The complexity of type inference for higher-order typed lambda calculi

Henglein, Fritz; Mairson, Harry G.

doi:10.1017/s0956796800001143

Cited by 27 publications

(30 citation statements)

References 34 publications

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“…Elucidating the contribution of this parameter to the complexity of the problem may illuminate why the problem is "easier to solve in practice" than one might expect from traditional complexity analysis. An example of this phenomena is the complexity of Type Inference in ML [HM,DF6]. Conversely, engineering practice may introduce (and fix) a parameter, that renders the complexity of the problem easier than one desires (e.g.…”

Section: Introductionmentioning

confidence: 99%

On the parameterized complexity of short computation and factorization

Cai

Chen

Downey

et al. 1997

Archive for Mathematical Logic

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SummaryA completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,DEF,FH,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in k steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word x can be derived in a grammar G in k steps, or whether a permutation has a factorization of length k over a given set of generators. These include a natural parameterized version of the famous Post Correspondence Systems. We show hardness and completeness for these problems for various levels of the W hierarchy. In particular, we show that Short TM Computation is complete for W [1]. This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes. The result could be viewed as one analogue of Cook's theorem and we believe provides strong evidence for the parameterized intractability of W [1].

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Section: Introductionmentioning

confidence: 99%

On the parameterized complexity of short computation and factorization

Cai

Chen

Downey

et al. 1997

Archive for Mathematical Logic

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“…The development of efficient parameterized algorithms has provided a new approach for practically solving problems that are theoretically intractable. For example, parameterized algorithms for the N P-hard problem vertex cover [9,13] have found applications in biochemistry [10], and variants thereof are applicable to problems arising in chip manufacturing [11,21,24], while parameterized algorithms in computational logic [35] have provided an effective method for solving practical instances of the ml type-checking problem, which is complete for the class exptime [30].…”

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confidence: 99%

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Fernau²,

et al. 2007

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Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving N P-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = N P, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363-384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.Key words. parameterized algorithm, planar graph, dominating set, vertex cover, independent set, kernel AMS subject classifications. 05C85, 68Q17 DOI. 10.1137/050646354 1. Introduction. Many practical algorithms for N P-hard problems start by applying data reduction subroutines to the input instances of the problem. The hope is that after the data reduction phase the instance of the problem has shrunk to a moderate size. This makes the applicability of a second phase, such as a branchand-bound phase, to the resulting instance more feasible. Weihe showed in [41] how a practical preprocessing algorithm for a variation of the dominating set problem, called the red/blue dominating set problem, resulted in breaking up input instances of the problem into much smaller instances. Langston, and Shanbhag [2], in their implementation of algorithms for the vertex cover problem,

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“…This property is a corollary of Henglein's subject invariance property of linear β-reductions for monomorphic program analyses [6][7][8]. Non-linear β-reductions, while they do preserve correctness of an analysis [17], are known not to preserve leastness.…”

Section: Related Workmentioning

confidence: 98%

CPS Transformation of Flow Information, Part II: Administrative Reductions

Damian¹,

Danvy

2002

BRICS

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We characterize the impact of a linear β-reduction on the result of a controlflow analysis. (By "a linear β-reduction" we mean the β-reduction of a linear λ-abstraction, i.e., of a λ-abstraction whose parameter occurs exactly once in its body.)As a corollary, we consider the administrative reductions of a Plotkin-style transformation into continuation-passing style (CPS), and how they affect the result of a constraint-based control-flow analysis and, in particular, the least element in the space of solutions. We show that administrative reductions preserve the least solution.

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The complexity of type inference for higher-order typed lambda calculi

Cited by 27 publications

References 34 publications

On the parameterized complexity of short computation and factorization

On the parameterized complexity of short computation and factorization

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

CPS Transformation of Flow Information, Part II: Administrative Reductions

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