Warszawa 199* Beta-Reduction as Unification Dedicated to the Memory of Professor Helena Rasiowa
Abstract. We define a new unification problem, which we call β-unification and which can be used ... more Abstract. We define a new unification problem, which we call β-unification and which can be used to characterize the β-strongnormalization of terms in the λ-calculus. We prove the undecidability of β-unification, its connection with the system of intersection types, and several of its basic properties.
We define a new unification problem, which we call β-unification and which can be used to charact... more We define a new unification problem, which we call β-unification and which can be used to characterize the β-strong normalization of terms in the λ-calculus. We prove the undecidability of β-unification, its connection with the system of intersection types, and several of its basic properties.
Addendum to "New Notions of Reduction and Non-Semantic Proofs of beta-Strong Normalization in Typed lambda-Calculi"
Background and Motivation" section to specify that we were discussing fi-SN proofs for type ... more Background and Motivation" section to specify that we were discussing fi-SN proofs for type systems more powerful than the simply-typed -calculus, the wording in the rest of the paper was not so careful and a comment which was intended to go at the beginning of Section 5 was placed at the beginning of Section 4 instead. As a result, the paper erroneously made false claims about the history of proving the fi-SN property for the simply-typed -calculus. Many readers have pointed out numerous methods for proving the fi-SN property for the simply-typed -calculus that are not variations on the methods of Tait or Girard. The methods mentioned have included proofs by Gandy, Nederpelt, Klop, de Groote, de Vrijer, van de Pol and Schwichtenberg [Gan80b, Ned73, Klo80, dG93, dV87, vdPS95]. Our intention was to discuss proofs of the fi-SN property for more powerful type systems such as system F,
Proceedings of the 1994 ACM conference on LISP and functional programming - LFP '94
We study the problem of type inference for a family of polymorphic type disciplines containing th... more We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the strati cation of the second-order lambda-calculus by \ r a n k " o f t ypes. We s h o w that typability is an undecidable problem at every rank k 3 of this strati cation. While it was already known that typability is decidable at rank 2, no direct and easy-to-implement algorithm was available. To design such an algorithm, we d e v elop a new notion of reduction and show h o w to use it to reduce the problem of typability at rank 2 to the problem of acyclic semi-uni cation. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-uni cation.
ACM SIGCOMM Computer Communication Review, Jul 1, 2004
The< i> science</i> of network service composition has emerged as one of the grand th... more The< i> science</i> of network service composition has emerged as one of the grand themes of networking research [17] as a direct result of the complexity and sophistication of emerging networked systems and applications. By" service composition" we mean that the performance and correctness properties local to the various constituent components of a service can be readily composed into global (end-to-end) properties without re-analyzing any of the constituent components in isolation, or as part of the whole ...
Interdomain routing on the Internet is performed using route preference policies specified indepe... more Interdomain routing on the Internet is performed using route preference policies specified independently and arbitrarily by each autonomous system (AS) in the network. These policies are used in the border gateway protocol (BGP) by each AS when selecting next-hop choices for routes to each destination. Conflicts between policies used by different ASs can lead to routing instabilities that, potentially, cannot be resolved regardless of how long BGP runs. The stable paths problem (SPP) is an abstract graph theoretic model of the problem of selecting next-hop routes for a destination. A solution to this problem is a set of next-hop choices, one for each AS, that is compatible with the policies of each AS. In a stable solution each AS has selected its best next-hop if the next-hop choices of all neighbors are fixed. BGP can be viewed as a distributed algorithm for solving an SPP instance. In this report we consider a family of restricted variants of SPP, which we call-SPP. We show that several natural variants of-SPP are NP-complete. This includes a variant in which each AS is restricted to one of only two policies, and each of these two policies is based on a monotonic path weight aggregation function. Furthermore, we show that for networks with particular topologies and edge weight distributions, there exist efficient centralized algorithms for solving-SPP.
The reassembling of a simple connected graph G = (V, E) is an abstraction of a problem arising in... more The reassembling of a simple connected graph G = (V, E) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps: (1) We cut every edge of G into two halves, thus obtaining a collection of n = V one-vertex components, such that for every v ∈ V the one-vertex component {v} has degree(v) half edges attached to it. (2) We splice the two halves of every edge together, not of all the edges at once, but in some ordering Θ of the edges that minimizes two measures that depend on the edge-boundary degrees of assembled components. A component A is a subset of V and its edge-boundary degree is the number of edges in G with one endpoint in A and one endpoint in V − A (which is the same as the number of half edges attached to A after all edges with both endpoints in A have been spliced together). The maximum edge-boundary degree encountered during the reassembling process is what we call the α-measure of the reassembling, and the sum of all edge-boundary degrees is its β-measure. The α-optimization (resp. β-optimization) of the reassembling of G is to determine an order Θ for splicing the edges that minimizes its α-measure (resp. β-measure). There are different forms of reassembling, depending on restrictions and variations on the ordering Θ of the edges. We consider only cases satisfying the condition that if the an edge between disjoint components A and B is spliced, then all the edges between A and B are spliced at the same time. In this report, we examine the particular case of linear reassembling, which requires that the next edge to be spliced must be adjacent to an already spliced edge. We delay other forms of reassembling to follow-up reports. We prove that α-optimization of linear reassembling and minimum-cutwidth linear arrangment (CutWidth) are polynomially reducible to each other, and that β-optimization of linear reassembling and minimum-cost linear arrangement (MinArr) are polynomially reducible to each other. The known NP-hardness of CutWidth and MinArr imply the NP-hardness of α-optimization and β-optimization.
