On a quasi-ordering on Boolean functions

  • Authors:

  • Miguel Couceiro;Maurice Pouzet

  • Affiliations:

  • Department of Mathematics and Statistics, University of Tampere, Kanslerinrinne 1, 33014 Tampere, Finland and Department of Philosophy, University of Tampere, Kanslerinrinne 1, 33014 Tampere, Finl ...;ICJ, Department of Mathematics, Université Claude-Bernard Lyon1, 43 Bd 11 Novembre 1918, 68622 Villeurbanne Cedex, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2008

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Abstract

It was proved few years ago that classes of Boolean functions definable by means of functional equations [O. Ekin, S. Foldes, P.L. Hammer, L. Hellerstein, Equational characterizations of boolean functions classes, Discrete Mathematics 211 (2000) 27-51], or equivalently, by means of relational constraints [N. Pippenger. Galois theory for minors of finite functions, Discrete Mathematics 254 (2002) 405-419], coincide with initial segments of the quasi-ordered set (@W,@?) made of the set @W of Boolean functions, suitably quasi-ordered. Furthermore, the classes defined by finitely many equations [O. Ekin, S. Foldes, P.L. Hammer, L. Hellerstein, Equational characterizations of boolean functions classes, Discrete Mathematics 211 (2000) 27-51] coincide with the initial segments of (@W,@?) which are definable by finitely many obstructions. The resulting ordered set (@W@?,@?) embeds into ([@w]^