Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-ityourself type theory. Formal Aspects of Computing, 1:19--84, 1989. 70  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this paper I shall use the term primitive recursive for the kind of recursion you have in type theory, which includes primitive recursive functionals and structural recursion on an arbitrary inductive ( inductively defined) set (or family) including transfinite recursion. Backhouse [3] et.al. [4] exhibited a schema for inductive sets which delimits a class of definitions admissible in Martin Lof s type theory which includes all the standard operations for forming small sets except the equality set. This schema extends Schroeder Heister s schema for the logical constants [13, 14] to the ....

R. Backhouse, P. Chisholm, G. Malcolm, and E. Saaman. Do-it-yourself type theory (part 1). Formal Aspects of Computing, 1:19--84, 1989.

....restriction to domains and continuous functions has serious consequences [34] Martin Lof s Type Theory is based on computation [22, 26] By the interpretation of propositions as types, a type can express a complete program specification. Developments and applications are proceeding rapidly [3]. However, the theory does not admit classical set theoretic arguments. Unwanted proof objects in types cause complications [32] Boyer and Moore use quantifier free first order logic with well founded induction and recursion [4] Although this combination gives unique simplicity and power, it ....

Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19--84, 1989.

....in this paper of describing TK in its entirety and this appears as an appendix. We will devote the rest of this introduction to a motivation for the current work and explain how it is related to similar research which has used MLTT as a basis for a programming logic [Abb 87] Con 86] Kha 86] Bac 89] The reasons for investigating and using systems like TK and MLTT are, by now, quite well known: program specifications are assertions (in MLTT qua type) and it is possible to prove them within the system. Such proofs show that they are, in principle, satisfiable specifications and it is ....

....of subtype were introduced into MLTT [Pet 84] For the W type Nordstrom introduced the acc type [Nor 87] In both cases these new types suppress the information which is deemed to be unnecessary from a computational point of view. Types such as these are known as types with information loss [Bac 89] Such additions undermine the desiderata of the theory (for example the principle of complete presentation) and we believe them to be incoherent extensions of MLTT if only for the reasons we now adumbrate. In general the extra information suppressed is not trivial and may be needed later in a ....

Backhouse, R. C. et al., Do-it-yourself type theory, Formal Aspects of Computing, 1, pp 19-84, 1989.

....methods specify a program by giving sets (carriers) of objects, operations on those objects, and axioms the operations and objects satisfy. The definitions of the operations are refined into a (usually functional) program, while keeping the axioms true. 2 ffl In Type theory a la Martin Lof [Bac89a] a specification is the set of its implementations. One element of that set is constructed. By an analogy between types and propositions, the construction produces the program and also a proof of its correctness. The programmer can read the mathematical specification, write the program, and then ....

Roland Backhouse. Do-it-yourself type theory. Formal Aspects of Computing, pages 19 -- 84, 1989.

....is reduced to writing programs in a pure functional language with dependent types, subject to some restrictions to ensure that the programs proofs are total. This is essentially the original approach of using Type Theory for program verification as proposed by Martin Lof and many others, e.g. see [NPS90, BCMS89]. However, it has been proposed to reintroduce a difference between proofs and programs and between data types and propositions in Type Theory either for pragmatic [PM89] or for philosophical reasons [Luo94] We shall attempt to show by means of example that a pure approach is not only feasible ....

Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Doit -yourself type theory. Formal Aspects of Computing, 1:19--84, 1989.

....Little other metamathematical development is undertaken, for soundness is the primary objective. 2 Background This section is a short survey of the important issues in the design and use of constructive type theories; for fuller descriptions and examples, see one or more of [CAB 86, NPS90, BCMS89] Some of the simpler systems that are subsystems of CTT include the typed calculus [HS86, LS86] and Godel s functionals of finite type (see [Ste72] ch. 7 or [Bar84] app. A) 3 There are also a family of impredicative type theories, the Calculus of Constructions and extensions thereof ....

