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Subrecursion as Basis for a Feasible Programming Language
 Proceedings of CSL'94, number 933 in LNCS
, 1994
"... We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of Sexpressions of LISP within t ..."
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Cited by 9 (8 self)
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We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of Sexpressions of LISP within the domain of natural numbers. In the process of doing this we introduce a missing stage in Grzegorczykbased hierarchies which solves the longstanding open problem of what is the precise relation between the small recursive classes and those of complexity theory. 1 Introduction We investigate subrecursive hierarchies based on pairing functions and solve a longstanding open problem in small recursive classes of what is the relationship between these and computational complexity classes (see [11]). The problem is solved by discovering that there is a missing stage in Grzegorczykbased hierarchies [7, 11]. The motivation for this research comes from our search for a good programming langu...
On Extensibility of Proof Checkers
 in Dybjer, Nordstrom and Smith (eds), Types for Proofs and Programs: International Workshop TYPES'94, Bastad
, 1995
"... This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. Howeve ..."
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Cited by 6 (2 self)
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This paper is about mechanical checking of formal mathematics. Given some formal system, we want to construct derivations in that system, or check the correctness of putative derivations; our job is not to ascertain truth (that is the job of the designer of our formal system), but only proof. However, we are quite rigid about this: only a derivation in our given formal system will do; nothing else counts as evidence! Thus it is not a collection of judgements (provability), or a consequence relation [Avr91] (derivability) we are interested in, but the derivations themselves; the formal system used to present a logic is important. This viewpoint seems forced on us by our intention to actually do formal mathematics. There is still a question, however, revolving around whether we insist on objects that are immediately recognisable as proofs (direct proofs), or will accept some metanotations that only compute to proofs (indirect proofs). For example, we informally refer to previously proved results, lemmas and theorems, without actually inserting the texts of their proofs in our argument. Such an argument could be made into a direct proof by replacing all references to previous results by their direct proofs, so it might be accepted as a kind of indirect proof. In fact, even for very simple formal systems, such an indirect proof may compute to a very much bigger direct proof, and if we will only accept a fully expanded direct proof (in a mechanical proof checker for example), we will not be able to do much mathematics. It is well known that this notion of referring to previous results can be internalized in a logic as a cut rule, or Modus Ponens. In a logic containing a cut rule, proofs containing cuts are considered direct proofs, and can be directly accepted by a proof ch...
Formalization of the Development Process
, 1998
"... Introduction 14.1.1 What is development? Software development encompasses many phases including requirements engineering, specification, design, implementation, verification or testing, and maintenance. In this chapter we concentrate on the intermediate tasks: the transition from requirements spec ..."
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Cited by 4 (3 self)
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Introduction 14.1.1 What is development? Software development encompasses many phases including requirements engineering, specification, design, implementation, verification or testing, and maintenance. In this chapter we concentrate on the intermediate tasks: the transition from requirements specification to verified design and design optimization; in particular techniques for developing correct designs as opposed to ad hoc or a posteriori methods in which a postulated design is later verified or tested. We view software development as the composition of correctness preserving refinement steps. Hence, a development is meaningful relative to a notion of refinement and correctness. There are many such notions (e.g., Chapter 6), each defining a type of problem transformation or reduction. For example specification refinement from a specification SP to SP 0 through a refinement map ae might be defined as co
A framework for defining logical frameworks
 University of Udine
, 2006
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Cited by 4 (1 self)
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
Constructibility and Decidability versus Domain Independence and Absoluteness
"... We develop a unified framework for dealing with constructibility and absoluteness in set theory, decidability of relations in effective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domainindependence and absolut ..."
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Cited by 4 (2 self)
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We develop a unified framework for dealing with constructibility and absoluteness in set theory, decidability of relations in effective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domainindependence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability. 1
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
"... ..."
A Framework for Formalizing Set Theories Based on the Use of Static Set Terms
"... To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity ..."
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Cited by 1 (1 self)
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To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two settheoretical principles: extensionality and comprehension (to which we add ∈induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x  ϕ} ↔ϕ, where {x  ϕ} is a legal set term of the theory. In order for {x  ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 1
Automating the Synthesis of Decision Procedures
, 1994
"... We present an approach to the automatic construction of decision procedures, via a detailed example in propositional logic. The approach adapts the methods of proofplanning, and particularly the technique of rippling [BSvH + 93], to a new domain, that of metatheoretic procedures. This approa ..."
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We present an approach to the automatic construction of decision procedures, via a detailed example in propositional logic. The approach adapts the methods of proofplanning, and particularly the technique of rippling [BSvH + 93], to a new domain, that of metatheoretic procedures. This approach starts by providing an alternative characterisation of validity; the proofs of the correctness of this characterisation, and the existence of a decision procedure, are then both amenable to automation in the way we describe. In this paper, the object logic is characterised semantically; as this characterisation is inherently recursive, we believe that the rippling technique is well suited to guide the automation of reasoning about logics presented in this way. Keywords: theorem proving, automatic programming, proof planning. Declaration: This paper has not already been accepted by and is not currently under review for a journal or another conference. Nor will it be submitted for s...