Feferman, S., Constructive theories of functions and classes, Logic Colloquium '78, M. Boffa et al (eds), pp 159-224, North Holland, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in the conclusion of [27] This further characterization of the basic feasible functionals underpins their importance as a key candidate for the notion of type two feasibility. 1 Introduction In this paper we deal with applicative theories in the spirit of Feferman s explicit mathematics (cf. [12, 13]) The paper is a successor to Strahm [27] cf. also [28] where so called bounded applicative theories with a strong relationship to classes of computational complexity have been introduced and analyzed. For a more detailed background on applicative theories, we refer the reader to [27] and the ....

....models of the combinatory axioms, which can easily be extended to models of PT. These include further recursiontheoretic models, term models, continuous models, generated models, and set theoretic models. For detailed descriptions and results the reader is referred to Beeson [2] Feferman [13], and Troelstra and van Dalen [30] 4 Bff in PT In this section we first clarify the notion of a provably total type two functional in a given applicative theory. Then we show that the basic feasible functionals of type two are provably total in PT. Indeed, this result already Actually, ....

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Bo#a, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224.

....92a] moreover can be extended in a simple way to provide a type checker for the present system. The system of [KK93] interprets an extended fragment of natural language where self reference and nominalisation are allowed. Martin Lof s type theory moreover in [Martin Lof 73] and Feferman s T 0 in [Feferman 79] present systems which have been extensively used in pl. Those systems too are related to the present one as we shall see below. accommodates most of the systems mentioned above, hence bringing in all the advantages. We will use oe and for types (two special instances of which are e the type ....

Feferman, S., Constructive theories of functions and classes, Logic Colloquium '78, M. Boffa et al (eds), 159-224, 1979.

....and logic. 14] moreover can be extended in a simple way to provide a type checker for the present system. The system of [18] interprets an extended fragment of natural language where self reference and nominalisation are allowed. Martin Lof s type theory moreover in [33] and Feferman s T 0 in [8], present systems which have been extensively used in pl. Those systems too are related to the present one as we shall see below. accommodates most of the systems mentioned above, hence bringing in all the advantages. We will use oe and for types (two special instances of which are e the type ....

Feferman, S., Constructive theories of functions and classes, Logic Colloquium '78, M. Boffa et al (eds), 159-224, 1979.

....in particular, it is shown that Cook and Urquhart s system PV is directly contained in a natural applicative theory of polynomial strength. 1 Introduction Theories with self application form the operational core of Feferman s systems of explicit mathematics, which have been introduced in [24, 25, 26]. The # Institut fur Informatik und angewandte Mathematik, Universitat Bern, Neubruckstrasse 10, CH 3012 Bern, Switzerland. Email: strahm iam.unibe.ch original aim of explicit mathematics was to provide a logical basis for Bishopstyle constructive mathematics. More generally, the explicit ....

....models of the combinatory axioms, which can easily be extended to models of B. These include further recursiontheoretic models, term models, continuous models, generated models, and set theoretic models. For detailed descriptions and results the reader is referred to Beeson [3] Feferman [26], and Troelstra and van Dalen [70] We will make use of the so called extensional term model of B in our upper bound arguments in Section 6; there we will define this model in some detail. We finish this subsection by spelling out the obvious axioms for word concatenation and word multiplication ....

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Bo#a, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224.

....order to model object oriented programming. This predicative object model will be our starting point for constructing a denotational semantics for Featherweight Java in a theory of types and names. Theories of types and names, or explicit mathematics, have originally been introduced by Feferman [14, 15] to formalize Bishop style constructive mathematics. In the sequel, these systems have gained considerable importance in proof theory, particularly for the proof theoretic analysis of subsystems of second order arithmetic and set theory. More recently, theories of types and names have been ....

Solomon Feferman. Constructive theories of functions and classes. In M. Bo#a, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78, pages 159--224. North Holland, 1979.

....types only. For such restricted forms of polymorphism, set theoretic models are readily available. We will show, how our two systems of predicative polymorphism may be embedded into two di#erent theories of so called explicit mathematics. Explicit mathematics was originally introduced by Feferman [Fef75, Fef79] as a formal framework for treating constructive mathematics. We will, however, be using a slight variation introduced by Jager [Jag88] 4 Introduction which mainly di#ers from Feferman s original approach by the use of a naming relation on types. Explicit mathematics itself features untyped ....

