Martin-Lo# f, P. (1968), Notes on Constructive Mathematics," Almqvist 6 Wiksell, Stockholm. 140 FRANK PFENNING  Home/Search   Document Not in Database   Summary   Related Articles   Check

....available online at http:##www.idealibrary.com on 84 0890 5401#00 #35.00 Copyright # 2000 by Academic Press All rights of reproduction in any form reserved. 1 This work was supported by NSF Grant CCR 9303383. Many proofs of cut elimination have been given in the literature (see, for example, ML68, Sch77, Dra87, Her95] Yet, to our knowledge, none of them have been formalized even though this is clearly possible in principle (see, for example, Matthews [Mat94] pencil and paper analysis of cut elimination for the ( 6 , c) fragment of classical propositional logic in FS 0 ) They are ....

....sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is Martin Lo# f s proof of admissibility [ML68] In Martin Lo# f s system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this ....

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Martin-Lo# f, P. (1968), Notes on Constructive Mathematics," Almqvist 6 Wiksell, Stockholm. 140 FRANK PFENNING

.... TO DEFINE MEASURE OF BOREL SETS THIERRY COQUAND Introduction In [12] is presented a constructive definition of Borel subsets of Cantor space Omega and their inclusion relation. They are seen as symbolic expressions built from the Boolean algebra of simple (closed and open) subsets of Omega by formal countable disjunction and conjunction. If X is such a ....

....to define by induction the formal complement X 0 of X: Classically one can think of symbolic expressions as sets of points, and define the inclusion relation extensionally. Constructively, it is still possible to define X Y for X; Y Borel subsets without mentioning points, and this is done in [12] using a suitable infinitary one sided sequent calculus. Using this approach the law Omega X [ X 0 for instance can be justified constructively. A theory of Lebesgue measure on Omega is also presented in [12] starting from a measure (b) 2 [0; 1] of simple (closed and open) subsets. ....

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P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiksell, Stockholm, 1968.

....i.e. one in which the various data structures (e.g. integers, reals, arrays, lists, trees, etc) and the arities of the procedures (e.g. lists to integers) are represented as types. Rich type structures, higher order logic and constructive proofs are combined in constructive type theories, Martin Lof, 1970 ] There are now a number of theorem provers based on constructive type theories, e.g. Coq [ Dowek et al. 1991 ] NUPRL [ Constable et al. 1986 ] LEGO [ Luo and Pollack, 1992 ] ALF, Augustsson et al. 1990 ] Due to the di#culty of automating interesting proofs in some of these logics, ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiksell, Stockholm, 1970.

....numbers as sequence of finite B adic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers Unlike Bishop (Bishop and Bridges, 1985) Martin Lof (Martin Lof, 1970) and Stolzenberg (Stolzenberg, 1987) we use here the classical real analysis as the framework to state properties of computable real numbers y , as in Rice fundamental paper (Rice, 1954) In this section we will use the Godel Kleene representation of recursive functions as recursive functions ....

Martin-Lof, P. (1970). Notes on Constructive Mathematics. Almqvist et Wiksell, Stockholm.

....Boolean algebra B there exists a unique oe algebra morphism f : BA B such that f ffi i = f: Proof. What is not direct in this statement is that the map i : A BA is an embedding, i.e. satisfies i(x) i(y) iff x y: For this, we can adapt the explicit description of BA presented in [11] in the special case where A = C (cf. Example 1.3) We consider infinitary propositional formulae over A OE = n OE n j n OE n j a 2 A We define negation :OE of formulae recursively using the de Morgan laws. We let sequents be finite set of formulae and we have the following derivation rules ....

.... a 1 Delta Delta Delta a n = 1 a 1 ; a n ; Gamma W n OE n 2 Gamma OE n 0 ; Gamma Gamma Delta Delta Delta OE n ; Gamma Delta Delta Delta V n OE n ; Gamma The main lemma is then that Gamma follows from Gamma; OE and Gamma; OE (cut elimination, [11]) We define recursively [ OE] 2 BA , by taking [ a] i(a) and, using cut elimination, we can show that OE 1 ; OE n iff [ OE i ] 1 2 BA : Notice that the inclusion i : A A 1 however does not preserve, in general, countable sup that may exist in A: The importance of this ....

