R. Statman. The typed #-calculus is not elementary recursive. Theoretical Computer Science, 9(1):73--81, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....There are estimations on the length of reduction sequences for various lambdacalculi (see [Bec01,Gan80,Sch82,Sch91] We adapt this to our purposes and argue that there is a computable upper bound on the size of a normal form of a term t. Note that there are also lower bounds for the complexity [Sta79,Bec01]. We need an estimation on the size of normal forms depending on the starting term. Let 2 0 (n) n and 2m (n) 2 2m 1 (n) for m 0. Let maxtypesize(t) be the maximal size of the types of subterms of t, i.e. maxtypesize(t) maxfsize( j 2 types(t)g. Lemma 2.15. Let t be a term. Then ....

Richard Statman. The typed -calculus is not elementary recursive. Theoretical Computer Science, 9:73-81, 1979.

....There are estimations on the length of reduction sequences for various lambdacalculi (see [Bec01,Gan80,Sch82,Sch91] We adapt this to our purposes and argue that there is a computable upper bound on the size of a normal form of a term t. Note that there are also lower bounds for the complexity [Sta79,Bec01]. We need an estimation on the size of normal forms depending on the starting term. Let 2 0 (n) n and 2m (n) 2 2m 1 (n) for m 0. Let maxtypesize(t) be the maximal size of the types of subterms of t, i.e. maxtypesize(t) maxfsize( j 2 types(t)g. Lemma 2.15. Let t be a term. Then ....

R. Statman. The typed -calculus is not elementary recursive. Theoretical Computer Science, 9:73-81, 1979.

....x are non terminating. We prove that the result of analysing a term involving x is equivalent to the result of analysing a certain nite unfolding of the term. Hence, all redexes predicted by the analysis will be met in a nite number of reduction steps. As a consequence of a theorem by Statman [30] the problem solved is nonelementary recursive, but we show that the analysis is decidable. While the analysis may not be of immediate practical interest, we believe that it gives a fundamental understanding of the nature of ow analysis. Thus, the analysis can be used both as a starting point in ....

....analysis for a higher order typed language with recursion. The analysis is proven to be exact: if the analysis predicts a redex, then there exists a reduction sequence such that this redex will be reduced. The analysis is decidable but the precision of the analysis implies (by a theorem by Statman [30]) that it is non elementary recursive. This is, however, no worse (or better) than the complexity of strictness analysis, and we believe that intersection based ow analysis (as strictness analysis) provides a good starting point for developing practical analyses. Finally, we believe that the ....

R. Statman. The typed -calculus is not elementary recursive. Theoretical Computer Science, 9(1):73-81, Juli 1979.

....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 ....

....Complexity of the higher order matching The cardinality of a type #, notation: card(#) is the number of its normal inhabitants, i.e. the number of closed terms in normal form of that type. A type # is inhabited if there exists a closed term of that type, i.e. if card(#) 0. Fact 1 (Statman [21, 20]) Consider lambda terms under an empty signature (i.e. without constants) Given a type #. 1. If order(#) 1, then card(#) 0. 2. If order(#) 2, then # = o k o for some k # 1, and card(#) k. 3. If order(#) # 3, then card(#) 0 or card(#) 1. 4. Moreover, a type # = # # ....

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Richard Statman, The typed #-calculus is not elementary recursive, Theoret. Comput. Sci., 9 (1979) 73--81.

....that allows us to see order in the graph reduction, it moreover serves as a lovely sort of optimizer that exchanges the work of parallel reduction for the work 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 ....

....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 ....

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Richard Statman. The typed -calculus is not elementary recursive. Theoretical Computer Science 9, 1979, pp. 73-81.

....algorithm (and hence with any optimal sharing graph algorithm) exceeds K (n) for any xed 0. The theorem shows that optimal reduction cannot be realized as a unit cost operation. However, it does not mean that optimal reduction, as a whole, is unfeasible. By a classical theorem of Statman [Sta79], any calculus machine has to take non elementary time in the size of a term to reach the normal form. In the case of optimal reduction, the non elementary bound (as a function of the number of shared reductions) is just a consequence of the fact that with this sharing technique the number of ....

....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 ....

Richard Statman. The typed -calculus is not elementary recursive. Theoretical Computer Science, 9:73-81, 1979.

....elimination corresponds exactly to normalisation. In N given a and an M one can nd relatively easily a proposition A such that M : A. Conversely, in L cf given the and the A one can nd relatively easily an M such that M : A. Still the complexity of this is PSPACE complete as shown by Statman [1979]. Using this approach, the Hauptsatz can be seen as a (canonical) consequence of the normalisation theorem for typeable lambda terms. Since all of this must have been known to several people, our contribution is mainly expository. But we have not seen the story told in this way. The emphasis is ....

....A; by 3.5. The proof can be depicted as follows. N ks L N L ;C N nf 3 L cf N cf L cf As it is clear that the proof implies that successive elimination of cuts leads to a method normalising terms typeable in N = the main result in Statman [1979] implies that the expense of such a procedure is beyond elementary time (Grzegorczyk class 4) Moreover, as a cut free derivation is of the same order of complexity as the corresponding normal lambda term, the size of a derivation after such a procedure may not be elementary in the size of the ....

Statman, R. [1979]. The typed -calculus is not elementary recursive, Theoretical Computer Science 9, pp. 73-81.

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R. Statman. The typed #-calculus is not elementary recursive. Theoretical Computer Science, 9(1):73--81, 1979.

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R. Statman. The typed -calculus is not elementary recursive. Theoretical Computer Science, 9(1):73--81, 1979.

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