....our claim holds by Lemma 2.4.4. # The following theorem about EET (T IN ) Tot) Ext) and EET (F IN ) Tot) Ext) sets these theories in relation to systems of arithmetic and therefore determines their prooftheoretical strength. The theorem can be constructed from various results given in [Fef79], Jag88] Mar93] and [JS95] Its proof is not in the scope of this thesis. Theorem 2.4.1 We have the following proof theoretical equivalences 1. The theories EET (T I N ) Tot) Ext) and PA. 2. The theories EET (F I N ) Tot) Ext) and # # CA. The interpretation of # in explicit ....

S. Feferman. Constructive theories of functions and classes. In M. Bo#a, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78, Studies in Logic and the Foundations of Mathematics, pages 159--224. North-Holland, 1979.

....Explicit mathematics, logic of partial terms, partial calculus, partial combinatory logic, explicit substitutions 1 Introduction Partial applicative theories form the basis of various formal systems for constructive mathematics and functional programming. Feferman introduced in [6] and [7] partial applicative theories of operations and classes in order to give a logical account to Research supported by the Swiss National Science Foundation. 1 Bishop s style of constructive mathematics (BCM) More recently, Feferman s systems of explicit mathematics were used to develop a unitary ....

....[22] 23] As far as the explicit representation of a theory with partial self application in the previous literature is concerned, people either took partial combinatory logic or the partial calculus (without the rule ) as the applicative basis. The former possibility was chosen in [2] 6] [7], 11] 16] 17] 23] 29] the latter in [8] 9] 10] and [22] At first sight, these two approaches seem to be completely equivalent, and sometimes they are treated as such in the literature. But they are only equivalent in the presence of a total logic, since then calculus (without ) can ....

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Boffa, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224.

....Our treatment of PTO is proof theoretic and very much in the spirit of reductive proof theory. 1 Introduction Theories with self application provide an elementary framework for many activities in (the foundations of) mathematics and computer science. They were first introduced by Feferman [11, 12] as a basis for his systems of explicit mathematics, e.g. the theory T 0 ; these theories are broadly discussed in the literature from a prooftheoretic and model theoretic point of view, cf. e.g. the textbooks Beeson [2] and Troelstra and Van Dalen [25] for a survey. It is the aim of the present ....

....are many more interesting models of the combinatory axioms, which can easily be extended to models of PTO. These include further recursion theoretic models, term models, generated models and set theoretic models. For detailed descriptions and results the reader is referred to Beeson [2] Feferman [12] and Troelstra and van Dalen [25] Let us finish this section by making some comments concerning polynomial time functionals. Cook and Urquhart [10] introduced a class BFF of basic feasible functionals in all finite types in order to provide functional interpretations of feasibly constructive ....

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Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Boffa, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224.

....of the bar rule, yielding systems of ordinal strength 0 and 20, respectively. Relevant use is made of xed point theories with ordinals plus bar rule. 1 Introduction In the past few years there have been rather extensive proof theoretic investigations on Feferman s explicit mathematics (cf. [6, 7]) with a predicatively justi ed quanti cation operator , cf. the papers Feferman and J ager [9, 10] Gla and Strahm [13] J ager and Strahm [17, 18] and Marzetta and Strahm [19] The systems studied in the context of range from pure rst order applicative frameworks to theories of types and ....

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Boa, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159-224.

....in particular, it is shown that Cook and Urquhart s system PV # is directly contained in a natural applicative theory of polynomial strength. 1 Introduction Theories with self application form the operational core of Feferman s systems of explicit mathematics, which have been introduced in [18, 19, 20]. The original aim of explicit mathematics was to provide a logical basis for # Institut fur Informatik und angewandte Mathematik, Universitat Bern, Neubruckstrasse 10, CH 3012 Bern, Switzerland. Email: strahm iam.unibe.ch 1 Bishop style constructive mathematics. More generally, the explicit ....

....models of the combinatory axioms, which can easily be extended to models of B. These include further recursiontheoretic models, term models, continuous models, generated models, and set theoretic models. For detailed descriptions and results the reader is referred to Beeson [2] Feferman [20], and Troelstra and van Dalen [53] We will make use of the so called extensional term model of B in our upper bound arguments in Section 6; there we will define this model in some detail. We finish this paragraph by spelling out the obvious axioms for word concatenation and word multiplication in ....

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Bo#a, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224. 44

.... operator and join. We make use of standard proof theoretic techniques such as cut elimination of appropriate semi formal systems and asymmetrical interpretations in standard structures for explicit mathematics. 1 Introduction Systems of explicit mathematics were introduced in Feferman [5, 9]. Two families of theories were presented there, namely the theories T 0 and T 1 together with their various subsystems. T 1 results from T 0 by adding the so called non constructive minimum operator, a predicatively justified quantification operator over the natural numbers. Complete ....

