Logic in Computer Science
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So far so good. It should be a list of all books that don't mention themselves. Now another question, just to check you did finish. Does your second book mention itself? Simple enough question you might think. It either does or it doesn't, and depending on the answer it must go into one of the two books, Book 1 or 2 that you've written. Let us suppose Book 2 lists itself for a moment.
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Then there is a problem because if it does mention itself, it shouldn't be listed in Book 2. It is only for books not listing themselves. Clearly it shouldn't mention itself then. This paradox, called Russell's Paradox lies at the heart of a result about what maths computers themselves or humans for that matter can actually do. The same problem leads to a whole series of Mission Impossible problems for computers. Newspaper puzzles like Sudoku and Kakuro are good examples of how lots of people find logic and proof a fun challenge.
All those commuters are doing it to wind down from work.
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Sudoku and similar puzzles have little to do with numbers and everything to do with logic. To solve them you have to extract the last drop of information from a situation. Programmers are doing the same kind of fun thinking every time they write code. When a program works you get the same kind of kick as when you write the last number into the Sudoku.
Some school Maths is really important if you are interested in specializing in particular areas of Computer Science. For example Computer Vision, Graphics and the games based on them uses a lot of matrix maths. It turns out that to work out what is going on in the world. Computer Security, and in particular encryption, has made use of some very obscure maths called Number theory about things like the prime numbers.
Once upon a time it was thought Number Theory was of no use to anyone - there for the challenge rather than to ever have any effect on the world. Turns out it is the key to e-commerce: sending money digitally and signing documents digitally so they can't be forged is only possible thanks to prime numbers. A different kind of maths again, known as Bayesian Reasoning, is used is the basis of making decisions when not all the facts are known. What is the likelihood of one thing happening given what you know about other things.
It can be used to determine the risks of a software project failing, about complex DNA evidence in court cases or even about who is most likely to win the World Cup So different areas of Computer Science that you could specialize in might or might not involve some branch of Maths. On the whole though the link is that Computer Science has some very simple and fun Maths at its core. Systems are modelled through Automata Theory for instance and are verified using logical tools. In the same motivation, computer scientists wanted to verify the soundness and prove properties about programs.
So we invented the Hoare Logic for imperative programs and more generally, Separation Logics.
By studying, the Curry-Howard isomorphism, a new logic emerged : Linear Logic which restricts structural rules weakening and contraction seen as the erasure and duplication operating in proofs and programs. The potential infinity of truth are explicited. It seems that this logic is a generalization of classical and intuitionistic logic and gives a whole new conception of Logic based on computation and a procedural paradigm.
Mathematical Logic for Computer Science
It is mostly studied by computer scientists. Linear Logic also comes from what we call Substructural logics rejecting structural rules of Logic. Relevant Logic and Affine Logic are examples for such systems. Any logic can be : propositional logic, first-order, second-order, third-order, All of these logics are coming underneath of Mathematical Logic. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.
These areas share basic results on logic, particularly first-order logic, and definability. In computer science particularly in the ACM Classification mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those. Moreover, if you want to know about the logic in general terms this article could be useful. Logic, originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.
A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What exactly is a logic? Ask Question. Asked 1 year, 6 months ago. Active 1 year, 5 months ago. Viewed 5k times.
What exactly does "Logic" mean in this context? How do we rigorously define the word "Logic"? Now coming to these logics. S : Thanks a ton for your kindness.
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Sheldon Kripke. Sheldon Kripke Sheldon Kripke 3 3 silver badges 5 5 bronze badges. I was pretty surprised myself when I encountered a formal definition of 'an algebra'. Syntax is a set of rules that determine what is and is not a formula.
What is a logic ?
Semantics is a set of rules that determine what formulae are "true" and what are "false". To a model theorist , this is expressed by relating formulas to the mathematical structures that they're true in; to a proof theorist , truth corresponds to provability from a chosen set of axioms with a chosen set of proof rules techniques.
Since reasoning is involved in most intellectual activities, logic is relevant to a broad range of pursuits. What exactly is a logic? Instructor Resources Downloadable instructor resources available for this title: lecture notes, exams, solutions, and file of figures in the book. It includes the logical and mathematical analysis of programs. It seems that this logic is a generalization of classical and intuitionistic logic and gives a whole new conception of Logic based on computation and a procedural paradigm. Start Expand Start Minimize Start.
David Richerby David Richerby A type system can be thought of as an alternative presentation of an equivalent logic, via the Howard-Curry correspondence. I recommend Pierce's book to get started. I don't know how I got this far, but this is the first time I ever considered the idea that Currying a function could be a reference to anything besides the spice. What is a logic? They behave like functions taking objects pairs, integer, string are arguments.
Think of the C language. In the second order logic, variable for predicates becomes kind of functions taking first-order ones. What is the difference between little endian and big endian data formats? What circumstances led to the rise of the big data ecosystem? What considerations are most important when deciding which big data solutions to implement? More of your questions answered by our Experts. Related Tags. Development Programming Tools Computer Science. Latest Articles. Reinforcement Learning Vs.