Deciding inclusion of set constants over infinite non-strict data structures
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 2, pp. 225-241.

Various static analyses of functional programming languages that permit infinite data structures make use of set constants like Top, Inf, and Bot, denoting all terms, all lists not eventually ending in Nil, and all non-terminating programs, respectively. We use a set language that permits union, constructors and recursive definition of set constants with a greatest fixpoint semantics in the set of all, also infinite, computable trees, where all term constructors are non-strict. This paper proves decidability, in particular DEXPTIME-completeness, of inclusion of co-inductively defined sets by using algorithms and results from tree automata and set constraints. The test for set inclusion is required by certain strictness analysis algorithms in lazy functional programming languages and could also be the basis for further set-based analyses.

DOI : https://doi.org/10.1051/ita:2007010
Classification : 68N18,  03B40,  68Q25,  68Q45
Mots clés : functional programming languages, lambda calculus, strictness analysis, set constraints, tree automata
@article{ITA_2007__41_2_225_0,
author = {Schmidt-Schauss, Manfred and Sabel, David and Sch\"utz, Marko},
title = {Deciding inclusion of set constants over infinite non-strict data structures},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {225--241},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
doi = {10.1051/ita:2007010},
mrnumber = {2350646},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ita:2007010/}
}
Schmidt-Schauss, Manfred; Sabel, David; Schütz, Marko. Deciding inclusion of set constants over infinite non-strict data structures. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 2, pp. 225-241. doi : 10.1051/ita:2007010. http://archive.numdam.org/articles/10.1051/ita:2007010/

[1] S. Abramsky and C. Hankin, Abstract interpretation of declarative languages. Ellis Horwood, (1987).

[2] A. Aiken, Set constraints: Results, applications, and future directions, in Second Workshop on the Principles and Practice of Constraint Programming, Orcas Island, Washington, Springer-Verlag. Lect. Notes Comput. Sci. 874 (1994) 171-179.

[3] A. Aiken, D. Kozen, M.Y. Vardi and E.L. Wimmers, The complexity of set constraints, in Proc. CSL 1993, Swansea, Wales (1993) 1-17. | Zbl 0953.68557

[4] Z.M. Ariola and S. Blom, Cyclic lambda calculi, in TACS, Sendai, Japan (1997) 77-106. | Zbl 0884.03008

[5] Z.M. Ariola and J.W. Klop, Lambda calculus with explicit recursion. Inform. Comput. 139 (1997) 154-233. | Zbl 0892.68015

[6] L. Bachmair, H. Ganzinger and U. Waldmann, Set constraints are the monadic class, in Proc. 8th Proc Symp. Logic in Computer Science, Swansea, Wales (1993) 75-83.

[7] G. Burn, Lazy Functional Languages: Abstract Interpretation and Compilation. Pitman, London, (1991). | MR 1156765 | Zbl 0809.68079

[8] G.L. Burn, C.L. Hankin and S. Abramsky, The theory for strictness analysis for higher order functions, in Programs as Data Structures, edited by H. Ganzinger and N.D. Jones. Lect. Notes Comput. Sci. 217 (1985) 42-62. | Zbl 0596.68009

[9] W. Charatonik and A. Podelski, Co-definite set constraints, in Proceedings of the 9th International Conference on Rewriting Techniques and Applications, edited by T. Nipkow, Springer-Verlag. Lect. Notes Comput. Sci. 1379 (1998) 211-225.

[10] D. Clark, C. Hankin, and S. Hunt, Safety of strictness analysis via term graph rewriting. In SAS 2000 (2000) 95-114. | Zbl 0966.68089

[11] H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, S. Tison and M. Tommasi. Tree automata techniques and applications. Available on: http://www.grappa.univ-lille3.fr/tata, 1997. release October, 1rst 2002.

[12] M. Coppo, F. Damiani and P. Giannini, Strictness, totality, and non-standard type inference. Theoret. Comput. Sci. 272 (2002) 69-112. | Zbl 0984.68028

[13] P. Cousot and R. Cousot, Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints, in Conference Record of the Fourth ACM Symposium on Principles of Programming Languages, ACM Press (1977) 252-252.

[14] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order. Cambridge University Press, Cambridge, (1992). | MR 1902334 | Zbl 1002.06001

[15] P. Devienne, J.-M. Talbot and S. Tison, Co-definite set constraints with membership expressions, in JICSLP'98: Proceedings of the 1998 joint international conference and symposium on Logic programming, Cambridge, MA, USA, MIT Press (1998) 25-39. | Zbl 0949.68017

[16] T.P. Jensen, inference of polymorphic and conditional strictness properties, in Symposium on Principles of Programming Languages, San Diego, ACM Press. (1998) 209-221.