Warszawa 199* Beta-Reduction as Unification Dedicated to the Memory of Professor Helena Rasiowa
Abstract. We define a new unification problem, which we call β-unification and which can be used ... more Abstract. We define a new unification problem, which we call β-unification and which can be used to characterize the β-strongnormalization of terms in the λ-calculus. We prove the undecidability of β-unification, its connection with the system of intersection types, and several of its basic properties.
We define a new unification problem, which we call β-unification and which can be used to charact... more We define a new unification problem, which we call β-unification and which can be used to characterize the β-strong normalization of terms in the λ-calculus. We prove the undecidability of β-unification, its connection with the system of intersection types, and several of its basic properties.
Addendum to "New Notions of Reduction and Non-Semantic Proofs of beta-Strong Normalization in Typed lambda-Calculi"
Background and Motivation" section to specify that we were discussing fi-SN proofs for type ... more Background and Motivation" section to specify that we were discussing fi-SN proofs for type systems more powerful than the simply-typed -calculus, the wording in the rest of the paper was not so careful and a comment which was intended to go at the beginning of Section 5 was placed at the beginning of Section 4 instead. As a result, the paper erroneously made false claims about the history of proving the fi-SN property for the simply-typed -calculus. Many readers have pointed out numerous methods for proving the fi-SN property for the simply-typed -calculus that are not variations on the methods of Tait or Girard. The methods mentioned have included proofs by Gandy, Nederpelt, Klop, de Groote, de Vrijer, van de Pol and Schwichtenberg [Gan80b, Ned73, Klo80, dG93, dV87, vdPS95]. Our intention was to discuss proofs of the fi-SN property for more powerful type systems such as system F,
Proceedings of the 1994 ACM conference on LISP and functional programming - LFP '94
We study the problem of type inference for a family of polymorphic type disciplines containing th... more We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the strati cation of the second-order lambda-calculus by \ r a n k " o f t ypes. We s h o w that typability is an undecidable problem at every rank k 3 of this strati cation. While it was already known that typability is decidable at rank 2, no direct and easy-to-implement algorithm was available. To design such an algorithm, we d e v elop a new notion of reduction and show h o w to use it to reduce the problem of typability at rank 2 to the problem of acyclic semi-uni cation. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-uni cation.
ACM SIGCOMM Computer Communication Review, Jul 1, 2004
The< i> science</i> of network service composition has emerged as one of the grand th... more The< i> science</i> of network service composition has emerged as one of the grand themes of networking research [17] as a direct result of the complexity and sophistication of emerging networked systems and applications. By" service composition" we mean that the performance and correctness properties local to the various constituent components of a service can be readily composed into global (end-to-end) properties without re-analyzing any of the constituent components in isolation, or as part of the whole ...
Interdomain routing on the Internet is performed using route preference policies specified indepe... more Interdomain routing on the Internet is performed using route preference policies specified independently and arbitrarily by each autonomous system (AS) in the network. These policies are used in the border gateway protocol (BGP) by each AS when selecting next-hop choices for routes to each destination. Conflicts between policies used by different ASs can lead to routing instabilities that, potentially, cannot be resolved regardless of how long BGP runs. The stable paths problem (SPP) is an abstract graph theoretic model of the problem of selecting next-hop routes for a destination. A solution to this problem is a set of next-hop choices, one for each AS, that is compatible with the policies of each AS. In a stable solution each AS has selected its best next-hop if the next-hop choices of all neighbors are fixed. BGP can be viewed as a distributed algorithm for solving an SPP instance. In this report we consider a family of restricted variants of SPP, which we call-SPP. We show that several natural variants of-SPP are NP-complete. This includes a variant in which each AS is restricted to one of only two policies, and each of these two policies is based on a monotonic path weight aggregation function. Furthermore, we show that for networks with particular topologies and edge weight distributions, there exist efficient centralized algorithms for solving-SPP.
The reassembling of a simple connected graph G = (V, E) is an abstraction of a problem arising in... more The reassembling of a simple connected graph G = (V, E) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps: (1) We cut every edge of G into two halves, thus obtaining a collection of n = V one-vertex components, such that for every v ∈ V the one-vertex component {v} has degree(v) half edges attached to it. (2) We splice the two halves of every edge together, not of all the edges at once, but in some ordering Θ of the edges that minimizes two measures that depend on the edge-boundary degrees of assembled components. A component A is a subset of V and its edge-boundary degree is the number of edges in G with one endpoint in A and one endpoint in V − A (which is the same as the number of half edges attached to A after all edges with both endpoints in A have been spliced together). The maximum edge-boundary degree encountered during the reassembling process is what we call the α-measure of the reassembling, and the sum of all edge-boundary degrees is its β-measure. The α-optimization (resp. β-optimization) of the reassembling of G is to determine an order Θ for splicing the edges that minimizes its α-measure (resp. β-measure). There are different forms of reassembling, depending on restrictions and variations on the ordering Θ of the edges. We consider only cases satisfying the condition that if the an edge between disjoint components A and B is spliced, then all the edges between A and B are spliced at the same time. In this report, we examine the particular case of linear reassembling, which requires that the next edge to be spliced must be adjacent to an already spliced edge. We delay other forms of reassembling to follow-up reports. We prove that α-optimization of linear reassembling and minimum-cutwidth linear arrangment (CutWidth) are polynomially reducible to each other, and that β-optimization of linear reassembling and minimum-cost linear arrangement (MinArr) are polynomially reducible to each other. The known NP-hardness of CutWidth and MinArr imply the NP-hardness of α-optimization and β-optimization.
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