Course Objectives. The goal of the course is to prepare students for using logic as a formal tool in computer science. By the end of course, students will be exposed to different fragments of logics and decision procedures for them, and connections between logic and automata theory and between computational complexity and descriptive complexity. They will have seen tools and techniques that exploit logic to come up with algorithms to solve problems, and establish lower bounds. Students will be informed of some of the major open problems in the field and current efforts to solve them.
Prerequisites. This course is aimed at advanced undergraduate and graduate students interested in logic and theoretical computer science. Familiarity with basic logic and discrete mathematics (CS 173), basic algorithms and theory of computation (CS 374), and mathematical maturity will be expected.
mathematical theory of computation zohar manna pdf 19
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The ultimate object of computer science was working programs,arguedMcCarthy, and a suitabletheory of computation would provide: first, a universal programminglanguage along the lines ofAlgol but with richer data descriptions; second, a theory of theequivalence of computationalprocesses, by which equivalence-preserving transformations would allowa choice of among variousforms of an algorithm, adapted to particular circumstances; third, aform of symbolic representationof algorithms that could accommodate significant changes in behavior bysimple changes in thesymbolic expressions; fourth, a formal way of representing computersalong with computation; andfinally a quantitative theory of computation along thelines of Claude Shannon's measure of information.(15)Note that as this list progresses,it sounds more andmore like engineering, and McCarthy's agenda (and itshistory) continued to echo in the software-engineeringliterature. In arguing in 1984 that "[p]rofessionalprogramming practice should be based on underlyingmathematical theories and follow the traditions ofbetter-established engineering disciplines," C.A.R.Hoare highlighted in a sidebar McCarthy's comparisonof physics and mathematical logic quoted above.(16)
Over the decade of the '60s theoreticalcomputer science achieved standing as a disciplinerecognized by both the mathematical and thecomputing communities, and it could point to bothapplications and mathematical elegance.(17)Yet, it tookthe form more of a family of loosely related researchagendas than of a coherent general theory validated byempirical results. No one mathematical model hadproved adequate to the diversity of computing, and thedifferent models were not related in any effective way. Whatmathematics one used depended on whatquestions one was asking, and for some questions nomathematics could account in theory for whatcomputing was accomplishing in practice. It was a farcry from Newton's mechanics, much less themathematical physics of the nineteenth century, and it remains so.
Those "factories" are also not likelyto hold a solution for the problems ofsoftware production that motivated the drivefor software engineering, but, then, neither isautomatic programming. Consider anotherversion of development phases of thesoftware life cycle (Fig. 1). We are on firmesttheoretical ground at the bottom of thediagram. That is where computer sciencehas achieved its most profound results, andthat is where theory has most effectivelytranslated into practical software tools. Butthe problems of air traffic control systems, ofnational weather systems, of airline bookingsystems all lie at the top of the diagram,where a real-world system must betransformed into a computational model. That is where softwareengineering is notabout software, indeed where it may not beabout engineering at all.
A Turing machine is a mathematical model of computation describing an abstract machine[1] that manipulates symbols on a strip of tape according to a table of rules.[2] Despite the model's simplicity, it is capable of implementing any computer algorithm.[3]
A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine (UTM, or simply a universal machine). A more mathematically oriented definition with a similar "universal" nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church-Turing thesis. The thesis states that Turing machines indeed capture the informal notion of effective methods in logic and mathematics, and provide a model through which one can reason about an algorithm or "mechanical procedure". Studying their abstract properties yields many insights into computer science and complexity theory.
Today, the counter, register and random-access machines and their sire the Turing machine continue to be the models of choice for theorists investigating questions in the theory of computation. In particular, computational complexity theory makes use of the Turing machine:
Depending on the objects one likes to manipulate in the computations (numbers like nonnegative integers or alphanumeric strings), two models have obtained a dominant position in machine-based complexity theory: 2ff7e9595c
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