....various subsystems. T 1 results from T 0 by adding the so called non constructive minimum operator, a predicatively justified quantification operator over the natural numbers. Complete proof theoretic information about T 0 and its various subsystems is available since 1983 by the work of Feferman [5, 9], Feferman and Sieg [14] Jager [20] and Jager and Pohlers [22] A crucial step in the proof theoretic analysis of subsystems of T 1 was established only recently in the two papers by Feferman and Jager [13, 12] 1 Whereas the first of these papers deals with pure applicative theories plus ....

[Article contains additional citation context not shown here]

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Boffa, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224.

....into fixed point theories with ordinals 81 4.5.3 The Church Rosser proof for i ae : 83 4. 6 On versus E : 86 Proof theoretic equivalences 87 List of symbols 89 2 Introduction In the mid seventies, Feferman [18, 20] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, is was the origin of Feferman s program to give a logical account to Bishop s style of constructive mathematics. Right from the beginning, systems of explicit ....

....for proof theory, mainly in connection with the proof theoretic analysis of subsystems of first and second order arithmetic and set theory. Complete proof theoretic information about the most prominent framework for explicit mathematics, T 0 , is available since 1983 by the work of Feferman [18, 20], Feferman and Sieg [30] Jager [45] and Jager and Pohlers [49] An excellent and uniform presentation of many of these results is contained, among other things, in Gla thesis [37] More recently, systems of explicit mathematics have been used to develop a general logical framework for ....

[Article contains additional citation context not shown here]

Feferman, S. Constructive theories of functions and classes. In Logic Colloquium '78, M. Boffa, D. van Dalen, and K. McAloon, Eds. North Holland, Amsterdam, 1979, pp. 159--224.

....be the least xed point. This is only given by the semantical interpretation, cf. e.g. Reade [25] In this paper we will present an applicative theory which allows to de ne a least xed point operator. Applicative theories build the rst order part of Feferman s systems of explicit mathematics [5, 6], which have originally been designed to formalize Bishop style constructive mathematics. More recently, these systems have been employed for the study of functional and objectoriented programming languages. In particular, they have been shown to provide a unitary axiomatic framework for ....

Solomon Feferman. Constructive theories of functions and classes. In M. Boa, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78, pages 159-224. North Holland, 1979.

....theories and show their syntactical expressive power. In particular, we present as a novelty a syntactical interpretation of ID 1 in a applicative truth theory based on supervaluation. 1 Introduction Applicative theories build the rst order part of Feferman s systems of explicit mathematics, [Fef75, Fef79]. They comprise type free combinatory logic, natural numbers, pairing and projection. A survey about the prooftheoretic results in the applicative framework can be found in [JKS99] In [Bee85] Beeson introduced a truth theory for applicative theories by adding a truth predicate T and appropriate ....

Solomon Feferman. Constructive theories of functions and classes. In M. Boa, D. van Dalen, and K. McAloon, editors, Logic Colloquium 78, pages 159-224. North{Holland, Amsterdam, 1979. 22

....carried out in these theories. So the result proven here underlines this arguments, and shows, that Martin Lof s type theory is important for the foundations of Mathematics. For more discussions on the constructivism as an approach towards better foundations of mathematics, see [Bee85] DT88] [Fef79], Fef82a]and for discussions on intuitionism [Tro73] It also justifies the use of it as basic theory for proof development systems. CHAPTER 1. INTRODUCTION 5 Precisely we calculate the proof theoretical strength of intensional Russel , extensional Tarski and extensional Russel version of ....

S. Feferman. Constructive theories of functions and classes. In M. Bo#a, D. v. Dalen, and K. McAloon, editors, Logic Colloquium 78, pages 159 -- 224, Amsterdam, 1979. North-Holland.

.... The basic notions of realizability were defined along the lines of BHK clauses with different constructive objects instead of proofs: computable functions and their codes (e.g. in [32] 33] computable operations of higher types (e.g. in [38] partial recursive operations (e.g. in [21] [22]) etc. For the references one may consult recent surveys on realizability and constructive semantics [8] 71] Note that the standard realizability semantics for Int is not adequate. First of all, following Kleene ( 32] one should distinguish between intuitionistic and classical understanding of ....

S. Feferman, "Constructive theories of functions and classes". In: M. Boffa, D. van Dalen, and K. McAloon, eds., Logic Colloquium '78, North Holland, pp. 159-224, 1979.

....#x # natF [x] for each formula F . We will also consider restricted form of induction, where F [x] must be of the form x # #. The following lemmas 1.1 and 1.2 are provable using only applicative axioms I; Lemma 1. 3 in addition calls for restricted induction on natural numbers (see, for example, [Fef79], Be85] or a review [JKS99] Lemma 1.1 # abstraction For every term t[x] there exists a term #x.t[x] such that #x.t[x]# and for every term s s## (#x.t[x] s # t[s] Abbreviation. We will use ID for ##x.x, #y.y#. Lemma 1.2 Recursion Theorem There exists a closed term rec such that ....