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P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiksell, Stockholm, 1968.

....= s i , for all i and C(x) 1 if x 62 fx i j i 2 Ng. 1 2 Computable and Uncomputable Reals The complexity of real numbers is a central topic in classical computability theory (see Turing [53] Rice [43] Calude [6] Soare [47] Odifreddi [39] Bridges [5] computable analysis (see Martin L of [38], Weihrauch [55] Pour El and Richards [42] Ko [34] Bridges [4] algorithmic information theory (see Chaitin [23, 25, 26] Martin L of [36] Calude [7] and information based complexity (see Traub, Wasilkowski, and Wo zniakowski [52] A important class of reals is certainly the set of ....

P. Martin-Lof. Notes on Constructive Mathematics, Almqvist & Wiksell, Stockholm, 1967.

.... used in Ko and Friedman n [12] 13] Turing [32] Wiedmer [36] A theory of computability of Baire spaces called type 2 recursion theory is used to study computability on real number in Kreitz and Weihrauch [14] Finally approximation spaces are introduced and used in Lacombe [15] Martin Lof [16], D. Scott [26] 26] Weihrauch [34] 1.1 Motivations for this work and results obtained. Although the theory of computable Analysis can be considered a well developed subject, there have been so far very few attempts of implementing computable Analysis on digital computers [6] 7] 10] ....

.... i) the end points a i ; b i of each interval s i are rational numbers, that is s i is a rational interval, ii) s i 1 s i , iii) lim i 1 (b i Gamma a i ) 0 iv) r = T i2IN s(i) This representation has been used by several authors which dealt with the 6 real number computability [15] [16], 26] 34] It can be considered for many aspects, the general form of real representation. Many other representations proposed in the literature differ from this one only in that they make use of a subset of the convergent sequences of rational intervals. Here are some examples: Definition 2 ....

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P. Martin-Lof, "Note on Constructive Mathematics." Almqvist and Wiksell, Stockholm (1970).

....proof, avoiding ordinals, from the ordinal theoretic proof in Rathjen and Weiermann 1993, see Seisenberger 2000. She restricts herself to the case of decidable relations on N. Another di erence with the present paper is that she avoids Brouwer s Thesis, thereby following a line recommended by P. Martin L of, see Martin L of 1970. Rather than invoking Brouwer s Thesis one might de ne a relation R to be almost full or unavoidable if and only if there exists a stump such that every nite sequence of natural numbers not belonging to meets R. This of course is a di erence in style mainly, the ....

....ordinals, from the ordinal theoretic proof in Rathjen and Weiermann 1993, see Seisenberger 2000. She restricts herself to the case of decidable relations on N. Another di erence with the present paper is that she avoids Brouwer s Thesis, thereby following a line recommended by P. Martin L of, see Martin L of 1970. Rather than invoking Brouwer s Thesis one might de ne a relation R to be almost full or unavoidable if and only if there exists a stump such that every nite sequence of natural numbers not belonging to meets R. This of course is a di erence in style mainly, the problem of how to prove ....

P. Martin-Lof (1970), Notes on Constructive Mathematics, (Almquist and Wiksell, Stockholm).

....objects. What follows is a brief survey of some of them. Some statements in combinatorics use infinite objects in their formulation. In some cases, however, the meaning of the statement can be expressed with inductive definitions yielding powerful proof principles. This was proposed already in [ML68] where the meaning of phrases 8m 1 m 2 : m n : 9nP (m 1 m 2 : m n ) for P a decidable property of finite sequences of natural numbers) is analyzed as being given by P bars [ where P bars l is defined inductively by P (l) is true P bars l P bars l:m for all natural numbers m ....

Per Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, 1968.

....Boolean algebra B there exists a unique oe algebra morphism f : BA B such that f ffi i = f: Proof. What is not direct in this statement is that the map i : A BA is an embedding, i.e. satisfies i(x) i(y) iff x y: For this, we can adapt the explicit description of BA presented in [11] in the special case where A = C (cf. Example 1.3) We consider infinitary propositional formulae over A OE = n OE n j n OE n j a 2 A We define negation :OE of formulae recursively using the de Morgan laws. We let sequents be finite set of formulae and we have the following derivation rules ....

