J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1. Does C have solutions (and what are they) 2. Does C imply (or entail) another system of constraints C # That is, is every solution of C also a solution of C # For (1) the basic algorithms for solving many natural forms of subtyping constraints are by now quite well understood (e.g. see [36]) For (2) there has been much less progress on subtype entailment, although entailment is as important as constraint resolution in applications of subtyping. For example, a type based program analysis extracts some system of constraints C from a program text; these constraints are the model of ....

....both a least type and a greatest type for any # . More details on structural and non structural subtyping can be found in [4, 19, 24] Despite extensive e#ort over many years, the exact complexity and even the decidability of entailment is open for non structural subtyping constraints [3, 12, 13, 17, 18, 22, 27, 28, 33, 34, 36, 42]. As we show in Section 2, the natural versions of entailment and subtyping constrained types can be encoded easily in the first order theory of subtyping, so to gain insight into and take a step towards resolving these di#cult problems, we study the full first order theory in this paper. The ....

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J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, 1998.

....of powerful simplification algorithms for typings. Such operations can be formulated on the basis of algorithms for subtype entailment. It then turned out that subtype entailment is a quite complex problem, even for unexpressive type languages where types are ordinary trees. Rehof and Henglein [13] clarified the situation for structural subtyping. This is a tree ordering that relates trees of the same shape only. It is induced by lifting an ordering on constants. If trees are built over the signature int, real, # , for instance, then structural subtyping is induced by the usual axiom ....

J. Rehof, The complexity of simple subtyping systems, Ph.D. thesis, DIKU, Copenhagen (1998). 44

....of powerful simplification algorithms for typings. Such operations can be formulated on the basis of algorithms for subtype entailment. It then turned out that subtype entailment is a quite complex problem, even for unexpressive type languages where types are ordinary trees. Rehof and Henglein [22] clarified the situation for structural subtyping. This is a tree ordering that relates trees of the same shape only. It is induced by lifting an ordering on constants. If trees are built over the signature int, real, # , for instance, then structural subtyping is induced by the usual axiom ....

J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, Copenh., 1998.

....algorithms for subtype entailment, i.e. entailment of subtype constraints [22, 16] First approaches towards subtype entailment seem to presuppose [22, 16] that the problem could be solved efficiently. But finding an efficient algorithm for subtype entailment turned out to surprisingly difficult [9, 20, 10, 18]. And in fact, it still remains open whether subtype entailment is decidable, even if restricted to an inexpressive type languages. The most prominent open problem is the decidability of entailment between non structural subtype constraints. Types. A simple type is finite tree built from a ....

.... who determined the complexity of structural subtype entailment: for simple types, it is coNP complete [9] and for recursive types it is PSPACE complete [10] However, the complexity of non structural subtype entailment remains open; it is at least PSPACE hard, both, for finite and infinite trees [20, 10]. It is even unknown whether non structural subtype entailment is decidable. Nevertheless, Rehof conjectures PSPACE completeness (see Conjecture 9.4.5 of [20] Contribution. In this paper, we investigate the source of complications underlying non structural subtype entailment. To this purpose, ....

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J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, University of Copenhagen, 1998.

..... 47 7.5 Recursive Feature Trees . 48 7. 6 Reversed Binary Tree with Prefix Closed Sets 48 8 Conclusion 49 1 Introduction Subtyping constraints are an important technique for checking and inferring program properties, used both in type systems and program analyses [34, 16, 13, 28, 23, 4, 3, 1, 2, 20, 41, 17, 54, 7, 8, 5, 42, 47, 19]. This paper presents a decision procedure for the firstorder theory of structural subtyping of non recursive types. This result solves (for the case of non recursive types) a problem left open in [48] 48] provides the decidability result for structural subtyping of only unary type ....

Jakob Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, Computer Science Department, Univ. of Copenhagen (DIKU), April 1998. 1

..... 47 7.5 Recursive Feature Trees . 47 7. 6 Reversed Binary Tree with Pre x Closed Sets 48 8 Conclusion 48 1 Introduction Subtyping constraints are an important technique for checking and inferring program properties, used both in type systems and program analyses [34, 16, 13, 28, 23, 4, 3, 1, 2, 20, 41, 17, 54, 7, 8, 5, 42, 47, 19]. This paper presents a decision procedure for the rstorder theory of structural subtyping of non recursive types. This result solves (for the case of non recursive types) a problem left open in [48] 48] provides the decidability result for structural subtyping of only unary type constructors, ....

Jakob Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, Computer Science Department, Univ. of Copenhagen (DIKU), April 1998.

....data, this paper shows that formal speci cation, design, and analysis lead to an ecient algorithm with exact complexity factors that can be measured in experiments. Clearly, there is a large body of work on all kinds of program analysis algorithms [37] from type inference algorithms, e.g. [45], to ecient xed point computation, e.g. 14] Precise and uni ed speci cation, design, and complexity analysis of all kinds of program analysis algorithms deserve much further study. We believe that such study can bene t greatly from the approach of Paige et al. 41, 39, 10, 40, 9] as ....

J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, University of Copenhagen, Copenhagen, Denmark, Apr. 1998.

....of dead code elimination on recursive data, this paper shows that formal speci cation, design, and analysis lead to an ef cient algorithm with exact complexity factors. Clearly, there is a large body of work on all kinds of program analysis algorithms [30] from type inference algorithms, e.g. [35], to ecient xed point computation, e.g. 12] Precise and uni ed speci cation, design, and complexity analysis of all kinds of program analysis algorithms deserve much further study. We believe that such study can bene t greatly from the approach of Paige et al. 34, 32, 8, 33, 7] as ....

J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, University of Copenhagen, Copenhagen, Denmark, Apr. 1998.

....C have solutions (and what are they) 2. Does C imply (or entail) another system of constraints C 0 That is, is every solution of C also a solution of C 0 For (1) the basic algorithms for solving many natural forms of subtyping constraints are by now quite well understood (e.g. see [37]) For (2) there has been much less progress on subtype entailment, although entailment is as This research was supported in part by the National Science Foundation grant No. CCR 0085949 and NASA Contract No. NAG21210. important as constraint resolution in applications of subtyping. For ....

....type and a greatest type , so that for any . More details on structural and non structural subtyping can be found in [4, 20, 25] Despite extensive e ort over many years, the exact complexity and even the decidability of entailment is open for non structural subtyping constraints [3, 13, 14, 18, 19, 23, 28, 29, 34, 35, 37, 43]. As we show in Section 2, the natural versions of entailment and subtyping constrained types can be encoded easily in the rst order theory of subtyping, so to gain insight into and take a step towards resolving these dif cult problems, we study the full rst order theory in this paper. The ....

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J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, DIKU, 1998.

.... cation, which aims at compacting constraint sets before they are copied [FM89, Cur90, Kae92, Smi94, EST95, Mos96, Pot96, TS96, FA96, AWP97, Reh97, FF97] It is unlikely that constraint simpli cation techniques alone will solve this problem, and complete simpli cation is a hard problem itself [Reh98, FF97]. 2.2 Problem 2: Demand driven ow computation Constraint copying methods for polymorphic subtyping systems are not demand driven [Mos96] For instance, if we 1 Function id can be given a most general typing without any subtyping constraints, but we choose the present typing for illustrative ....

J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, Dept. of Computer Science, University of Copenhagen, Denmark, April 1998.

.... simpli cation, which aims at compacting constraint sets before they are copied [FM89, Cur90, Kae92, Smi94, EST95, Pot96, TS96, FA96, AWP97, Reh97, FF97] It is unlikely that constraint simpli cation techniques alone will solve this problem, and complete simpli cation is a hard problem itself [Reh98, FF97]. 1.2 A new method based on instantiation constraints In this paper, we tackle the constraint copying problem in a new way. Our starting point is a new presentation of polymorphic subtyping, based on instantiation constraints, which we call POLYFLOW CFL . Instead of constrained types 8 :C ) ....

J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, Dept. of Computer Science, University of Copenhagen, Denmark, April 1998.

.... simpli cation, which aims at compacting constraint sets before they are copied [FM89, Cur90, Kae92, Smi94, EST95, Pot96, TS96, FA96, AWP97, Reh97, FF97] It is unlikely that constraint simpli cation techniques alone will solve this problem, and complete simpli cation is a hard problem itself [Reh98, FF97]. 1.2 A New Method Based on Instantiation Constraints In this article, we tackle the constraint copying problem in a new way. Our starting point is a new presentation of polymorphic subtyping, based on instantiation constraints, which we 1 Function id can be given a most general typing without ....

J. Rehof. The Complexity of Simple Subtyping Systems. PhD thesis, Dept. of Computer Science, University of Copenhagen, Denmark, April 1998.

....to our case. Their axiomatization leads to a complete algorithm for entailment, but since it runs in exponential time and consumes exponential space they do not give a tight classification of complexity. Due to space limitations, details of some proofs had to be left out. They can be found in [19, 18]. 2 Preliminaries Let Sigma be the ranked alphabet of constructors, Sigma = B[f Theta; g; here L = B; B ) is a lattice of constants (constructors of arity 0, to be thought of as base types such as, e.g. int, real) and Theta; are binary constructors. The order on L defines the subtype ....

....in T f Sigma . If L(A) 6= ff; sg , then v j= C f A with v(ff 0 ) 6 v(fi) for some v in T Sigma , by the argument in Theorem 1; by passing to vd k for sufficiently large k (using the property above) one can then show C f A 6j= ff 0 fi in T f Sigma . More details can be found in [19, 18]. The problem of giving an upper bound on nonstructural entailment remains open. In Section 5.2 we will show that structural recursive subtype entailment is in PSPACE. We believe that an elaboration of the ideas in the PSPACE algorithm sketched there can be used to give a PSPACE decision ....

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J. Rehof. The complexity of simple subtyping systems. Technical report, DIKU, Dept. of Computer Science, University of Copenhagen, Denmark. Available at http://www.diku.dk/research-groups/topps/personal/ rehof/publications.html, April 1998. Thesis submitted for the Ph.D. degree.

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