A. R. Meyer. The inherent computational complexity of theories of ordered sets. In Proceedings of the International Congress of Mathematicians. , pp. 477--482, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on expressive power. Only a fragment of PTIME is expressible this way (called the extended polynomials) and this does not illustrate the full capabilities of TLC. That more expressive power is possible follows from the fact that provably hard decision problems can be embedded in TLC, see [39, 45, 37], and that different typings allow exponentiation [17] One way of expressing all of PTIME, while avoiding the anomalies associated with representations over Church numerals was recently demonstrated by Leivant and Marion [35] By augmenting the simply typed lambda calculus with a pairing ....

A. R. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. Proceedings of the International Congress of Mathematicians (1974), pp. 477--482.

....outputs. Under this nonuniform typed inputs convention Statman s theorem that deciding equivalence of normal forms of two simply typed terms is not elementary recursive [37] becomes an expressibility result. The proof in [37] uses Meyer s theorem on the complexity of higher order type theory [30]. For a simple proof of both see [28] Under this convention it is possible to have inputs that are finite structures containing an equality predicate on the domain of each structure. This makes it possible to interpret Statman s and Meyer s theorems as expressibility of the Elementary sets in ....

A. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. In Proceedings of the International Congress of Mathematicians, pp. 477-- 482, 1975.

....iteration lemma to get lower bounds on the problem by reworking certain technical machinery in the 8 This is k rather than 0 because PM,x contributes k to the minimal rank. analysis of a closely related problem, called the decision problem for type theory, due to Rick Statman and Albert Meyer [Mey74, Sta79]: Let D0 = true, false , and define D k 1 = powerset(D k ) Let x k , y k , z k be variables allowed to range over D k ; we define the prime formulas as x 0 , true # y 1 , false # y 1 , and x k # y k 1 . Now consider a formula # built up out of prime formulas, the ....

A. R. Meyer. The inherent computational complexity of theories of ordered sets. In Proceedings of the International Congress of Mathematicians. , pp. 477--482, 1974.

....programming implementation of quanti er elimination for higher order logic over a nite base type is employed to simulate arbitrary Kalm ar elementary time bounded computation. That the decision problem for this higher order logic has nonelementary complexity was originally proven by Meyer [Mey74]. It is very easy to give a brief description of the proof of our lower bound, if the reader has a nodding familiarity with sharing graphs. We de ne the Kalm ar elementary functions K (n) as K 0 (n) n, and K t 1 (n) 2 K t (n) Kal43] 2 Main Theorem. Let 0 be any xed integer. Then ....

....Section 3 a description of the expansion method. Section 4 shows how to describe succinctly generic elementary time bounded computation in higher order logic, and how to compile expressions in this logic into short typed terms these comprising the essence of the theorems of Statman and Meyer [Sta79, Mey74], as fundamentally reconstructed in [Mai92] Section 5 contains the main results of the paper. Finally, for those interested in the algorithmics of Lamping s technology, Section 6 describes the basic graph constructions involving sharing nodes that allow huge computations to be simulated by so few ....

[Article contains additional citation context not shown here]

Albert R. Meyer. The inherent computational complexity of theories of ordered sets. Proceedings of the International Congress of Mathematicians, 1974, pp. 477-482.

....and propagating all sharing nodes to the base type. Theorem 1 For any simply typed term M , the total number of shared reductions in the graph normalization of (M) is limited by the size of (M) The second ingredient is obtained from Mairson s proof [Mai92] of theorems of Statman and Meyer [Sta79, Mey74]. De ne D1 = ftrue; falseg, and Dk 1 = powerset(Dk ) The decision problem for propositional calculus can be naturally generalized to higher order types by allowing variables and quanti ers to range over values of Dk , for k 1. Let x k , y k , z k be variables ranging over Dk ; we de ne ....

Albert R. Meyer. The inherent computational complexity of theories of ordered sets. In Proceedings of the International Congress of Mathematicians, pages 477-482, 1974.