.... a 1 Delta Delta Delta a n = 1 a 1 ; a n ; Gamma W n OE n 2 Gamma OE n 0 ; Gamma Gamma Delta Delta Delta OE n ; Gamma Delta Delta Delta V n OE n ; Gamma The main lemma is then that Gamma follows from Gamma; OE and Gamma; OE (cut elimination, [11]) We define recursively [ OE] 2 BA , by taking [ a] i(a) and, using cut elimination, we can show that OE 1 ; OE n iff [ OE i ] 1 2 BA : Notice that the inclusion i : A A 1 however does not preserve, in general, countable sup that may exist in A: The importance of this ....

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P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiksell, Stockholm, 1968.

.... We recall [Halmos] that a oe complete Boolean algebra is a Boolean algebra A with an infinitary operation W x n 2 A for (x n ) sequence of elements of A such that W x n is a least upper bound of the family x n in A : we have x n W x n for any n and if x n y for all n; then W x n y: In [Martin Lof] it is explained how to build the free oe complete Boolean algebra A 1 ; i : A A 1 over a given Boolean algebra A (also called the oe completion of A) in the special case where A is the Boolean algebra of closed open subset of Cantor space. This Boolean algebra is characterised by the following ....

.... if B is any oe complete Boolean algebra and f : A B any morphism of Boolean algebras, then there is one and only one morphism f : A 1 B (preserving countable suprema) such that f ffi i = f: Furthermore it can be shown, using for instance the explicit construction of the oe completion in [Martin Lof], that x y in A iff i(x) i(y) in A 1 so that A may be considered as a subalgebra of A 1 1 . In general we can by Stone duality consider elements of A as symbols for closed open subsets of the space X of ultrafilters of A [Halmos] It follows from Loomis representation theorem [Halmos] that ....

P. Martin-Lof. Notes on Constructive Mathematics. Almquist and Wiskell, Stockholm, 1968. 3

.... We recall [Halmos] that a oe complete Boolean algebra is a Boolean algebra A with an infinitary operation W x n 2 A for (x n ) sequence of elements of A such that W x n is a least upper bound of the family x n in A : we have x n W x n for any n and if x n y for all n; then W x n y: In [Martin Lof] it is explained how to build the free oe complete Boolean algebra A 1 ; i : A A 1 over a given Boolean algebra A (also called the oe completion of A) in the special case where A is the Boolean algebra of closed open subset of Cantor space. This Boolean algebra is characterised 1 In the first ....

.... if B is any oe complete Boolean algebra and f : A B any morphism of Boolean algebras, then there is one and only one morphism f : A 1 B (preserving countable suprema) such that f ffi i = f: Furthermore it can be shown, using for instance the explicit construction of the oe completion in [Martin Lof], that x y in A iff i(x) i(y) in A 1 so that A may be considered as a subalgebra of A 1 3 . In general we can by Stone duality consider elements of A as symbols for closed open subsets of the space X of ultrafilters of A [Halmos] It follows from Loomis representation theorem [Halmos] that ....

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P. Martin-Lof. Notes on Constructive Mathematics. Almquist and Wiskell, Stockholm, 1968. 5

....i.e. one in which the various data structures (e.g. integers, reals, arrays, lists, trees, etc) and the arities of the procedures (e.g. lists to integers) are represented as types. Rich type structures, higher order logic and constructive proofs are combined in constructive type theories, Martin Lof, 1970 ] There are now a number of theorem provers based on constructive type theories, e.g. Coq [ Dowek et al. 1991 ] NUPRL [ Constable et al. 1986 ] LEGO [ Luo and Pollack, 1992 ] ALF, Augustsson et al. 1990 ] Due to the difficulty of automating interesting proofs in some of these ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiksell, Stockholm, 1970.

