|
Logical Framework
In logic, a logical framework provides a means to define (or present) a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory. This approach has been used successfully for (interactive) automated theorem proving. The first logical framework was Automath; however, the name of the idea comes from the more widely known Edinburgh Logical Framework, LF. Several more recent proof tools like Isabelle are based on this idea. Unlike a direct embedding, the logical framework approach allows many logics to be embedded in the same type system. Overview A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typed lambda calculus. Syntax is treated in a style similar to, but more general than Per Martin-Löf's system of arities. To describe a logical framework, one must provide the following: # A characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Harper (computer Scientist)
Robert William "Bob" Harper, Jr. (born ) is a computer science professor at Carnegie Mellon University who works in programming language research. Prior to his position at Carnegie Mellon, Harper was a research fellow at the University of Edinburgh. Career Harper made major contributions to the design of the Standard ML programming language and the LF logical framework. Harper was named an ACM Fellow in 2005 for his contributions to type systems for programming languages. In 2021, he received the ACM SIGPLAN Programming Languages Achievement Award for his "foundational contributions to our understanding of type theory and its use in the design, specification, implementation, and verification of modern programming languages". Books *Robin Milner, Mads Tofte, Robert Harper, and David MacQueen. ''The Definition of Standard ML (Revised)''. MIT Press, 1997. *Robert Harper (editor). Types in Compilation'. Springer-Verlag Lecture Notes in Computer Science, volume 2071, 2001. *Robert H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Pfenning
Frank Pfenning is a German-American professor of computer science, adjunct professor in the department of philosophy, and head of the Computer Science Department at Carnegie Mellon University. Education and career Pfenning grew up in Rüsselsheim in Germany. He studied mathematics and computer science at Technische Universität Darmstadt in Germany. He then moved to the US and studied at Carnegie Mellon University, where he received his M.S. and Ph.D. in the Department of Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ... in 1987, for his dissertation entitled ''Proof Transformations in Higher-Order Logic''. He was a student of Peter B. Andrews. His research includes work in the area of programming languages, logic and type theory, logical frameworks, automa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Schwichtenberg
Helmut Schwichtenberg (born 5 April 1942 in Żagań) is a German mathematical logician. Schwichtenberg studied mathematics from 1961 at the FU Berlin and from 1964 at the University of Münster, where he received his doctorate in 1968 from Dieter Rödding. He then worked as an assistant and then as a professor in Münster, and since 1978 has been professor of mathematical logic at the Ludwig-Maximilians-Universität Munich (successor of Kurt Schütte). Schwichtenberg deals with, among other things, proof theory, theory of computability, lambda calculus and applications of logic in computer science. He is a member of the Bavarian Academy of Sciences. Selected publications * * (2nd edition 2000: ) * * References External links Homepageat Ludwig-Maximilians-Universität Munich The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: link=no, Ludwig-Maximilians-Universität München) is a public research university in Munich, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turnstile (symbol)
In mathematical logic and computer science the symbol \vdash has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails". Interpretations The turnstile represents a binary relation. It has several different interpretations in different contexts: * In epistemology, Per Martin-Löf (1996) analyzes the \vdash symbol thus: "... e combination of Frege's , judgement stroke and , content stroke €” came to be called the assertion sign." Frege's notation for a judgement of some content ::\vdash A :can then be read ::''I know is true''. :In the same vein, a conditional assertion ::P \vdash Q :can be read as: ::''From , I know that '' * In metalogic, the study of formal languages; the turnstile represents syntactic consequence (or "derivability"). This is to say, that it shows that one string can be derived from another in a single step, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grammatical Framework
Grammatical Framework (GF) is a programming language for writing grammars of natural languages. GF is capable of parsing and generating texts in several languages simultaneously while working from a language-independent representation of meaning. Grammars written in GF can be compiled into a platform independent format and then used from different programming languages including C and Java, C#, Python and Haskell. A companion to GF is the ''GF Resource Grammar Library'', a reusable library for dealing with the morphology and syntax of a growing number of natural languages. Both GF itself and the GF Resource Grammar Library are open-source. Typologically, GF is a functional programming language. Mathematically, it is a type-theoretic formal system (a logical framework to be precise) based on Martin-Löf's intuitionistic type theory, with additional judgments tailored specifically to the domain of linguistics. Language features * a static type system In computer program ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meta-logic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, 2001, p. 336. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived ''about'' the languages and systems that are used to express truths. Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic', University of California Press, 1973 The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory. Overview Formal language A ''formal language'' is an organized set of symbols, the symbols of which precise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carnegie Mellon University
Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania. One of its predecessors was established in 1900 by Andrew Carnegie as the Carnegie Technical Schools; it became the Carnegie Institute of Technology in 1912 and began granting four-year degrees in the same year. In 1967, the Carnegie Institute of Technology merged with the Mellon Institute of Industrial Research, founded in 1913 by Andrew Mellon and Richard B. Mellon and formerly a part of the University of Pittsburgh. Carnegie Mellon University has operated as a single institution since the merger. The university consists of seven colleges and independent schools: The College of Engineering, College of Fine Arts, Dietrich College of Humanities and Social Sciences, Mellon College of Science, Tepper School of Business, Heinz College of Information Systems and Public Policy, and the School of Computer Science. The university has its main campus located 5 miles (8 km) from Downto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twelf
Twelf is an implementation of the logical framework LF developed by Frank Pfenning and Carsten Schürmann at Carnegie Mellon University. It is used for logic programming and for the formalization of programming language theory. Introduction At its simplest, a Twelf program (called a "signature") is a collection of declarations of type families (relations) and constants that inhabit those type families. For example, the following is the standard definition of the natural numbers, with standing for zero and the successor operator. nat : type. z : nat. s : nat -> nat. Here is a type, and and are constant terms. As a dependently typed system, types can be indexed by terms, which allows the definition of more interesting type families. Here is a definition of addition: plus : nat -> nat -> nat -> type. plus_zero : plus M z M. plus_succ : plus M (s N) (s P) <- plus M N P. The type family is read as a relation between three na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