R. Backhouse, P. Chisholm, G. Malcom, and E. Saaman. Doit -yourself type theory (part 1). Formal Aspects of Computing, 1:19--84, 1989.

....trees. This datatype appears to require us to introduce all the difficulties necessary for dealing with the general case: difficulties which were not at all apparent in [Hen93] 2. 1 General background Transformational programming, like program derivation in a constructive type theory [Con86] [Bac89] [HaN87] Hen89a] Tho91] is a methodology in which programs are obtained by reasoning from specifications. However, it is well known that the calculus of transformations is unsound. Thus, strictly speaking, every transformation induces a correctness proof obligation. Some attempts to ....

Backhouse, R. C. et al, Do-it-yourself type theory, Formal Aspects of Computing, 1, pp 19-84, 1989.

....the proposition it verifies. Martin Lof in [Mar75] p.73 expresses this as follows: 2 This certainly includes testing. There are different inherent limitations in both strategies: verification and testing. 3 Apart from Martin Lof s own work e.g. Mar75] Mar84] good accounts can be found in [BCMS89] and [NPS90] Chapter 1. Introduction 11 The language of the theory is richer than the language of traditional intuitionistic systems in permitting proofs to appear as parts of propositions so that the propositions of the theory can express properties of proofs (and not only individuals, like ....

Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19--84, 1989. Bibliography 174

....also those members which only differ in the order in which their elements were added. Specifying carrier types for ADTs that have axiomatized equalities requires a quotient type constructor such as Nuprl s, or extensions to the underlying type theory in the spirit of those suggested by Backhouse [4]. For example, using the quotient type we can implement multi sets as lists where the quotienting equality is equivalence under permutation. One complication of packaging ADT declarations as Sigma types is that we must be able to open them and access the components of their members. To this end, ....

....declarations. In [40] Paulin Mohring details how induction and computation principles can be automatically generated from declarations of constructor functions and their types and arities within the Calculus of Constructions; The INRIA implementation of CoC contains such a facility. In [4], Backhouse gives a similar account for predicative type theories (e.g. Martin Lof s) and he also explains how these rules can incorporate quotiented equality relations like the kind needed for multi sets. For brevity of presentation, we will not explicitly give the induction and computation ....

Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1(1):19--84, 1989.

....encode Type Theory. There are a number of good introductions to Type Theory available now: lecture notes by Martin Lof were once published as a book [ML84] which is, alas, not available anymore. A very nice tutorial on Type Theory with a similar spirit as ours has been published by Backhouse et al. [BCMS89]. The book by Nordstrom et al. [NPS90] also emphasises the programming view of Type Theory, one of the main differences to our exposition is that they do not use pattern matching. A more recent alternative to Nordstrom et al. is Simon Thompson s book [Tho91] Zhaohui Luo s book [Luo94] concentrates ....

Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19--84, 1989.

....,ucc 0 , which in a more triional syle c be expressed m preds.succ.n = preds.n) gn preds 0 = or, informally, preds n = 0, 1, n 1] Catamorphisms on snoc lists are also known as left reduces, and another way of writing ( 1D is 1 (BIRD[9. 3] Thus, ac = 1 preds Peferences [1] Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do it yourself type theory. Formal Aspects of Computing, 1:19 84, 1989. 2] Palchard S. Bird. An introduction to the theory of lists. In M. Broy, editor, Logic of Programming and Calculi of Discrete Design, volume F36 of NATO ASI ....

....functor (qb (fid)t) in. fid) F fc = c (qb. f id)t)f (in idt) 3)v: is a functor; id is identity (both sides) qbo (fid) in = o in)fin (9) f F is a bijection (qb. f fid) f. in End of proof. The substitution : qb] gives the weaker version (22) [1] id) 1] in UEP FOR PARAMORPHISMS: g ( fid)t = in) q6. gfid)t = g. in) of the same form as for the promotion law, here we find a divergence. The = 13) Catamorphism (weak version) in End of proof. based on theory about some generically defined functions being uniquely ....