....iteration lemma to get lower bounds on the problem by reworking certain technical machinery in the 8 This is k rather than 0 because PM;x contributes k to the minimal rank. analysis of a closely related problem, called the decision problem for type theory, due to Rick Statman and Albert Meyer [Mey74, Sta79]: Let D 0 = ftrue; falseg, and define D k 1 = powerset(D k ) Let x k ; y k ; z k be variables allowed to range over D k ; we define the prime formulas as x 0 , true 2 y 1 , false 2 y 1 , and x k 2 y k 1 . Now consider a formula Phi built up out of prime formulas, the usual ....

A. R. Meyer. The inherent computational complexity of theories of ordered sets. In Proceedings of the International Congress of Mathematicians., pp. 477--482, 1974.

.... Hardest Natural Decidable Theory y Sergei Vorobyov Max Planck Institut fur Informatik Im Stadtwald, Saarbrucken, D 66123, Germany sv mpi sb.mpg.de Abstract We prove that any decision procedure for a modest fragment of L. Henkin s theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2 s of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2 s and since mid seventies it was an open problem whether natural decidable theories requiring more ....

....exceeding a tower of 2 s of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2 s and since mid seventies it was an open problem whether natural decidable theories requiring more than that exist [12, 2]. We give the affirmative answer. As an application of this today s strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus. 1. Introduction In his survey paper [12] A. Meyer mentioned (p. 479) as a curious empirical ....

[Article contains additional citation context not shown here]

A. R. Meyer. The inherent computational complexity of theories of ordered sets. In International Congress of Mathematicians, pages 477--482, Vancouver, 1974.

.... the terms have the same normal form requires nonelementary time [Sta79, Mai92b] The proof of Statman s theorem is a reduction from deciding the truth of expressions in higher order logic, where quantification is allowed not only over Boolean values, but over higher order functions over Booleans [Mey74]. Every formula in higher order logic is transformed, using the reduction, into a term that fi reduces to the standard term t: oe:f : oe:t coding true if and only if the formula is true, and otherwise to the term t: oe:f : oe:f coding false. We wish to emphasize both the abstract structural ....

....the polynomial time hierarchy, and in the limit, the problem of quantified boolean formulas, complete for polynomial space [SM73, GJ79] Finally, if we allow quantification over functions of Booleans, functions of functions of Booleans, etc. we get a problem complete for nonelementary time [Mey74, Sta79]. The theorems described in this paper follow much the same pattern. First order unification is complete for polynomial time [DKM84] corresponding to the complexity of first order type inference. The progressively stronger lower bounds in this paper are derived by similarly allowing greater and ....

A. R. Meyer. The inherent computational complexity of theories of ordered sets. Proceedings of the International Congress of Mathematicians, 1974, pp. 477--482.

....problems can be embedded into TLC. This follows from a theorem of Statman that deciding equivalence of normal forms of two well typed terms is not elementary recursive [43] The proof in [43] uses a result of Meyer concerning the complexity of decision problems in higher order type theory [37]. A simple proof of both these results appears in [35] However, there are a number of difficulties when one tries to turn these proofs into frameworks for computations. They do not separate the fixed program (representing a function) from the variable data (representing the input) They use ....

...., in the sense of expressing queries of interest. The additional requirement (3) is important if one wishes to consider the typed calculus as a functional database query language operating by reduction. We call embeddings that satisfy (1 3) PTIME embeddings . It is implicit in the literature [35, 37, 43] that, under our input output conventions but without considering an efficient reduction strategy, all elementary functions are expressible (where this class of functions includes PTIME, NP, PSPACE, EXPTIME, k EXPTIME, etc. 40] For all practical purposes, ELEMENTARY is a powerful complexity ....

A. R. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. In Proceedings of the International Congress of Mathematicians, pp. 477--482, 1975.

....University, Waltham, Massachusetts. Supported in part by NSF Grant CCR9017125, and grants from Texas Instruments and from the Tyson Foundation. This paper appeared in Theoretical Computer Science 103 (1992) pp. 387 394. As shown by Meyer, this decision problem requires nonelementary time [Mey74]. Statman s theorem is a reduction to this problem: it shows how to use typed lambda calculus to simulate the logical connectives as well as a quantifier elimination procedure to reduce Phi, in the logical and calculus sense, to either true or false. We indicate how list iteration is a ....

A. R. Meyer. The inherent computational complexity of theories of ordered sets. Proceedings of the International Congress of Mathematicians, 1974.

