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DOI: 10.4230/LIPIcs.TYPES.2014.251

Restricted Positive Quantification Is Not Elementary

Authors: Aleksy Schubert, Pawel Urzyczyn, and Daria Walukiewicz-Chrzaszcz

Published in: LIPIcs, Volume 39, 20th International Conference on Types for Proofs and Programs (TYPES 2014)

Abstract

We show that a restricted variant of constructive predicate logic with positive (covariant) quantification is of super-elementary complexity. The restriction is to limit the number of eigenvariables used in quantifier introductions rules to a reasonably usable level. This construction suggests that the known non-elementary decision algorithms for positive logic may actually be best possible.

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Aleksy Schubert, Pawel Urzyczyn, and Daria Walukiewicz-Chrzaszcz. Restricted Positive Quantification Is Not Elementary. In 20th International Conference on Types for Proofs and Programs (TYPES 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 39, pp. 251-273, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{schubert_et_al:LIPIcs.TYPES.2014.251,
  author =	{Schubert, Aleksy and Urzyczyn, Pawel and Walukiewicz-Chrzaszcz, Daria},
  title =	{{Restricted Positive Quantification Is Not Elementary}},
  booktitle =	{20th International Conference on Types for Proofs and Programs (TYPES 2014)},
  pages =	{251--273},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-88-0},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{39},
  editor =	{Herbelin, Hugo and Letouzey, Pierre and Sozeau, Matthieu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2014.251},
  URN =		{urn:nbn:de:0030-drops-55002},
  doi =		{10.4230/LIPIcs.TYPES.2014.251},
  annote =	{Keywords: constructive logic, complexity, automata theory}
}

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DOI: 10.4230/LIPIcs.CSL.2012.243

Bounded Combinatory Logic

Authors: Boris Düdder, Moritz Martens, Jakob Rehof, and Pawel Urzyczyn

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Abstract

In combinatory logic one usually assumes a fixed set of basic combinators (axiom schemes), usually K and S. In this setting the set of provable formulas (inhabited types) is PSPACE-complete in simple types and undecidable in intersection types. When arbitrary sets of axiom schemes are considered, the inhabitation problem is undecidable even in simple types (this is known as Linial-Post theorem). k-bounded combinatory logic with intersection types arises from combinatory logic by imposing the bound k on the depth of types (formulae) which may be substituted for type variables in axiom schemes. We consider the inhabitation (provability) problem for k-bounded combinatory logic: Given an arbitrary set of typed combinators and a type tau, is there a combinatory term of type tau in k-bounded combinatory logic? Our main result is that the problem is (k+2)-EXPTIME complete for k-bounded combinatory logic with intersection types, for every fixed k (and hence non-elementary when k is a parameter). We also show that the problem is EXPTIME-complete for simple types, for all k. Theoretically, our results give new insight into the expressive power of intersection types. From an application perspective, our results are useful as a foundation for composition synthesis based on combinatory logic.

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Boris Düdder, Moritz Martens, Jakob Rehof, and Pawel Urzyczyn. Bounded Combinatory Logic. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 243-258, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{dudder_et_al:LIPIcs.CSL.2012.243,
  author =	{D\"{u}dder, Boris and Martens, Moritz and Rehof, Jakob and Urzyczyn, Pawel},
  title =	{{Bounded Combinatory Logic}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{243--258},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.243},
  URN =		{urn:nbn:de:0030-drops-36763},
  doi =		{10.4230/LIPIcs.CSL.2012.243},
  annote =	{Keywords: Intersection types, Inhabitation, Composition synthesis}
}

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Documents authored by Urzyczyn, Pawel

Restricted Positive Quantification Is Not Elementary

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Bounded Combinatory Logic

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