....that the measure does not depend on the presentation of the set. 3 1.2.1 Post Canonical Systems The canonical systems were invented by Post [32] in an attempt to find the most general form of a formal system. An economical presentation is given in Martin Lof s notes on constructive mathematics [21]. As shown for instance in Lorenzen s of Martin Lof s book [20, 21] the use of canonical Post systems allows for a short and elegant presentation of the basic notions of recursivity. 1.2.2 Tarski s Truth Definition Natural examples of inductive definitions appeared in Tarski s work on truth ....

....3 1.2.1 Post Canonical Systems The canonical systems were invented by Post [32] in an attempt to find the most general form of a formal system. An economical presentation is given in Martin Lof s notes on constructive mathematics [21] As shown for instance in Lorenzen s of Martin Lof s book [20, 21], the use of canonical Post systems allows for a short and elegant presentation of the basic notions of recursivity. 1.2.2 Tarski s Truth Definition Natural examples of inductive definitions appeared in Tarski s work on truth definitions. First, one needed to give a precise definition of the ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm, 1968.

....be needed if we can to capture the case X = 0; 1] or X = 0; 1] 2 : We prove now the existence of an invariant measure. But we should first precise what we mean by a measure on a compact space, since, in a constructive framework, it is not always possible to assign a measure to any open sets [5]. Even if we consider only a basis of open sets of a compact space, it does not seem possible to assign constructively a measure to them. Instead, the right notion of measure seems to be of an integral that is a continuous linear functional If defined for f 2 C(X) set of continuous function over ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell ed., 1968.

....Metamathematical Analysis 2.3.1 Post Canonical Systems The canonical systems were invented by Post [26] in an attempt to find the most general form of a formal system, such as for example Principia Mathematica. An economical presentation is given in Martin Lof s notes on constructive mathematics [16]. As shown for instance in [15, 16] the use of canonical Post systems allows for a short and elegant presentation of the basic notions of recursivity. 2.3.2 Tarski s Truth Definition Natural examples of inductive definitions appeared in Tarski s work on truth definitions. First, one needed to ....

....Post Canonical Systems The canonical systems were invented by Post [26] in an attempt to find the most general form of a formal system, such as for example Principia Mathematica. An economical presentation is given in Martin Lof s notes on constructive mathematics [16] As shown for instance in [15, 16], the use of canonical Post systems allows for a short and elegant presentation of the basic notions of recursivity. 2.3.2 Tarski s Truth Definition Natural examples of inductive definitions appeared in Tarski s work on truth definitions. First, one needed to give a precise definition of the ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm, 1968.

....for being the less than relation on N. We want to express in type theory, extended with inductive definitions, what it means for a relation R over a set B to be well. To this end, we define GoodR (b 0 Delta Delta Delta b m ) to be 9i j m: b i R b j . We use an inductive definition of bar [ML68] to express that for any infinite sequence b 0 b 1 Delta Delta Delta, GoodR (oe) will eventually hold for an initial segment oe of b 0 b 1 Delta Delta Delta. Definition 6. Given a set B and a predicate P over the lists of B, we define inductively when the predicate P bars oe, written P j ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, 1968.

.... Theorem 1: C 0 is an ideal of C: 1 The fact that any continuous map [N F 2 ] are of the form OE(x) for one x 2 C can be proved using a classical metalanguage, or using some form of bar induction [18] 2 This is similar to cut elimination results for infinitary logic; see for instance [13]. 2 Proof: Using lemma 1, it is enough to show that C 0 is closed by binary sup. If x 2 C 0 and y 2 C 0 then 1x = x 2 C 0 and (x y) 1 Gamma x) y 2 C 0 . By lemma 2, we have that x y = 1(x y) 2 C 0 : Corollary: x Gamma x 2 C 0 for all x: Proof: We have directly y = x(1 Gamma x ) ....

