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for expressions and types. Adding subtyping means that a partial order 
on types is defined and that the typing rules are extended to the effect that expression 
7, the maximal size semigroup w.r.t. its size among all (transformation) semigroups which can be generated with two generators, is generated by a permutation with two cycles (of appropriate lengths) and a non-bijective mapping merging elements from these two cycles. As a by-product of our investigations we determine the maximal size among all semigroups generated by two transformations, where one is a permutation with a single cycle and the other is a non-bijective mapping.
doi:10.1016/S0890-5401(03)00140-8 
Copyright © 2003 Elsevier Inc. All rights reserved.
Non-structural subtype entailment in automata theory
Joachim Niehren
, 
and Tim Priesnitz
Received 14 March 2002;
Abstract
Decidability of non-structural subtype entailment is a long-standing open problem in programming language theory. In this paper, we apply automata theoretic methods to characterize the problem equivalently by using regular expressions and word equations. This characterization induces new results on non-structural subtype entailment, constitutes a promising starting point for further investigations on decidability, and explains for the first time why the problem is so difficult. The difficulty is caused by implicit word equations that we make explicit.
Author Keywords: Programming languages; Subtyping; Finite automata; Word equations
References
[1]. R.M. Amadio and L. Cardelli, Subtyping recursive types. ACM Transactions on Programming Languages and Systems 15 4 (1993), pp. 575–631. View Record in Scopus | Cited By in Scopus (99)
[2]. F. Baader and K. Schulz, Unification in the union of disjoint equational theories: combining decision procedures. Journal of Symbolic Computation 21 (1996), pp. 211–243. Abstract | 
PDF (755 K)
| View Record in Scopus | Cited By in Scopus (28)
[3]. H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, S. Tison, M. Tommasi, Tree automata techniques and applications, October 1999. Available from http://www.grappa.univ-lille3.fr/tata
[4]. V. Durnev, Unsolvability of positive 
3-theory of free groups. Sibirsky mathematichesky jurnal 36 5 (1995), pp. 1067–1080 in Russian, also exists in English translation .
[5]. J. Eifrig, S. Smith and V. Trifonov, Sound polymorphic type inference for objects. In: ACM Conference on Object-Oriented Programming: Systems, Languages, and Applications (1995), pp. 169–184. View Record in Scopus | Cited By in Scopus (18)
[6]. J. Eifrig, S. Smith, V. Trifonov, Type inference for recursively constrained types and its application to object-oriented programming, Electronic Notes in Theoretical Computer Science 1
[7]. Y. Fuh and P. Mishra, Type inference with subtypes. Theoretical Computer Science 73 2 (1990), pp. 155–175. Abstract | PDF (2390 K)
| View Record in Scopus | Cited By in Scopus (12)
[8]. F. Henglein and J. Rehof, Constraint automata and the complexity of recursive subtype entailment. In: 25th International Colloquium on Automata, Languages, & ProgrammingLecture Notes in Computer Science vol. 1443, Springer-Verlag (1998), pp. 616–627.
[9]. F. Henglein and J. Rehof, The complexity of subtype entailment for simple types. In: Proceedings of the 12th IEEE Symposium on Logic in Computer science, IEEE Computer Society Press (1997), pp. 362–372.
[10]. D. Kozen, J. Palsberg and M.I. Schwartzbach, Efficient inference of partial types. Journal of Computer and System Sciences 49 2 (1994), pp. 306–324. Abstract | PDF (798 K)
| View Record in Scopus | Cited By in Scopus (14)
[11]. D. Kozen, J. Palsberg and M.I. Schwartzbach, Efficient recursive subtyping. Mathematical Structures in Computer Science 5 (1995), pp. 1–13.
[12]. G. Makanin, The problem of solvability of equations in a free semigroup, Mathematics of the USSR Sbornik 32 (translated from Russian)
[13]. J.C. Mitchell, Type inference with simple subtypes. The Journal of Functional Programming 1 3 (1991), pp. 245–285.
[14]. M. Müller, J. Niehren and R. Treinen, The first-order theory of ordering constraints over feature trees. In: IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press (1998), pp. 432–443.
[15]. J. Niehren and T. Priesnitz, Entailment of non-structural subtype constraints. In: Asian Computing Science ConferenceLecture Notes in Computer Science vol. 1742, Springer-Verlag (1999), pp. 251–265.
[16]. J. Niehren, M. Müller and J.-M. Talbot, Entailment of atomic set constraints is PSPACE-complete. In: 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press (1999), pp. 285–294. View Record in Scopus | Cited By in Scopus (4)
[17]. J. Niehren and T. Priesnitz, Non-structural subtype entailment in automata theory. In: Fourth International Symposium on Theoretical Aspects of Computer SoftwareLecture Notes in Computer Science vol. 2215, Springer-Verlag (2001), pp. 360–384.
[18]. J. Palsberg and P. O’Keefe, A type system equivalent to flow analysis. ACM Transactions on Programming Languages and Systems 17 4 (1995), pp. 576–599. View Record in Scopus | Cited By in Scopus (31)
[19]. J. Palsberg, M. Wand and P. O’Keefe, Type inference with non-structural subtyping. Formal Aspects of Computing 9 (1997), pp. 49–67. View Record in Scopus | Cited By in Scopus (7)
[20]. W. Plandowski, Satisfiability of word equations with constants is in PSPACE. In: Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press (1999), pp. 495–500. View Record in Scopus | Cited By in Scopus (21)
[21]. F. Pottier, Simplifying subtyping constraints. In: ACM SIGPLAN International Conference on Functional Programming, ACM Press, New York (1996), pp. 122–133. View Record in Scopus | Cited By in Scopus (19)
[22]. F. Pottier, Type inference in the presence of subtyping: from theory to practice, Ph.D. thesis, Institut de Recherche d’Informatique et d’Automatique, 1998
[23]. F. Pottier, A framework for type inference with subtyping. In: Proceedings of the third ACM SIGPLAN International Conference on Functional programming, ACM Press, New York (1998), pp. 228–238. View Record in Scopus | Cited By in Scopus (8)
[24]. J. Rehof, Minimal typings in atomic subtyping. In: ACM Symposium on Principles of Programming Languages, ACM Press, New York (1997), pp. 278–291. View Record in Scopus | Cited By in Scopus (3)
[25]. J. Rehof, The complexity of simple subtyping systems, Ph.D. thesis, DIKU, Copenhagen, 1998
[26]. K.U. Schulz, Makanin’s algorithm for word equations – two improvements and a generalization. In: Word Equations and Related TopicsLecture Notes in Computer Science vol. 572, Springer-Verlag (1991), pp. 85–150.
[27]. Z. Su, A. Aiken, J. Niehren, T. Priesnitz and R. Treinen, First-order theory of subtyping constraints. In: The 29th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, ACM Press (2002), pp. 203–216. View Record in Scopus | Cited By in Scopus (3)
[28]. R. Treinen, Predicate logic and tree automata with tests. In: Foundations of Software and Computer StructuresLecture Notes in Computer Science vol. 1784, Springer-Verlag (2000), pp. 329–343.
[29]. Y. Vazhenin and B. Rozenblat, Decidability of the positive theory of a free countably generated semigroup. Mathematics of the USSR Sbornik 44 (1983), pp. 109–116.
