Harry Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387-394, September 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 provably hard decision problems can be embedded in TLC, see [37, 33], and that different typings allow exponentiation [19] One way of expressing all of PTIME, while avoiding the anomalies associated with representations over Church numerals was recently demonstrated by Leivant and Marion [31] By augmenting the simply typed lambda calculus with a pairing ....

....Unlike circuit complexity, the size of the program computing parity is constant, because the iterative machinery is taken from the data, i.e. the list L. List iteration is a powerful programming technique, which can be used in the context of TLC and TLC = to encode any elementary recursion [37, 33]. However, some care is needed if one is to maintain well typedness [21] 3 Representing Databases and Queries 3.1 Databases as Lambda Terms Relations are represented in our framework as follows. Let O = fo 1 ; o 2 ; g be the set of constants of the TLC = calculus. For convenience, we ....

H. Mairson. A Simple Proof of a Theorem of Statman. Theoretical Comput. Sci., 103 (1992), pp. 387--394.

....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 TLC [28] and to code Ptime in TLC of ....

....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 TLC [28] and to code Ptime in TLC of order 5 [21] Simple types limit language uniform (but not program uniform or nonuniform) expressibility. This has provided motivation for examining more expressive typed calculi, such as the Girard Reynolds secondorder calculus [12, 34] polymorphism via type ....

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H. Mairson. A Simple Proof of a Theorem of Statman. Theoretical Computer Science, 103 (1992), pp. 387-- 394.

.... 5 Polymorphic Iteration and Expressiveness Theorems Given two # terms of size n that are both typable in System I k , how hard is it to decide if they have the same normal form In this paper, we omit many technical details relating to the complexity analysis of this decision problem; see [Sta79, Mai92, AM98]. Instead, we outline the analysis at a high level. The technical details are not di#cult, and amount to a fairly mundane form of functional programming. The above decision problem is a simple form of detecting program equivalence. We can use the polymorphic iteration lemma to get lower bounds on ....

....c, such that #M,x is true i# M accepts x. The Boolean universe D t c log # n is just big enough to code such a complex computation. Statman went on to show that in the simply typed # calculus, one could use # reduction to implement quantifier elimination for this logic, a point clarified in [Mai92], where the quantifier elimination uses a style of primitive recursion that is obvious to any functional programmer. As a consequence, deciding if two simply typed terms of size n (including type annotations) have the same normal form was not Kalmar elementary the decision problem was ....

H. G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387--394, Sept. 1992.

....with delta functions is undecidable. Some other extensions of matching are also known to be undecidable, e.g. matching in G odel s system T and Girard s system F [7] Statman is also the author of a theorem stating that deciding equality of simply typed lambda terms is not elementary recursive [21, 14]. This result can be immediately used to prove the same lower bound for higher order matching, as noted by Sergei Vorobyov [24] All the mentioned facts are summarized in Table 1. Statman [23] showed that any k th order unification problem containing constants of order not greater than m may be ....

....of the # reduction process cannot show off in solving that problem. However redexes may stay latent until searching for solutions begins. Thus it turns out that matching is at least as hard as testing equality of simply typed lambda terms, which is known to be non elementary: Theorem 1 (Statman [21, 14]) Given two simply typed closed lambda terms M and N, the second one in normal form. It is not elementary recursive to decide whether N is a normal form of M. This theorem immediately implies: Corollary 2 (Vorobyov [24] Higher order pure matching is not elementary recursive. Since terms M and ....

Harry G. Mairson, A simple proof of a theorem of Statman, Theoret. Comput. Sci., 103 (1992) 387-- 394.

....reducing all the new preliminary redexes 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 , ....

....2 y 2 , and x k 2 y k 1 . Finally, let a formula be built up out of prime formulas, the propositional connectives , and the quanti ers 8 and 9. Statman showed how to reduce the truth of a higherorder formula to the reduction to a given normal form of a suitable typed term. Mairson [Mai92] showed how to simplify the proof of Statman with a di erent encoding based on the same basic idea the quanti er elimination procedure but much simpler and easy to understand for the use of list iteration as a quanti er elimination procedure. Theorem 2 A higher order formula is true if and ....

