Reverse Mathematics of Matroids

  • Jeffry L. Hirst
  • Carl Mummert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10010)


Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for \(\varSigma ^0_2\) formulas.


Reverse mathematics Matroid Induction Graph Connected component 

MSC Subject Class (2000)

03B30 03F35 05B35 


  1. 1.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Ann. Pure Appl. Logic 163(8), 968–1008 (2012). doi: 10.1016/j.apal.2011.12.020 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles, weak computability. J. Symb. Log. 76(1), 143–176 (2011). doi: 10.2178/jsl/1294170993. MR2791341 (2012c:03186)
  3. 3.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symb. Log. 1(1), 73–117 (2011). doi: 10.2178/bsl/1294186663 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. Am. Math. Soc. 368, 1321–1359 (2014). doi: 10.1090/tran/6465 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Crossley, J.N., Remmel, J.B.: Undecidability and recursive equivalence II. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds.) Computation and Proof Theory. LNM, vol. 1104, pp. 79–100. Springer, Heidelberg (1984). doi: 10.1007/BFb0099480. MR775710CrossRefGoogle Scholar
  6. 6.
    Downey, R.: Abstract dependence and recursion theory and the lattice of recursively enumerable filters, Ph.D. thesis, Monash University, Victoria, Australia (1982)Google Scholar
  7. 7.
    Downey, R.: Abstract dependence, recursion theory, and the lattice of recursively enumerable filters. Bull. Aust. Math. Soc. 27, 461–464 (1983). doi: 10.1017/S0004972700025958 CrossRefMATHGoogle Scholar
  8. 8.
    Downey, R.: Nowhere simplicity in matroids. J. Aust. Math. Soc. Ser. A 35(1), 28–45 (1983). doi: 10.1017/S1446788700024757. MR697655MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Friedman, H.M., Simpson, S.G., Smith, R.L.: Countable algebra and set existence axioms. Ann. Pure Appl. Logic 25(2), 141–181 (1983). doi: 10.1016/0168-0072(83)90012-X. MR725732MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Friedman, H.M., Simpson, S.G., Smith, R.L.: Addendum to: “Countable algebra and set existence axioms". Ann. Pure Appl. Logic 28(3), 319–320 (1985). doi: 10.1016/0168-0072(85)90020-X. MR790391MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gura, K., Hirst, J.L., Mummert, C.: On the existence of a connected component of a graph. Computability 4(2), 103–117 (2015). doi: 10.3233/COM-150039. MR3393974MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Harrison-Trainor, M., Melnikov, A., Montalbán, A.: Independence in computable algebra. J. Algebra 443, 441–468 (2015). doi: 10.1016/j.jalgebra.2015.06.004. MR3400410MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hirst, J.L.: Connected components of graphs and reverse mathematics. Arch. Math. Logic 31(3), 183–192 (1992). doi: 10.1007/BF01269946. MR1147740MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Metakides, G., Nerode, A.: Recursively enumerable vector spaces. Ann. Math. Logic 11(2), 147–171 (1977). doi: 10.1016/0003-4843(77)90015-8. MR0446936MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Metakides, G., Nerode, A.: Recursion theory on fields and abstract dependence. J. Algebra 65(1), 36–59 (1980). doi: 10.1016/0021-8693(80)90237-9. MR578794MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nerode, A., Remmel, J.: Recursion theory on matroids. Stud. Logic Found. Math. 109, 41–65 (1982). doi: 10.1016/S0049-237X(08)71356-9. Patras Logic Symposion (Patras, 1980). MR694252MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. Cambridge University Press, Association for Symbolic Logic, Cambridge, Poughkeepsie (2009). MR2517689 (2010e:03073)CrossRefMATHGoogle Scholar
  18. 18.
    Whitney, H.: On the abstract properties of linear dependence. Am. J. Math. 57(3), 509–533 (1935). doi: 10.2307/2371182. MR1507091MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Appalachian State UniversityBooneUSA
  2. 2.Marshall UniversityHuntingtonUSA

Personalised recommendations