RLVE: Scaling Up Reinforcement Learning for Language Models with Adaptive Verifiable Environments — part 5/5
Cleaned-text code points [60527, 71633). Bibliography omitted here; source.md includes it.
(8) BAJBytecomputer , (10) BannedPointSupersetPathCounting , (11) BanyanHeart , (30) CRT , (33) Canon , (42) CirculatingDecimalCounting , (46) Clique_IndependentSet_Partitioning_Counting , (53) ConvexHull , (55) CountdownClose , (57) CowDanceShow , (79) DoubleStackSorting , (80) DynDynamite , (85) EvenDegreeGraphPartitioning , (86) Expression_AddingParenthese_Counting , (89) FactorialTrailingZeroCount , (94) FixedModK_Selection_Counting , (98) FutoshikiPuzzle , (106) GraMinimaGame , (115) GridTriangleCounting , (117) HalvingChainCounting , (123) ImpParty , (125) IntegerFactorizationCounting , (131) JugPuzzle , (151) LandAcquisition , (152) LandformGenerationCounting , (165) MakingGrade , (167) MatrixPermutation_BothDiagonalOne , (171) Matrix_BinaryExponentiation , (187) MaxXorSet , (197) Maximum_IndependentSet_Tree , (226) MinimumSpanningTree , (234) Minimum_DominatingInterval , (245) MostComponentTreeRemovingTwoPaths , (253) NextPalindromic , (266) PalindromePartitionCounting , (271) PipelineArrangement , (274) PolynomialInterpolation , (280) PrefixConcatenation , (287) PythagoreanGraph_IndependentSetCounting , (315) SetSplitting , (323) SkyscraperPuzzle , (326) SmallestBinaryMultiple , (337) StoneIntervalsGame , (343) SubgraphIsomorphism , (344) SubmatrixSumDivisibleCounting , (347) SubsetSumSequence , (355) SumProductDivisorNum , (360) SumXorDivisorNum , (361) Sum_DivisorNum , (369) TopologicalSort_MinimalLexicographicalOrder .
For each held-out test environment, we randomly generate 50 distinct problems, with the difficulty level d d evenly distributed within the range [ 0 , 4 ] [0,4] . This results in a total of 2,500 problems for the 50 held-out environments.
We sample one model output per problem for all evaluations on the in-distribution (ID) test set in 4.1 and on 𝒟 ood \mathcal{D}_{\text{ood}} .
## Appendix E Details of Training Environment Collection
As described in Section 4 , we construct four collections of training environments, denoted as 𝒞 1 \mathcal{C}_{1} , 𝒞 4 \mathcal{C}_{4} , 𝒞 16 \mathcal{C}_{16} , and 𝒞 256 \mathcal{C}_{256} , and each larger collection strictly contains all smaller ones, i.e., 𝒞 1 ⊂ 𝒞 4 ⊂ 𝒞 16 ⊂ 𝒞 256 \mathcal{C}_{1}\subset\mathcal{C}_{4}\subset\mathcal{C}_{16}\subset\mathcal{C}_{256} . All four collections are from the full suite of 400 verifiable environments introduced in Appendix B.1 , excluding the 50 held-out test environments.
For clarity, we list below the specific composition of each collection and its incremental difference from the preceding one.
𝒞 1 \mathcal{C}_{1} contains only a single environment (249) Multiplication .
𝒞 4 \mathcal{C}_{4} expands 𝒞 1 \mathcal{C}_{1} by including 3 additional environments, for a total of 4. The incremental environments 𝒞 4 − 𝒞 1 \mathcal{C}_{4}-\mathcal{C}_{1} are: (75) Division , (84) EuclidGame , (328) Sorting .
𝒞 16 \mathcal{C}_{16} further expands 𝒞 4 \mathcal{C}_{4} by adding 12 more environments, for a total of 16. The incremental environments 𝒞 16 − 𝒞 4 \mathcal{C}_{16}-\mathcal{C}_{4} are: (100) GCDOne_Counting , (118) HamiltonianPath , (150) LampChanging , (153) LargestConvexPolygon , (262) PCPPermutation , (269) Path_NoGoingBack_Counting , (303) SAT , (317) ShortestPath , (329) SpiralMatrix , (345) SubsequenceReversalLNDS , (387) UndamagedSubmatrixCounting , (394) WYRLevelingGround .