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Harry G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387-394, September 1992.

.... 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 ....

....typed lambda calculus , known to be not elementary recursive (R. Statman [15] in fact requires time F (1; cn) exp 1 (cn) 2 2 Delta Delta Delta 2 oe cn ; as opposed to a far less impressive F (n; c log(log(n) 2 2 Delta Delta Delta 2 n oe c log(log(n) implicit in [15, 11]. Our improved lower bound exp 1 (cn) matches (with a different constant) with the known exp 1 (dn) upper bound for fi(j) equality in the simply typed lambda calculus due to W. Tait [18, 6] Moreover, the known exp 1 (cn) lower bound for any normalization algorithm in (H. Schwichtenberg [14] ....

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H. Mairson. A simple proof of a theorem of Statman. Theor. Comput. Sci., 103:387--394, 1992.

.... 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 is somehow too restrictive to bring this expressive CHAPTER 1. ....

....of the typed calculus. It has long been known that there is some hyperexponential connection between the order of terms and their expressive power, measured either as the ability to generate long reduction sequences [44] long normal forms [19] or simulate generic Turing machine computations [37]. We show that, unlike the somewhat confusing picture for functions over Church numerals, this connection can be made very precise for computations over finite structures: there is a natural correspondence between fixed order classes of terms and a hierarchy of query complexity classes ranging ....

[Article contains additional citation context not shown here]

H. Mairson. A Simple Proof of a Theorem of Statman. Theoretical Computer Sci., 103 (1992), pp. 387--394.

....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 ....

....) The variable x in the loop body x : ff: Succ serves to absorb the current element of L; the successor function is then applied to the accumulator value. List iteration is a powerful programming technique, which can be used in the context of TLC and TLC = to encode any elementary recursion [38, 43]. However, some care is needed if one is to maintain well typedness (cf. the type laundering technique of [25] 3 Representing Databases and Queries 3.1 Databases as Lambda Terms Relations are represented in our setting as simple generalizations of Church numerals. Let O = fo 1 ; o 2 ; ....

H. Mairson. A Simple Proof of a Theorem of Statman. Theoretical Comput. Sci., 103 (1992), pp. 387-- 394.

....mathematical interest, could then be shown to be computationally unnecessary. The means to this demonstration was a shift in the computation paradigm: instead of considering numerical computations, Paris suggested coding database queries. His suggestion was inspired by a lambda calculus encoding [Mai92] of the decision problem for higher order logic, which Paris proposed to extend to an encoding of the complex object algebra [AB88] a powerful database query language. We briefly discuss some relevant details. Quantified Boolean Formulas allows quantification only over Booleans, but can be ....

H. G. Mairson. A Simple Proof of a Theorem of Statman. Theoretical Computer Science 103:2 (September 1992), pp. 387--395.

....sets, tuples and lists, as well as the important concept of list iteration. In Section 5.2 we present, as an extended example, the embedding of the relational algebra. Our analysis of the expressive power of the simply typed lambda calculus is based on the analysis of reduction sequences of [40, 34]. We refer to [25] for the more advanced fixpoint and powerset features. Note that these features allow the embedding of higher order queries, e.g. the secondorder queries of [22, 17] Note that we only use the very core of programming language type systems, i.e. only simple types without any ....

H. Mairson. A Simple Proof of a Theorem of Statman. TCS, 103 (1992), pp. 387-- 394.

....in Core ML (or TLC) are all ELEMENTARY functions (where this class of functions includes PTIME, NP, PSPACE, EXPTIME, 2 EXPTIME, etc. see [41] 1 . We include some of the basic expressibility analysis in Section 3. For more details on the analysis of expressibility we refer the reader to [44, 45, 17, 37, 23, 24, 25]. The expressive power of TLC was originally analyzed in terms of computations on simply typed Church numerals (see, e.g. 4, 17, 44] Unfortunately, the simply typed Church numeral input output convention imposes severe limitations on expressive power. Only a fragment of PTIME is expressible ....

