": "theorem" }, { "id": "Lax916827.TwoWayOfRegular", "title": "Every regular function is computed by a two-way transducer", "type": "theorem" }, { "id": "Lax916827.TwoWayRationalPrecomposition", "title": "Two-way transducers are closed under pre-composition with rational functions", "type": "theorem" }, { "id": "Lax916827.TwoWayTransducers", "title": "Two-way transducers", "type": "definition" } ], "proofs": [ "Lax916827Proofs.Results.continuous_duplicate", "Lax916827Proofs.Results.continuous_mapLift", "Lax916827Proofs.Results.continuous_of_isRegularFun", "Lax916827Proofs.Results.continuous_of_isTwoWay", "Lax916827Proofs.Results.continuous_reverse", "Lax916827Proofs.Results.decidable_twoWayCodeRel_eq", "Lax916827Proofs.Results.exists_isRegularFun_sum", "Lax916827Proofs.Results.isRationalFun_enc", "Lax916827Proofs.Results.isRegular_encOutputLang", "Lax916827Proofs.Results.isRegularFun_comp", "Lax916827Proofs.Results.isRegularFun_concat", "Lax916827Proofs.Results.isRegularFun_ite", "Lax916827Proofs.Results.isRegularFun_mapLift", "Lax916827Proofs.Results.isRegularFun_of_isSST", "Lax916827Proofs.Results.isRegularFun_of_isTwoWay", "Lax916827Proofs.Results.isRegularFun_snakeOut", "Lax916827Proofs.Results.isSST_iff_isRegularFun", "Lax916827Proofs.Results.isSST_of_isRegularFun", "Lax916827Proofs.Results.isTwoWay_comp", "Lax916827Proofs.Results.isTwoWay_comp_isMealy", "Lax916827Proofs.Results.isTwoWay_comp_isRationalFun", "Lax916827Proofs.Results.isTwoWay_iff_isRegularFun", "Lax916827Proofs.Results.isTwoWay_of_isRegularFun" ] }, { "id": "lax-979537", "state": "draft", "environment": "v4.30.0", "title": "The Immerman–Vardi theorem", "concepts": [ { "id": "Lax979537.FixedPointEvaluation", "title": "Polynomial-time evaluation of fixed-point queries", "type": "theorem" }, { "id": "Lax979537.FixedPointSemantics", "title": "Semantics and definability in FO(LFP)", "type": "definition and theorem" }, { "id": "Lax979537.FixedPointSyntax", "title": "First-order logic with least fixed points", "type": "definition" }, { "id": "Lax979537.ImmermanVardi", "title": "The Immerman–Vardi theorem", "type": "theorem" }, { "id": "Lax979537.LeastFixedPoints", "title": "Least fixed points on finite relations", "type": "definition and theorem" }, { "id": "Lax979537.OrderedStructures", "title": "Finite ordered relational structures and queries", "type": "definition" }, { "id": "Lax979537.PolynomialTime", "title": "Polynomial-time queries on ordered structures", "type": "definition" }, { "id": "Lax979537.StructureEncoding", "title": "Binary encodings of ordered structures and tuples", "type": "definition and theorem" } ], "proofs": [ "Lax979537Proofs.FixedPointEvaluation.evaluationInP", "Lax979537Proofs.ImmermanVardi.capturesPtime", "Lax979537Proofs.ImmermanVardi.ptimeDefinable", "Lax979537Proofs.LeastFixedPoints.finiteConvergence", "Lax979537Proofs.LeastFixedPoints.fixedPoint", "Lax979537Proofs.LeastFixedPoints.least", "Lax979537Proofs.Positivity.positiveBodyMonotone", "Lax979537Proofs.StructureEncoding.encodeInjective", "Lax979537Proofs.StructureEncoding.encodeLength" ] } ] }