Starting from the idea that simple functions will show some regular behaviour, we introduce the notion of sequential representation to formalize a nonuniform way of decribing such regularities, e.g. periodically repea...
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This paper presents a reflection about function construction through well-founded recursion in the type theory known as Calculus of Inductive Constructions. It shows the expressive power of this calculus when dealing ...
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ISBN:
(纸本)3540290028
This paper presents a reflection about function construction through well-founded recursion in the type theory known as Calculus of Inductive Constructions. It shows the expressive power of this calculus when dealing with concepts as accesibility and noetherianity among others. The properties of the General Recursion Scheme ([2]) and its relation with Structural Recursion in inductive types are analyzed. As a consequence, a methodology arises which is illustrated with some examples. We use the INRIA's compiler of the Calculus of Inductive Constructions: Coq [6].
Our goal is to help people construct software that does what they wish. We develop tools and algorithms that span static and dynamic verification, constraint solving, and program synthesis. I will outline the current ...
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ISBN:
(纸本)9783662439487
Our goal is to help people construct software that does what they wish. We develop tools and algorithms that span static and dynamic verification, constraint solving, and program synthesis. I will outline the current state our verification and synthesis system, Leon, which translates software into a functional language and uses SMT solvers to reason about paths in programs and specifications. Certain completeness results partly explain the effectiveness of verification and synthesis procedures implemented within Leon, in particular results on decidability of sufficiently surjective abstraction functions, and the framework of complete functional synthesis.
In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive definitions of functions are shown to terminate automatically. In standard non-formalised termination proofs of recursive functions, a dec...
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ISBN:
(数字)9783540481645
ISBN:
(纸本)3540665404
In proof checkers and theorem provers (e.g. Coq [4] and ProPre [13]) recursive definitions of functions are shown to terminate automatically. In standard non-formalised termination proofs of recursive functions, a decreasing measure is sometimes used. Such a decreasing measure is usually difficult to find. By observing the proof trees of the proofs of termination of recursive functions in ProPre (the system used in Cog's proofs of termination), [14] finds a decreasing measure which could be used to show termination in the standard non-formalised way. This is important because it establishes a method to find decreasing measures that help in showing termination. As the ProPre system made heavy use of structural rather than inductive rules, an extended more powerful version has been built with new proof trees based on new rules. In this article, we show that the ordinal measures found in [14] lose the decreasing property in the extended ProPre system and then, set out to show that the extended ProPre system will still be suitable for finding measures required by other systems (e.g. NQTHM). We do this by showing that exist other measures that can be associated to the proof trees developed in the extended ProPre system that respect the decreasing property. We also show that the new parameterised measure functions preserve the decreasing property up to a simple condition.
Many useful calculation rules, such as fusion and tupling, rely on well-structured functions, especially in terms of inputs and outputs. For instance, fusion requires that well-produced outputs should be connected to ...
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ISBN:
(纸本)3540356312
Many useful calculation rules, such as fusion and tupling, rely on well-structured functions, especially in terms of inputs and outputs. For instance, fusion requires that well-produced outputs should be connected to well-consumed inputs, so that unnecessary intermediate data structures can be eliminated. These calculation rules generally fail to work unless functions are well-structured. In this paper, we propose a new calculation rule called 10 swapping. 10 swapping exchanges call-time computations (occurring in the arguments) and return-time computations (occurring in the results) of a function, while guaranteeing that the original and resulting function compute the same value. 10 swapping enables us to rearrange inputs and outputs so that the existing calculation rules can be applied. We present new systematic derivations of efficient programs for detecting palindromes, and a method of higher-order removal that can be applied to defunctionalize function arguments, as two concrete applications.
A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn} is convergent on a metric space I subset of R, then it is possible to ...
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A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn} is convergent on a metric space I subset of R, then it is possible to observe this behaviour on the set D subset of Q of all numbers represented in a computer. However, as D is not complete, the representation of fn on D is subject to an error. Then f(n) and f(m) are considered equal when its differences computed on D are equal or lower than the sum of error of each f(n) and f(m). An example is given to illustrate the use of the theorem.
SMT solvers have recently been extended with techniques for finding models of universally quantified formulas in some restricted fragments of first-order logic. This paper introduces a translation that reduces axioms ...
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ISBN:
(纸本)9783319402291
SMT solvers have recently been extended with techniques for finding models of universally quantified formulas in some restricted fragments of first-order logic. This paper introduces a translation that reduces axioms specifying a large class of recursive functions, including terminating functions, to universally quantified formulas for which these techniques are applicable. An evaluation confirms that the approach improves the performance of existing solvers on benchmarks from three sources. The translation is implemented as a preprocessor in the CVC4 solver and in a new higher-order model finder called Nunchaku.
The paper investigates whether it is possible to learn every enumerable classes of recursive functions from "typical" examples. "Typical" means, there is a computable family of finite sets, such th...
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ISBN:
(纸本)3540667482
The paper investigates whether it is possible to learn every enumerable classes of recursive functions from "typical" examples. "Typical" means, there is a computable family of finite sets, such that for each function in the class there is one set of examples that can be used in any suitable hypothesis space for this class of functions. As it will turn out, there are enumerable classes of recursive functions that are not learnable from "typical" examples. The learnable classes are characterized. The results are proved within an abstract model of learning from examples, introduced by Freivalds, Kinber and Wiehagen. Finally, the results are interpreted and possible connections of this theoretical work to the situation in real life classrooms are pointed out.
In this paper, we report some empirical studies of students evaluating recursive functions defined according to the rules of the functional programming language Miranda, and describe the misconceptions and processing ...
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In this paper, we report some empirical studies of students evaluating recursive functions defined according to the rules of the functional programming language Miranda, and describe the misconceptions and processing strategies observed. We then discuss the implications of these observations as regards teaching content.
Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted Vir...
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ISBN:
(纸本)9783319284750;9783319284743
Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted Virus Machines are unbounded Virus Machines, in the sense that no restriction on the number of hosts, the number of instructions and the number of viruses contained in any host along any computation is placed on them. The computational completeness of these machines has been obtained by simulating register machines. In this paper, Virus Machines as function computing devices are considered. Then, the universality of non-restricted virus machines is proved by showing that they can compute all partial recursive functions.
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