Results 1  10
of
26
A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally e ..."
Abstract

Cited by 39 (16 self)
 Add to MetaCart
(Show Context)
Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Towards Selfverification of HOL Light
 In International Joint Conference on Automated Reasoning
, 2006
"... Abstract. The HOL Light prover is based on a logical kernel consisting of about 400 lines of mostly functional OCaml, whose complete formal verification seems to be quite feasible. We would like to formally verify (i) that the abstract HOL logic is indeed correct, and (ii) that the OCaml code does c ..."
Abstract

Cited by 28 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The HOL Light prover is based on a logical kernel consisting of about 400 lines of mostly functional OCaml, whose complete formal verification seems to be quite feasible. We would like to formally verify (i) that the abstract HOL logic is indeed correct, and (ii) that the OCaml code does correctly implement this logic. We have performed a full verification of an imperfect but quite detailed model of the basic HOL Light core, without definitional mechanisms, and this verification is entirely conducted with respect to a settheoretic semantics within HOL Light itself. We will duly explain why the obvious logical and pragmatic difficulties do not vitiate this approach, even though it looks impossible or useless at first sight. Extension to include definitional mechanisms seems straightforward enough, and the results so far allay most of our practical worries. 1 Introduction: quis custodiet ipsos custodes? Mathematical proofs are subjected to peer review before publication, but there
Efficient Interpretation by Transforming Data Types and Patterns to Functions
 In Trends in Functional Programming, volume 7. Intellect
, 2007
"... In this paper we present the stepwise construction of an efficient interpreter for lazy functional programming languages like Haskell and Clean. The interpreter is realized by first transforming the source language to the intermediate language SAPL (Simple Application Programming Language) consistin ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
(Show Context)
In this paper we present the stepwise construction of an efficient interpreter for lazy functional programming languages like Haskell and Clean. The interpreter is realized by first transforming the source language to the intermediate language SAPL (Simple Application Programming Language) consisting of pure functions only. During this transformation algebraic data types and patternbased function definitions are mapped to functions. This eliminates the need for constructs for Algebraic Data Types and Pattern Matching in SAPL. For SAPL a simple and elegant interpreter is constructed using straightforward graph reduction techniques. This interpreter can be considered as a prototype implementation of lazy functional programming languages. Using abstract interpretation techniques the interpreter is optimised. The performance of the resulting interpreter turns out to be very competitive in a comparison with other interpreters like Hugs, Helium, GHCi and Amanda for a number benchmarks. For some benchmarks the interpreter even rivals the speed of the GHC compiler. Due to its simplicity and the stepwise construction this implementation is an ideal subject for introduction courses on implementation aspects of lazy functional programming languages. 1
Constructor subtyping
, 1999
"... Constructor subtyping is a form of subtyping in which an inductive type is viewed as a subtype of another inductive type Ï if Ï has more constructors than. As suggested in [5, 12], its (potential) uses include proof assistants and functional programming languages. In this paper, we introduce and ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Constructor subtyping is a form of subtyping in which an inductive type is viewed as a subtype of another inductive type Ï if Ï has more constructors than. As suggested in [5, 12], its (potential) uses include proof assistants and functional programming languages. In this paper, we introduce and study the properties of a simply typed Î»calculus with record types and datatypes, and which supports record subtyping and constructor subtyping. In the first part of the paper, we show that the calculus is confluent and strongly normalizing. In the second part of the paper, we show that the calculus admits a wellbehaved theory of canonical inhabitants, provided one adopts expansive extensionality rules, includingexpansion, surjective pairing, and a suitable expansion rule for datatypes. Finally, in the third part of the paper, we extend our calculus with unbounded recursion and show that confluence is preserved.
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Monadic Style Control Constructs For Inference Systems
, 2002
"... Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and sidee#ects in functional and imperative languages.
(http://www. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Recent advances in programming languages study and design have established a standard way of grounding computational systems representation in category theory. These formal results led to a better understanding of issues of control and sidee#ects in functional and imperative languages.
(http://www.lccapital.com/~jmc/articles/monadic1.pdf)
Automation of HigherOrder Logic
 THE HANDBOOK OF THE HISTORY OF LOGIC, EDS. D. GABBAY & J. WOODS; VOLUME 9: LOGIC AND COMPUTATION, EDITOR JÖRG SIEKMANN
, 2014
"... ..."
Proof Nets for Intuitionistic Logic
 SAARBRÜCKEN, GERMANY
, 2006
"... Until the beginning of the 20th century, there was no way to reason formally about proofs. In particular, the question of proof equivalence had never been explored. When Hilbert asked in 1920 for an answer to this very question in his famous program, people started looking for proof formalizations.
..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Until the beginning of the 20th century, there was no way to reason formally about proofs. In particular, the question of proof equivalence had never been explored. When Hilbert asked in 1920 for an answer to this very question in his famous program, people started looking for proof formalizations.
Natural deduction and sequent calculi, which were invented by Gentzen in 1935, quickly became two of the main tools for the study of proofs. Gentzen’s Hauptsatz on normal forms for his sequent calculi, and later on Prawitz’ analog theorem for natural deduction, put forth a first notion of equivalent proofs in intuitionistic and classical logic.
However, natural deduction only works well for intuitionistic logic. This is why Girard invented proof nets in 1986 as an analog to natural deduction for (the multiplicative fragment of) linear logic. Their universal structure made proof nets also interesting for other logics. Proof nets have the great advantage that they eliminate most of the bureaucracy involved in deductive systems and so are probably closer to the essence of a proof. There has recently been an increasing interest in the development of proof nets for various kinds of logics. In 2005 for example, Lamarche and Straßburger were able to express sequent proofs in classical logic as proof nets.
In this thesis, I will, starting from proof nets for classical logic, turn the focus back on intuitionistic logic and propose proof nets that are suited as an extension of natural deduction. I will examine these nets and characterize those corresponding to natural deduction proofs. Additionally, I provide a cut elimination procedure for the new proof nets and prove termination and confluence for this reduction system, thus effectively a new notion of the equivalence of intuitionistic proofs.
Functions, Frames, and Interactions  completing a λcalculusbased purely functional language with respect to programminginthelarge and interactions with runtime environments
, 1998
"... The original aim of the work that led to this dissertation was to extend an existing, purely functional language with facilities for input/output and modular programming. The language is based on an untyped calculus, i.e., program execution is defined as program transformation according to a fixed ..."
Abstract
 Add to MetaCart
The original aim of the work that led to this dissertation was to extend an existing, purely functional language with facilities for input/output and modular programming. The language is based on an untyped calculus, i.e., program execution is defined as program transformation according to a fixed set of reduction rules including fireduction. Consistently, the implementation comprises an interactive reduction system which is integrated with a syntaxoriented editor: any subexpression or program result can be submitted for (stepwise) reduction. There is no distinguished main program, no `global' environment and no explicit static part of the language  in particular, there is no static type system. It is therefore not clear how to add one of the known solutions for input/output or modular programming to such a programming environment. Furthermore, simply adding features to the language would lead to a complex language design with weakly integrated parts, thus losing much of the appe...