How the principle of mathematical induction works. Proof:We use mathematical induction. Recall that this as- sumption is called the induction hypothesis; it is the driving force of our argument. Since we successfully showed the base case and the inductive step, we can use mathematical induction to infer that all natural numbers n have the property stated in the theorem above.
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Logic in computer science explained Logic in computer science covers the overlap between the field of logic and that of computer science.
The topic can essentially be divided into three main areas: Theoretical foundations and analysis Use of computer technology to aid logicians Use of concepts from logic for computer applications Theoretical foundations and analysis Logic plays a fundamental role in computer science.
Some of the key areas of logic that are particularly significant are computability theory formerly called recursion theory , modal logic and category theory. The theory of computation is based on concepts defined by logicians and mathematicians such as Alonzo Church and Alan Turing. This has direct application to theoretical issues relating to the feasibility of proving the completeness and correctness of software. This theory established a precise correspondence between proofs and programs.
In particular it showed that terms in the simply-typed lambda-calculus correspond to proofs of intuitionistic propositional logic. Category theory represents a view of mathematics that emphasizes the relations between structures. It is intimately tied to many aspects of computer science: type systems for programming languages, the theory of transition systems, models of programming languages and the theory of programming language semantics.
Shaw, and Herbert Simon in One of the things that a logician does is to take a set of statements in logic and deduce the conclusions additional statements that must be true by the laws of logic. For example, If given a logical system that states "All humans are mortal" and "Socrates is human" a valid conclusion is "Socrates is mortal". Of course this is a trivial example. In actual logical systems the statements can be numerous and complex.
It was realized early on that this kind of analysis could be significantly aided by the use of computers. The Logic Theorist validated the theoretical work of Bertrand Russell and Alfred North Whitehead in their influential work on mathematical logic called Principia Mathematica. In addition, subsequent systems have been utilized by logicians to validate and discover new logical theorems and proofs. From the beginning of the field it was realized that technology to automate logical inferences could have great potential to solve problems and draw conclusions from facts.
Ron Brachman has described first-order logic FOL as the metric by which all AI knowledge representation formalisms should be evaluated. There is no more general or powerful known method for describing and analyzing information than FOL.
The reason FOL itself is simply not used as a computer language is that it is actually too expressive, in the sense that FOL can easily express statements that no computer, no matter how powerful, could ever solve. For this reason every form of knowledge representation is in some sense a trade off between expressivity and computability.
The more expressive the language is, the closer it is to FOL, the more likely it is to be slower and prone to an infinite loop. Rather than arbitrary formulas with the full range of logical operators the starting point is simply what logicians refer to as modus ponens. As a result, rule-based system s can support high-performance computation, especially if they take advantage of optimization algorithms and compilation.
They also used them to transform the specifications into efficient code on diverse platforms and to prove the equivalence between the implementation and the specification.
However, in specific domains with appropriate formalisms and reusable templates the approach has proven viable for commercial products. The appropriate domains are usually those such as weapons systems, security systems, and real time financial systems where failure of the system has excessively high human or financial cost.
An error in a chip is catastrophic. As a result, there is commercial justification for using formal methods to prove that the implementation corresponds to the specification. This allows specialized theorem provers called classifiers to analyze the various declarations between sets, subsets, and relations in a given model.
In this way the model can be validated and any inconsistent definitions flagged. The classifier can also infer new information, for example define new sets based on existing information and change the definition of existing sets based on new data. The level of flexibility is ideal for handling the ever changing world of the Internet.
Classifier technology is built on top of languages such as the Web Ontology Language to allow a logical semantic level on to the existing Internet. This layer of is called the Semantic web.
Logic in computer science explained
Logic in Computer Science Michael Huth, Mark Ryan