....use of Boolean cover in topos theory in the proof of Barr s theorem [3, 14] Conclusion We think that the result of theorem 2, that C 0 is an ideal, is interesting in itself. Notice that its proof is quite close to the proof of cut elimination of infinitary propositional logic (see for instance [13]) On the other hand, as we already noticed, it can be seen as a reformulation of the classical clopen Ramsey s theorem [8] and provide a direct constructive proof of this result. We find it remarkable that this formulation comes from a problem that seems at first quite distinct, which was to ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiskell, Stockholm, 1970

....The starting motivation of this work was a comparison between the work of Tait Schutte and the work of Lorenzen Novikov on cut elimination. From a constructive perspective, the analysis of sequent calculus is most elegantly expressed in Lorenzen Novikov s way, as the admissibility of the cut rule [2, 3]. Tait Schutte s approach introduces two extra parameters ff ae Gamma which measures the depth of the proof tree and the maximal degree of the cut in the proof (see for instance [1] While this may be extremely interesting for an evaluation of the complexity of the cut elimination processes, ....

....since our metalanguage is constructive. 2 Proof: This means that we have Gamma; A; Delta; A Gamma; Delta; the proof is similar to the usual proof of admissibility of cut, by doing an induction first on the cut formula A, and then on the proofs of Gamma; A and Delta; A (see [2, 3]) The following lemma is a kind of partial inversion for the introduction rule of B(oe) Lemma 2: If Gamma; B(oe) in (S 0 ) and Gamma is positive then Gamma; G ff (oe) in (S 0 ) for one ff: Proof: In the case of the closure rule, we have by induction hypothesis a sequence ff n such ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell ed., 1968

....theorems for being the less than relation on N. We want to express in type theory, extended with inductive definitions, what it means for a relation R over a set B to be a wqr. To this end, we define GoodR (b 0 ; b m ) to be 9i j m: b i R b j . We use an inductive definition of bar [ML68] to express that for any infinite sequence b 0 ; b 1 ; GoodR (oe) will eventually hold for an initial segment oe of b 0 ; b 1 ; Definition 2.4 Given a predicate R on a set B, we define inductively when the predicate GoodR bars oe, written Good R j oe: GoodR (oe) Good R j oe 8b: ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, 1968.

....due to the presence of roundoff errors. Moreover, they are inappropriate for problems whose solution is sensitive to small variations on the input. As a consequence, exact real number computation has been advocated as an alternative solution (see e.g. 4, 5, 25] on the practical side and e.g. [2, 20, 21, 24, 26, 27, 28] on the foundational side) However, work on exact real number computation has focused on representations of real numbers and has neglected the issue of data types for real numbers. In particular, programming languages for exact real number computation with an explicit distinction between ....

....computable w.r.t. decimal representation. In fact, any base has essentially the same problem [28] Let us consider binary expansions of numbers in the unit interval. In this case, a solution for the above problem is to allow the digit 1 2 in addition to the digits 0 and 1. According to Martin Lof [21], this kind of solution goes back to Brouwer. For an 2 f0; 1 2 ; 1g, the sequence a 1 a 2 Delta Delta Delta an Delta Delta Delta represents the number X n1 an 2 Gamman : Therefore the operations a 1 a 2 Delta Delta Delta an Delta Delta Delta 7 0 a 1 a 2 Delta Delta Delta ....

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P. Martin-Loef. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm, 1970.

....and the universality result follows. In order to obtain our definability results, we consider a domain equation like structure on the real numbers data type. The Real PCF notion of computability induces classical notions of computability on real numbers and real valued functions of real variables [3, 11, 12, 16, 19, 20], but this material is not included in this extended abstract due lack of space. Also, we only consider the unit interval type of Real PCF, although we indicate how the type for the whole real line can be handled. Contents 1 The real numbers domains 2 2 Effectively given coherent domains 5 3 ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm, 1970.

....giving better and better partial results converging to its value. Keywords: Exact real number computation. Domain Theory and Interval Analysis. Denotational and operational semantics. 1 Introduction There are several practical and theoretical approaches to exact real number computation (see e.g. [7 9,18,21,24,26,30,31,37,39,40,42]) However, the author is not aware of any attempt to give denotational and operational semantics to an implementable programming language with a data type for exact real numbers. Most approaches to exact real number computation are based on representation of real numbers in other data types, such ....

P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm, 1970.

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Martin-Lof, P., Notes on Constructive Mathematics, Almqvist & Wiksell, 1968.

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P. Martin-Lof. Notes on Constructive Mathematics. Almqvist and Wiskell, Stockholm, 1970

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