S. Feferman. Constructive theories of functions and classes. In: Logic Colloquium '78, 159--224, 1979

..... The R interpretation is an adaptation of Kleene s recursive realizability, and is applicable only to intuitionistic theories. Introduction Systems of Explicit Mathematics were introduced by S. Feferman in the 70 es as a logical framework for Bishop style constructive mathematics (see [Fef75] [Fef79]) In [Fef79] he gave an embedding of the basic theory T 0 into a subsystem # 1 2 CA BI of 2 nd order arithmetic and conjectured that the converse also holds. In [Ja83] G. Jager carried out a necessary well ordering proof in T 0 , which together with [JP82] completed its proof theoretical ....

....is an adaptation of Kleene s recursive realizability, and is applicable only to intuitionistic theories. Introduction Systems of Explicit Mathematics were introduced by S. Feferman in the 70 es as a logical framework for Bishop style constructive mathematics (see [Fef75] Fef79] In [Fef79] he gave an embedding of the basic theory T 0 into a subsystem # 1 2 CA BI of 2 nd order arithmetic and conjectured that the converse also holds. In [Ja83] G. Jager carried out a necessary well ordering proof in T 0 , which together with [JP82] completed its proof theoretical analysis and ....

[Article contains additional citation context not shown here]

S. Feferman. Constructive theories of functions and classes. In: Logic Colloquium '78, 159--224, 1979

....a larger class of TK formulae. Again, this is a standard notion in the theory of realizability. There are a number of different classifications in the literature which arise because of slightly different underlying theories (for example: negative, almost negative [Bee 85] essentially ( free [Fef 79] Harrop [Dum 77] type zero [HaN 87] in TK we refer to safe formulae. Safe formulae for TK are closest to the type zero formulae of the PX system and extend the Harrop formulae of Intuitionistic Logic. 2.5 Definition (Safe formulae) i) Safe(a) ii) If Safe(j) and Safe(y) then Safe(j y) ....

Feferman, S., Constructive theories of functions and classes, Logic Coll. '78, pp 159-224, North Holland, 1979.

....be the least fixed point. This is only given by the semantical interpretation, cf. e.g. Reade [25] In this paper we will present an applicative theory which allows to define a least fixed point operator. Applicative theories build the first order part of Feferman s systems of explicit mathematics [5, 6], which have originally been designed to formalize Bishop style constructive mathematics. More recently, these systems have been employed for the study of functional and object oriented programming languages. In particular, they have been shown to provide a unitary axiomatic framework for ....

Solomon Feferman. Constructive theories of functions and classes. In M. Bo#a, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78, pages 159--224. North Holland, 1979.

....in order to model object oriented programming. This predicative object model will be our starting point for constructing a denotational semantics for Featherweight Java in a theory of types and names. Theories of types and names, or explicit mathematics, have originally been introduced by Feferman [13, 14] to formalize Bishop style constructive mathematics. In the sequel, these systems have gained considerable importance in proof theory, particularly for the proof theoretic analysis of subsystems of second order arithmetic and set theory. More recently, theories of types and names have been ....

Solomon Feferman. Constructive theories of functions and classes. In M. Bo#a, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78, pages 159--224. North Holland, 1979.

....I favor my own, and I will sketch that now. What I did in (Feferman 1975) was to introduce some formal systems of Explicit Mathematics , to begin with, a theory T 0 which is constructive in a suitable sense of the word, and then an extension T 1 which incorporates predicative systems. The paper (Feferman 1979) elaborates on the uses of T 0 and its metamathematical properties. We conceive its universe of discourse, V to be rather rich: it includes the natural numbers, is closed under pairing, and includes elements which are regarded as partial functions. Then partial functions can apply to natural ....

Feferman, Solomon, "Constructive theories of functions and classes", in Logic Colloquium `78 (Amsterdam: North-Holland, 1979), 159--224.

No context found.

Feferman, S., Constructive theories of functions and classes, Logic Colloquium '78, M. Boffa et al (eds), pp 159-224, North Holland, 1979.

No context found.

Solomon Feferman. Constructive theories of functions and classes. In M. Bo#a, D. van Dalen, and K. McAloon, editors, Logic Colloquium '78, pages 159--224. North Holland, 1979.

No context found.

Feferman, S., Constructive theories of functions and classes, Logic Colloqium 78, North-Holland, Amsterdam, 159-224.

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