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The formal method known as B and a sketch for its implementation
, 2002
"... This thesis provides a reconstruction of the Bmethod and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (firstorder) logic is also conside ..."
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This thesis provides a reconstruction of the Bmethod and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (firstorder) logic is also considered: both its development and important points relevant to formal methods. Automated reasoning, particularly its theoretical limits as well as unification and resolution, is discussed. The main part of this thesis is a systematic reconstruction of the Bmethod, starting from its version of untyped predicate calculus and typed set theory, continuing with the Generalized Substitution Language (GSL) and finishing with the Abstract Machine Notation (AMN). Specification, refinement and implementation of a simple example using the Bmethod is presented. Both validation and verification of specifications, refinements and implementations using the Bmethod is discussed. The thesis concludes with a report of the current state of the effort (by the author) to implement the tool support of the Bmethod, as the Ebba Toolset. The main design decisions are discussed. The use of Unicode as the primary input encoding of AMN and GSL in Ebba is described.
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, 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). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
The History and Concept of Mathematical Proof
, 2007
"... A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Intern ..."
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A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Internet design, engineering, chemistry, biological science, social science, anthropology, genetics, warfare, cryptography, plastic surgery, security analysis, data manipulation, computer science, and in many other disciplines and endeavors as well. The unique feature that sets mathematics apart from other sciences, from philosophy, and indeed from all other forms of intellectual discourse, is the use of rigorous proof. It is the proof concept that makes the subject cohere, that gives it its timelessness, and that enables it to travel well. The purpose of this discussion is to describe proof, to put it in context, to give its history, and to explain its significance. There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject. This is the reason that we can depend on mathematics that was done by Euclid 2300 years ago as readily as we believe in the mathematics that is done today. No other discipline can make such an assertion.
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"... According to Curtis Franks ’ preface, this book bundles his historical, philosophical and logical research to center around what he thinks are “the most important and most overlooked aspects of Hilbert’s program … a glaring oversight of one truly unique aspect of Hilbert’s thought, ” namely that “qu ..."
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According to Curtis Franks ’ preface, this book bundles his historical, philosophical and logical research to center around what he thinks are “the most important and most overlooked aspects of Hilbert’s program … a glaring oversight of one truly unique aspect of Hilbert’s thought, ” namely that “questions about mathematics that arise in philosophical reflection⎯questions about how and why its methods work⎯might best be addressed mathematically … Hilbert’s program was primarily an effort to demonstrate that. ” The standard “wellrehearsed ” story for these oversights is said to be that Hilbert’s philosophical vision was dashed by Gödel’s incompleteness theorems. But Franks argues to the contrary that Gödel’s remarkable early contributions to metamathematics instead drew “significant attention to the then fledgling discipline, ” a field that has since proved to be exceptionally productive scientifically (even through the work of such logicians as Tarski, who mocked Hilbert’s program). One may well ask how the author’s effort to put this positive face on the patent failure of Hilbert’s program can possibly succeed in showing that mathematical knowledge is autonomous, that mathematics has only to look to itself for its proper foundations. Let us see.
Effectiveness
, 2011
"... We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. ..."
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We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis.
Universality, Turing Incompleteness and Observers
"... The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics a ..."
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The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics and construct a finite set of axioms that are strong enough to prove all proper theorems, but no more. Thus a proof of consistency and a proof of completeness were required. These proofs should be carried only by strictly finitary means so as to be beyond any reasonable criticism. As Hilbert pointed out [19], to carry out this project one needs to develop a better understanding of proofs as objects of mathematical discourse: To reach our goal, we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves. Furthermore, Hilbert hoped to find a single, mechanical procedure that would, at least in principle, provide correct answers to all welldefined questions
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.