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Roland Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19-84, 1989.

....[20] to H.J. Boom. It plays a central role in the development that follows. The formal properties stated here allude to a complete formalisation based on Martin Lof s theory of types [19] originally developed for finite sets by Chisholm [12] and later extended by the author, Chisholm and Malcolm [3, 4]. Some readers may suffer disquiet over the fact that no notational distinction has been made between the constructors for trees, lists, bags and sets. It is common practice to use [a] and fag instead of :a for lists and 14 sets, respectively, and to use [ instead of for set join. What the ....

R.C. Backhouse, P. Chisholm, and G. Malcolm. Do-it-yourself type theory, part 2. EATCS Bulletin, 35, June 1988.

....[20] to H.J. Boom. It plays a central role in the development that follows. The formal properties stated here allude to a complete formalisation based on Martin Lof s theory of types [19] originally developed for finite sets by Chisholm [12] and later extended by the author, Chisholm and Malcolm [3, 4]. Some readers may suffer disquiet over the fact that no notational distinction has been made between the constructors for trees, lists, bags and sets. It is common practice to use [a] and fag instead of :a for lists and 14 sets, respectively, and to use [ instead of for set join. What the ....

R.C. Backhouse, P. Chisholm, and G. Malcolm. Do-it-yourself type theory, part 1. EATCS Bulletin, 34, February 1988.

....where space constraints force us to omit the proofs. Ultimately the intention is to write a document that does emphasise the relevance to programming and is accessible to the uninitiated. The work reported here has its origins in the first named author s interest in constructive type theory. In [2] the importance and relevance of constructive type theory to program design was argued we shall not reiterate the arguments here . The paper concluded as follows: Finally, the relationship between the work reported here and categorical accounts of type structures is one that we have only ....

R.C. Backhouse, P. Chisholm, G. Malcolm, and E. Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19--84, 1989.

....proof, but it involves considerably less writing. Indeed, the calculation has been so simplified that it is hardly necessary to write down the four initial steps. It is now time to shift the direction of this lecture for a short while towards type theory. Elsewhere, I and my colleagues [4] have argued that the prevalent notion that type j statically checkable is a major obstacle to further progress. How much so became abundantly clear to me when John Hughes of Glasgow University visited us and talked about polymorphic functions of type ffy ffz for all ff, for some functors y ....

R.C. Backhouse, P. Chisholm, G. Malcolm, and E. Saaman. Do-ityourself type theory. Formal Aspects of Computing, 1:19--84, 1989.

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Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-ityourself type theory. Formal Aspects of Computing, 1:19--84, 1989. 70

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Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19--84, 1989.

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Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-ityourself type theory. Formal Aspects of Computing, 1:184, 1989.

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Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-ityourself type theory. Formal Aspects of Computing, 1:19-84, 1989.

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R. C. Backhouse, P. Chisholm, G. Malcolm, and E. Saaman. Do-it-yourself type theory (part I). Formal Aspects of Computing, 1:19--84, 1989.

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Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-ityourself type theory. Formal Aspects of Computing, 1:19-84, 1989.

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R. C. Backhouse et al. Do-it-yourself type-theory. Formal Aspects of Computing, 1:19--84, 1989.

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R.Backhouse, P.Chisholm, and G.Malcolm, Do-it-yourself Type Theory, notes for the International Summer School on Constructive Methods in Computer Science, Marktoberdorf 1988. Bibliography 230

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R. C. Backhouse et al. Do-it-yourself type-theory. Formal Aspects of Computing, 1:19--84, 1989.

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R. C. Backhouse, P. Chisholm, G. Malcolm, and E. Saaman. Do-it-yourself type theory (part I). Formal Aspects of Computing, 1:19--84, 1989.

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