[17] S.P. Jones, Haskell 98 Language and Libraries. Cambridge University Press (2003). www.haskell.org | MR 1989220

[18] Tsun-Ming Kuo and P. Mishra, Strictness analysis: A new perspective based on type inference, in Functional Programming Languages and Computer Architecture, ACM Press, (1989) 260-272.

[19] K. Lackner Solberg Gasser, H. Riis Nielson and F. Nielson, Strictness and totality analysis. Sci. Comput. Programming 31 (1998) 113-145. | Zbl 0941.68021

[20] J. Launchbury and S. Peyton Jones, State in Haskell. Lisp Symbolic Comput. 8 (1995) 293-341.

[21] L. Mauborgne, Improving the representation of infinite trees to deal with sets of trees, in ESOP '00: Proceedings of the 9th European Symposium on Programming Languages and Systems. Lect. Notes Comput. Sci. 1782 (2000) 275-289. | Zbl 0960.68041

[22] A.K.D. Moran, D. Sands and M. Carlsson, Erratic fudgets: A semantic theory for an embedded coordination language. Sci. Comput. Programming 46 (2003) 99-135. | Zbl 1026.68091

[23] A. Mycroft, Abstract Interpretation and Optimising Transformations for Applicative Programs. Ph.D. thesis, University of Edinburgh (1981).

[24] E. Nöcker, Strictness analysis using abstract reduction. Technical Report 90-14, Department of Computer Science, University of Nijmegen (1990).

[25] E. Nöcker, Strictness analysis by abstract reduction in orthogonal term rewriting systems. Technical Report 92-31, University of Nijmegen, Department of Computer Science (1992).

[26] E. Nöcker, Strictness analysis using abstract reduction. In Functional Programming Languages and Computer Architecture, ACM Press, (1993) 255-265.

[27] D. Pape, Higher order demand propagation. In Implementation of Functional Languages (IFL '98) London, edited by K. Hammond, A.J.T. Davie and C. Clack, Springer-Verlag. Lect. Notes Comput. Sci. 1595 (1998) 155-170.

[28] D. Pape, Striktheitsanalysen funktionaler Sprachen. Ph.D. thesis, Fachbereich Mathematik und Informatik, Freie Universität Berlin, (2000). In German. | Zbl 0951.68526

[29] R. Paterson, Compiling laziness using projections, in Static Analysis Symposium, Aachen, Germany. Lect. Notes Comput. Sci. 1145 (1996) 255-269.

[30] R. Plasmeijer and M. Van Eekelen, The concurrent Clean language report: Version 1.3 and 2.0. Technical report, Dept. of Computer Science, University of Nijmegen, 2003. http://www.cs.kun.nl/~clean/ | Zbl 0633.68003

[31] P. Rychlikowski and T. Truderung, in Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 20-24, 2004, Proceedings, Set constraints on regular terms, edited by J. Marcinkowski and A. Tarlecki, Springer. Lect. Notes Comput. Sci. 3210 (2004) 458-472. | Zbl 1095.68056

[32] M. Schmidt-Schauß, S. Eric Panitz and M. Schütz, Strictness analysis by abstract reduction using a tableau calculus, in Proc. of the Static Analysis Symposium. Lect. Notes Comput. Sci. 983 (1995) 348-365.

[33] M. Schmidt-Schauß, M. Schütz and D. Sabel, On the safety of Nöcker's strictness analysis. Technical Report Frank-19, Institut für Informatik. J.W. Goethe-University (2004).

[34] M. Schmidt-Schauß, M. Schütz and D. Sabel, A complete proof of the safety of Nöcker's strictness analysis. Technical Report Frank-20, Institut für Informatik. J.W. Goethe-University, (2005).

[35] M. Schütz, Analysing demand in nonstrict functional programming languages. Dissertation, J.W.Goethe-Universität Frankfurt, 2000. Available at http://www.ki.informatik.uni-frankfurt.de/papers/marko

[36] H. Seidl, Deciding equivalence of finite tree automata. SIAM J. Comput. 19 (1990) 424-437. | Zbl 0699.68075

[37] W. Thomas, Automata on infinite objects. In Handbook of Theoretical Computer Science, Formal Models and Semantics (B), edited by J. van Leeuwen, Elsevier (1990) 133-192. | Zbl 0900.68316

[38] P. Wadler, Strictness analysis on non-flat domains (by abstract interpretation over finite domains). In Abstract Interpretation of Declarative Languages, Chap. 12. Edited by S. Abramsky and C. Hankin, Ellis Horwood Limited, Chichester (1987).

[39] P. Wadler and J. Hughes, Projections for strictness analysis. In Functional Programming Languages and Computer Architecture. Lect. Notes Comput. Sci. 274 (1987) 385-407. | Zbl 0625.68014