....hard decision problems can be embedded into the typed calculus. For example, Statman [45] shows that deciding whether two typed terms have the same normal form is at least as hard as deciding the truth of a formula in higher order type theory, a decision problem known to be non elementary [39]. Mairson s instructive re proof of Statman s and Meyer s results (in [37] actually shows how to simulate any elementary time Turing machine computation using typed terms. Thus, it appears that the typed calculus can express quite powerful computations, but that the domain of Church numerals ....

A. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. In Proceedings of the International Congress of Mathematicians, pp. 477--482, 1975.

....severe limitations on expressive power. Only a fragment of PTIME is expressible this way (i.e. the extended polynomials) This does not illustrate the full capabilities of TLC. That more expressive power is possible follows from the fact that hard decision problems can be embedded in TLC, see [38, 39, 43], and that different typings allow exponentiation [18] However, very few connections have been established between complexity theory and the calculus. One such connection was recently demonstrated by Leivant and Marion [36] who express all of PTIME while avoiding the anomalies associated with ....

A. R. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. In Proceedings of the International Congress of Mathematicians, pp. 477--482, 1975.

.... nonelementary complexity (not solvable in any 2 Given a closed propositional formula F where each variable X is 8 or 9 bound, is F true under the naive interpretation fixed stack of exponentials) and can be used to show that deciding equivalence of two typed lambda terms is nonelementary [Mey74, Sta79]. In the related complex object algebra, quantification ranges over atomic constants, sets, and tuples, expressing exactly the generic elementary database queries that are computable in some fixed stack of exponentials. By implementing the complex objects algebra in the simply typed lambda ....

A. R. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. Proceedings of the International Congress of Mathematicians, pp. 477--482, 1975.

....programming implementation of quantifier elimination for higher order logic over a finite base type is employed to simulate arbitrary Kalm ar elementary time bounded computation. That the decision problem for this higher order logic has nonelementary complexity was originally proven by Meyer [Mey74]. It is very easy to give a brief description of the proof of our lower bound, if the reader has a nodding familiarity with sharing graphs. Definition 1.2 We define the Kalm ar elementary functions K (n) as K 0 (n) n, and K t 1 (n) 2 K t (n) Kal43] Dually, we define the iterated ....

....Section 3 a description of the j expansion method. Section 4 shows how to describe succinctly generic elementary time bounded computation in higher order logic, and how to compile expressions in this logic into short typed terms these comprising the essence of the theorems of Statman and Meyer [Sta79, Mey74], as fundamentally reconstructed in [Mai92] Section 5 contains the main results of the paper. Finally, for those interested in the algorithmics of Lamping s technology, Section 6 describes the basic graph constructions involving sharing nodes that allow huge computations to be simulated by so few ....

[Article contains additional citation context not shown here]

Albert R. Meyer. The inherent computational complexity of theories of ordered sets. Proceedings of the International Congress of Mathematicians, 1974, pp. 477--482.

....on expressive power. Only a fragment of PTIME is expressible this way (called the extended polynomials) and this does not illustrate the full capabilities of TLC. That more expressive power is possible follows from the fact that provably hard decision problems can be embedded in TLC, see [39, 45, 37], and that different typings allow exponentiation [17] One way of expressing all of PTIME, while avoiding the anomalies associated with representations over Church numerals was recently demonstrated by Leivant and Marion [35] By augmenting the simply typed lambda calculus with a pairing operator ....

A. R. Meyer. The Inherent Computational Complexity of Theories of Ordered Sets. Proceedings of the International Congress of Mathematicians, 1974, pp. 477--482.

No context found.

A. R. Meyer. The inherent computational complexity of theories of ordered sets. In Proceedings of the International Congress of Mathematicians. , pp. 477--482, 1974.

No context found.

A.R. Meyer. The inherent computational complexity of theories of ordered sets. In International Congress of Mathematicians, pages 477--482, 1974.

No context found.

M Albert R. Meyer, "The Inherent Computational Complexity of Theories of Ordered Sets", in Proc. 1974 Intl. Cong. of Mathematicians, Vancouver, B.C., Canada (1974), 477--482.

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