𝒞 256 \mathcal{C}_{256} extends 𝒞 16 \mathcal{C}_{16} by adding 240 additional environments, for a total of 256. The incremental environments 𝒞 256 − 𝒞 1 \mathcal{C}_{256}-\mathcal{C}_{1} are: (1) ABProgramSimulation , (2) AddMultiple_Divisible_Counting , (3) AdditionTable , (4) AlmostCompleteGraphCycleCounting , (6) AntiPalindromicSubstringCounting , (9) BEZMinimalistSecurity , (13) Binario , (14) Binario_NoAdjacencyRequirement , (15) BinaryAlternation , (16) BinaryLinearEquation_SolutionCounting , (17) BinaryTreeLeafNumExpectation , (18) BitAndZero_PathCounting , (19) BitEquationCounting , (20) BitwiseOperationSequenceCounting , (21) BlockImage , (23) BoundedIntervalIntersection , (24) BoundedMeanSubarrayCounting , (25) BoundedSubarrayCounting , (26) BoxScheduling , (27) Bridge , (31) CampfireParty , (35) CapitalCityEffect , (36) CardColoringCounting , (38) CheckAllCycleXorZero , (40) Cinema , (41) Circuit , (43) CirculatingGrid , (44) CleaningUp , (47) CoinSquareGame , (48) ColoringCounting , (50) ConcatenationPartitionCountingSum , (51) CongruentEquation , (54) Cornfield , (56) CountdownEqual , (58) Cryptarithmetic , (59) Cube_FixedLocalMaximumCounting , (60) CycleCounting , (61) DecreasingDigitCounting , (62) DegreeFixed_SpanningTree , (63) DeltaMinPopcount , (64) DeltaNimGame , (66) DifferenceConstraintSystem , (69) Differentiate , (71) DiscreteLogarithm , (72) Disinfection , (76) DivisorFlipExpectation , (77) DoubleCrossCounting , (78) DoublePalindromicStringCounting , (81) EightDigitPuzzle , (87) FBI_BinaryTree , (88) FaceRightWay , (90) Fibonacci , (92) Fibtrain , (95) FixedOneEdgeNum_SpanningTree , (96) FractionalProgramming , (97) FractionalProgramming_BipartiteGraphMatching , (99) GCDFibonacciProduct , (102) GasFireExtinguishers , (104) GcdLcmCounting , (107) GradeRankingCounting , (108) GraphContainTreeCounting , (109) GraphIsomorphism , (110) GridBFS , (111) GridColoringCounting , (112) GridComponent , (113) GridLocalMinimumCounting , (116) HURWarehouseStore , (119) HamiltonianPathExistence , (121) HitoriPuzzle , (122) HungryRabbit , (124) IndividualSumBounded_SequenceCounting , (126) IntegerProgramming , (127) Integral , (128) InversionPair , (129) InversionPairK_Counting , (130) Josephus , (132) KPartition , (133) KUR , (136) KingSorting , (137) KloBlocks , (138) Knapsack , (140) KosDicing , (141) KthSubsequence , (142) Kth_BinaryTree , (143) Kth_SemiBalancedBracketSequence , (144) LAS , (145) LASLaser , (146) LCM , (147) LDSTwo_Counting , (149) LIZ_Lollipop , (154) LargestRectangle_AmongPoints , (155) LightUpPuzzle , (156) LinkBeads , (157) LongestMaxDiffBoundedInterval , (159) Longest_DoublePalindrome , (160) Longest_MatchingSubsequence , (161) Longest_RepeatedPalindrome , (164) MagicSquarePuzzle , (166) MatrixPermutationEquivalence , (169) MatrixPooling , (170) MatrixRMQCounting , (172) MaxDifferentGroupPairDivision , (173) MaxGridPathIntersection , (175) MaxMultSplit , (178) MaxPermutation , (179) MaxRMQExpectation , (180) MaxSegmentCoverageConstraint , (181) MaxSumLDS , (182) MaxThreeSquareSum , (185) MaxWeightPalindromicSubstring , (186) MaxXorPath , (188) Max_NonAdjacent_KElementSum , (193) MaximumIndependentSetGrid , (194) MaximumLexicographicalOrderSubsequence , (195) MaximumPointSegmentMatching , (196) MaximumWeightMatching , (198) Maximum_SubsequenceNum , (200) MinConversionToCycleCost , (201) MinCostReducingLNDS , (202) MinCostTreeCoverage , (203) MinCubeAssignment , (204) MinDivisionSumXor , (206) MinKDivisorNumber , (207) MinNoSolutionLinearDiophantineEquation , (208) MinNonsubstring , (209) MinPairSumMultiplicationPermutation , (212) MinSumDistanceSquare , (214) MinSwapTwoPermutations , (215) MinXorPair , (216) Minesweeping , (218) MinimumChromaticNumber , (219) MinimumChromaticNumber_SegmentOverlap , (221) MinimumDirectedSpanningTree , (222) MinimumFibonacciRepresentation , (223) MinimumHarmoniousChromaticNumber , (225) MinimumRatioPath , (227) MinimumSpanningTreeCounting , (229) MinimumSumDifferenceSubmatrix , (230) MinimumTreeWeightedDominatingAncestor , (232) MinimumWeightedSpanningTree , (233) Minimum_CrossingEdges_GraphPartition , (235) Minimum_DominatingSet , (237) Minimum_MaxAbsSlicer , (238) Minimum_MaxSlicer , (240) MitterTransportation , (241) MixedGraphEulerianCircuit , (243) MonochromeBlockCounting , (247) MultiDrink , (248) MultipleFlippingGame , (251) NegativeBase , (255) NoAdjacentGirlCounting , (256) NoDoubleTripleCounting , (258) NumberPartitionCounting , (259) Numbrix , (260) ODLDistance , (263) POLPolarization , (264) PairMoreOneCounting , (265) PalembangBridges , (267) PalindromicSubstringNumberCounting , (268) PanSolarPanels , (270) Patrol , (273) PolynomialFactorization , (276) PolynomialRemainder , (277) PowerCycle , (278) PowerNest , (279) PowerShortcut , (281) PrefixProductMODDistinctPermutation , (282) PrefixSumMODDistinctPermutation , (283) Prefixuffix , (284) PreorderTraversal , (285) PrimeGraph_MinimumChromaticNumber , (290) QuantumLockPuzzle , (291) QueenPlacement , (292) RandomRangeMaxExpectation , (293) RangeConstrained_IncreasingSequence_Counting , (294) RangeFourSequenceConstruction , (299) RootExtraction , (300) RoundRobin , (301) RoundTableAssignment , (302) RoyalLockCounting , (304) SCC_Sequence_Counting , (307) SaladBar , (308) SalesmanFatigue , (310) SecretCowCode , (311) SegmentMinLengthEqual_Counting , (312) SegmentTreeSortingCounting , (313) SelfPowerSequenceMOD , (316) SharedSubstringCounting , (319) ShortestUnicolorSubstring , (321) SingleStackSorting , (324) SkyscraperSumPuzzle , (325) SlidingWindow , (327) SmallestCircle , (330) SplittingGame , (331) SpyNetwork , (332) SquSquarks , (334) StarBattle , (335) StirlingSecond , (336) StoneGame , (338) StringPartitionShuffle , (346) SubsetSum , (348) Sudoku , (350) SumGCDWithIndividual , (351) SumLCM , (352) SumMOD , (353) SumManhattan_CurvedSurface , (354) SumPHIInterval , (356) SumPseudoEuclidean , (359) SumTriangleArea , (362) SurvoPuzzle , (363) TakingPrimeGame , (364) TaskArrangement , (365) TetrisAttack , (367) ThreeVertexCycleCounting , (370) Tournament_LongestPath , (371) TransmissionDelay , (372) TreeAddOneEdgeDiameter , (373) TreeCenter , (374) TreeChangeOneEdgeDiameter , (375) TreeColoring , (376) TreeDynamic_XORZeroPath , (377) TreeElimination_Expectation , (378) TreeEvenPartitioning , (379) TreeMaximumVisitedVertex , (380) TreeRandomWalkExpectation , (381) TreeTopologicalSequenceCounting , (382) Tree_DistanceEqualTriad_Counting , (383) TriumphalArch , (384) TwiddlePuzzle , (385) TwoSAT , (386) TwoSet_AllCoprime_Counting , (388) ValueDiminishingSelection , (389) Vertex_KCenter , (391) VisibleLine , (392) WIL , (393) WYC , (396) WeightedBinaryTree , (397) WeightedLIS , (399) XorEquationCounting , (400) ZeroPrefixSubsetCounting .
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