Skip to main content
SpringerLink
Log in

Reverse Mathematics of Matroids

  • Chapter
  • First Online:
Computability and Complexity

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. MR775710

    Chapter  Google Scholar 

  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. 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

    Article  MATH  Google Scholar 

  8. Downey, R.: Nowhere simplicity in matroids. J. Aust. Math. Soc. Ser. A 35(1), 28–45 (1983). doi:10.1017/S1446788700024757. MR697655

    Article  MathSciNet  MATH  Google Scholar 

  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. MR725732

    Article  MathSciNet  MATH  Google Scholar 

  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. MR790391

    Article  MathSciNet  MATH  Google Scholar 

  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. MR3393974

    Article  MathSciNet  MATH  Google Scholar 

  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. MR3400410

    Article  MathSciNet  MATH  Google Scholar 

  13. Hirst, J.L.: Connected components of graphs and reverse mathematics. Arch. Math. Logic 31(3), 183–192 (1992). doi:10.1007/BF01269946. MR1147740

    Article  MathSciNet  MATH  Google Scholar 

  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. MR0446936

    Article  MathSciNet  MATH  Google Scholar 

  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. MR578794

    Article  MathSciNet  MATH  Google Scholar 

  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). MR694252

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  18. Whitney, H.: On the abstract properties of linear dependence. Am. J. Math. 57(3), 509–533 (1935). doi:10.2307/2371182. MR1507091

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Appalachian State University, Boone, North Carolina, 28608, USA

    Jeffry L. Hirst

  2. Marshall University, Huntington, West Virginia, 25755, USA

    Carl Mummert

Authors
  1. Jeffry L. Hirst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carl Mummert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeffry L. Hirst .

Editor information

Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Hirst, J.L., Mummert, C. (2017). Reverse Mathematics of Matroids. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50062-1_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation