Mitchell Wand and Patrick O'Keefe. On the complexity of type inference with coercion. In Conference on Functional Programming Languages and Computer Architecture, 1989. 32  Home/Search   Document Not in Database   Summary   Related Articles   Check

...., and ffl C = fx : ff : fi i ] j fi i 2 fig. In a type system with overloading but no subtyping, the restriction to overloading by constructors allows the satisfiability problem to be solved efficiently. On the other hand, for a system with subtyping but no overloading, it is shown in [20] and [9] that testing the satisfiability of a set of atomic inclusions is NP complete. Testing the satisfiability of a set of arbitrary inclusions is shown in [16] to be PSPACE hard, and [17] gives a DEXPTIME algorithm. In our system, which has both overloading and subtyping, the restriction to ....

Mitchell Wand and Patrick O'Keefe. On the complexity of type inference with coercion. In Conference on Functional Programming Languages and Computer Architecture, 1989. 32

....problems for the case of simply typed lambda calculus with subtyping. The reader is referred to J. Mitchell s paper [Mitchell84] for introduction to that area as well as the basic reduction of the original problem of type reconstruction to the problem of poset satisfiability. O Keefe and Wand [OKeefeWand89] treat a similar reduction. The NP hardness result of section 2 of this paper has been used by Mitchell and Lincoln [MitchellLincoln92] to show that the type reconstruction problem for simply typed calculus with subtyping is NP hard for certain posets of atomic subtypes. The second author has ....

Mitchell Wand and Patrick M. O'Keefe. On the Complexity of Type Inference with Coercion. In Conf. on Functional Programming Languages and Computer Architecture, 1989.

....subgroup problem becomes NP complete. Does this sequence of observations lead to a classification of the problems in CSP as polynomially solvable or NP complete 2 Preliminaries A large class of problems in AI and other areas of computer science can be viewed as constraint satisfaction problems [9, 30, 36, 37, 38, 39, 41]. This includes 3 problems in machine vision, belief maintenance, scheduling, temporal reasoning, type reconstruction, graph theory, and satisfiability. We start with some definitions. A vocabulary is a set V = f(R 1 ; k 1 ) R t ; k t )g of relation names and their arities. A ....

....For strict partial orders, even if the strict partial order is bipartite, the problem is equivalent to the bipartite graph retract problem and hence to all of CSP. For partial orders, there are applications to type reconstruction, see Mitchell [37] Mitchell and Lincoln [38] O Keefe and Wand [39]. Pratt and Tiuryn [41] showed that the bipartite partial orderretract problem is polynomially solvable if the underlying graph is a tree (in fact in NLOGSPACE) NP complete otherwise. We can give here an alternative proof of this result here. If the underlying graph is a tree, we have a directed ....

M. Wand and P. M. O'Keefe, "On the complexity of type inference with coercion," in Conf. on Functional Programming Languages and Computer Architecture (1989).

.... 1995, Tiuryn 1997, Frey 1997) When the subtyping relation on atomic types is identity, the whole subtype relation is also identity, and the SSCP becomes the unification problem for simple types, known to be PTIME complete (Dwork, Kanellakis Mitchell 1984) Lincoln Mitchell 1992) improving (Wand O Keefe 1989), demonstrated that for some fixed ordering of atomic types the SSCP is NP hard. Tiuryn 1992, Pratt Tiuryn 96) improved it further to PSPACE hardness for some simple fixed orderings of atomic types called crowns. Lincoln Mitchell 1992, Tiuryn 1992) gave the NEXPTIME upper bound for the ....

Wand, M. & O'Keefe, P. (1989), On the complexity of type inference with coercion, in `Functional Programming Languages and Computer Architecture'89', pp. 293--298.

....a valuation oe(fl) 2 fff; fig for fl such that ff oe(fl) and fi oe(fl) where is the partial order defined by ff ff, fi fi, fl fl, ff fl, and fi fl, which is clearly impossible. The most obvious solution to this problem, which is called po sat and is (implicitly) shown to be NP complete in [45], is of course to try all possible maps oe from # to # 1 , but this algorithm is far from efficient, and we would rather find an algorithm which is polynomial most of the time . To this end, we define the functional Phi from the finite complete lattice 3 # P(# 1 ) into itself as follows 8v ....

....2 # is a pre solution. Consequently, the existence of a pre solution is a prerequisite to the existence of a solution 4 , and if G is the maximum pre solution, then every solution oe of the problem is such that oe(v) 2 G(v) for every v 2 #. Unfortunately, as implied by the complexity result of [45], the existence of a pre solution is not sufficient to prove the existence of a solution. The simplest counter example that we have found is given figure 7.2, with # 1 = fa; b; c; dg and # 2 = fac; ad; bc; bd; cd; abg. This problem (which is built by simplifying the set of convex subsets of fa; b; ....

M. Wand, P. O'Keefe. On the complexity of type inference with coercion. Proc. of FPCA'89 (1989) 293--298

....this has only impact on the solvability of subtype constraints, not their generation and simplification. 7 Computing principal typings reduces the problem of typability over a particular subtype theory to one of deciding satisfiability of subtyping constraints. There has been an array of results [WO89, LM92, Tiu92, Ben93, HM95, PT95] that characterize when solving subtyping constraints is feasible and when not. Subtyping has recently been studied intensively in a setting with recursive types and combinatorially simple primitive subtype theories: Palsberg et al. e.g. KPS92, KPS93, Pal94, PWO95] have worked on efficient type ....

M. Wand and P. O'Keefe. On the complexity of type inference with coercion. In Proc. ACM Conf. on Functional Programming and Computer Architecture, 1989.

....statement that describes the set of possible typings with respect to any subtype hierarchy. Various aspects of the algorithm have been studied by other authors, with Fuh and Mishra elaborating algorithmic alternatives [FM90] and Wand and O Keefe studying the computational complexity of typability [WO89]. An extension with polymorphic record operations [JM88] has been implemented by Jategaonkar [Jat89] Unfortunately, the straightforward implementation of the type inference algorithm with subtypes requires exponential time, even in the absence of polymorphic let declarations (see [KMM91] This ....

....that it is np hard to decide whether a lambda expression with constants has a type, given a set of subtyping relationships between ground (atomic) types. This applies to polymorphic and monomorphic languages, and languages without functional constants. This lower bound improves the main result of [WO89], which requires a constant with a polymorphic type. We also observe that if type parameters and subtype assumptions are given explicitly in the syntax of terms, it follows from the results in [Tiu91] that deciding whether an explicitly typed term has a type is pspace hard. We give two algorithms ....

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M. Wand and P. O'Keefe. On the complexity of type inference with coercion. In Proc. ACM Conf. Functional Programming and Computer Architecture, pages 293--298, 1989.

....fl by first running W ( fl; p; and, if it succeeds with (C; b ; V ) then checking whether C is satisfiable with respect to the partial ordering of security levels. Checking the satisfiability of a flat set of subtyping inequalities with respect to a partial order has been studied previously [15, 18]. It is NPcomplete, in general, but can sometimes be done efficiently, for example, if the partial order is a disjoint union of lattices. 6.1 Principal Types In addition to checking typability, type inference gives us the ability to compute principal types, that document all possible types of ....

Wand, M. and O'Keefe, P., On the Complexity of Type Inference with Coercion, Proc. ACM Conf. on Functional Programming Languages and Computer Architecture, pp. 293--298, 1989.

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