....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 ....

[Article contains additional citation context not shown here]

H. G. Mairson. A Simple Proof of a Theorem of Statman. TCS, 103 (1992), pp. 387--394.

.... Science 792 (Spring Term, 2000) Computational Logic Course instructor: Harry Mairson (mairson cs.bu.edu) MCS 283, phone 353 8926. Oce hours 11am 12pm Tuesday and Friday, and by arrangement. I especially encourage you to communicate with me via electronic mail, for fastest and most reliable responses to your questions. I try to read e mail every 5 minutes, 24 hours a day. Time ....

Harry Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science 103 (1992), pp. 387-394.

....of sharing. Our result then follows from Statman s theorem that deciding equivalence of typed terms is not elementary recursive [Sta79] We emphasize in Statman s theorem the generic simulation of timebounded computation. In particular, we stress the straightforward but powerful technology of [Mai92], where a functional 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 ....

.... 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 parallel steps. 2 In choosing this ....

[Article contains additional citation context not shown here]

Harry G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science 103 (1992), pp. 387-394.

....to true. 5 Polymorphic Iteration and Expressiveness Theorems Given two terms of size n that are both typable in System I k , how hard is it to decide if they have the same normal form In this paper, we omit many technical details relating to the complexity analysis of this decision problem; see [Sta79, Mai92, AM98]. Instead, we outline the analysis at a high level. The technical details are not difficult, and amount to a fairly mundane form of functional programming. The above decision problem is a simple form of detecting program equivalence. We can use the polymorphic iteration lemma to get lower bounds ....

....c, such that Phi M;x is true iff M accepts x. The Boolean universe D t c log n is just big enough to code such a complex computation. Statman went on to show that in the simply typed calculus, one could use fi reduction to implement quantifier elimination for this logic, a point clarified in [Mai92], where the quantifier elimination uses a style of primitive recursion that is obvious to any functional programmer. As a consequence, deciding if two simply typed terms of size n (including type annotations) have the same normal form was not Kalmar elementary the decision problem was dtime[K(t; ....

H. G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387--394, Sept. 1992.

.... The nonelementary lower bound for F type inference should immediately call to mind a wellknown theorem of Statman: the theorem states that if we have two terms typable in the first order typed lambda calculus, deciding whether 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 ....

H. G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science 103 (1992), pp. 387--394.

.... 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 computational overkill for lower complexity classes. ....

...., 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 ....

[Article contains additional citation context not shown here]

H. G. Mairson. A Simple Proof of a Theorem of Statman. Theoretical Comp. Sci., 103 (1992), pp. 387--394.

....of sharing. Our result then follows from Statman s theorem that deciding equivalence of typed terms is not elementary recursive [Sta79] We emphasize in Statman s theorem the generic simulation of timebounded computation. In particular, we stress the straightforward but powerful technology of [Mai92], where a functional 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 ....

.... 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 parallel fi steps. 2 In choosing this ....

[Article contains additional citation context not shown here]

Harry G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science 103 (1992), pp. 387--394.

....expressed in Core ML (or TLC) are all ELEMENTARY functions (where this class of functions includes PTIME, NP, PSPACE, EXPTIME, 2 EXPTIME etc, see [41] 4 . We include some of the basic expressibility analysis in Section 3. For more details on the analysis of expressibility we refer the reader to [44, 45, 17, 37, 23, 24, 25]. The expressive power of TLC was originally analyzed in terms of computations on simply typed Church numerals (see, e.g. 4, 17, 44] Unfortunately, the simply typed Church numeral input output convention imposes severe limitations on expressive power. Only a fragment of PTIME is expressible ....

....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 ....

[Article contains additional citation context not shown here]

H.G. Mairson. A Simple Proof of a Theorem of Statman. TCS, 103 (1992), pp. 387--394.

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Harry Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387-394, September 1992.

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Harry Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387-394, September 1992.

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H. G. Mairson. A simple proof of a theorem of Statman. Theoretical Computer Science, 103(2):387--394, Sept. 1992.

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