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Dialectica Interpretation
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic arithmetic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal ''Dialectica'', where Gödel's paper was published in a 1958 special issue dedicated to Paul Bernays on his 70th birthday. Motivation Via the Gödel–Gentzen negative translation, the consistency of classical Peano arithmetic had already been reduced to the consistency of intuitionistic Heyting arithmetic. Gödel's motivation for developing the dialectica interpretation was to obtain a relative consistency proof for Heyting arithmetic (and hence for Peano arithmetic). Dialectica interpretation of intuitionistic logic The interpretation has two components: a formula translation and a proof translation. The formula translation describes how ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel–Gentzen Negative Translation
In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic, typically by translating formulas to formulas which are classically equivalent but intuitionistically inequivalent. Particular instances of double-negation translation include Glivenko's translation for propositional logic, and the Gödel–Gentzen translation and Kuroda's translation for first-order logic. Propositional logic The easiest double-negation translation to describe comes from Glivenko's theorem, proved by Valery Glivenko in 1929. It maps each classical formula φ to its double negation ¬¬φ. Glivenko's theorem states: :If φ is a propositional formula, then φ is a classical tautology if and only if ¬¬φ is an intuitionistic tautology. Glivenko's theorem implies the more general statement: :If ''T'' is a set of propositional formulas and φ a propositional formul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic that represents Mathematical proof, proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as Recursive data type, inductively-defined data structures such as list (computer science), lists, boxed lists, or Tree (data structure), trees, which are constructed according to the axioms and rule of inference, rules of inference of the logical system. Consequently, proof theory is syntax (logic), syntactic in nature, in contrast to model theory, which is Formal semantics (logic), semantic in nature. Some of the major areas of proof theory include structural proof theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-order Function
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: * takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), * returns a function as its result. All other functions are ''first-order functions''. In mathematics higher-order functions are also termed ''operators'' or '' functionals''. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (other). In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form (\tau_1\to\tau_2)\to\tau_3. General examples * ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulrich Kohlenbach
Ulrich Wilhelm Kohlenbach (born 27 July 1962 in Frankfurt am Main) is a German mathematician and professor of algebra and Mathematical logic, logic at the Technische Universität Darmstadt. His research interests lie in the field of proof mining. Kohlenbach was president of the Deutsche Vereinigung für mathematische Logik und für Grundlagenforschung der exakten Wissenschaften, German Association for Mathematical Logic and for Basic Research in the Exact Sciences (DVMLG) from 2008 to 2012 and president of the Association for Symbolic Logic from 2016 to 2018. Life He graduated ('Abitur') from Lessing-Gymnasium, Frankfurt, Lessing-Gymnasium (High School) in 1980 and completed his studies of mathematics, philosophy, and linguistics with a diplom from the Goethe University Frankfurt. During his studies he received a scholarship from the Studienstiftung des deutschen Volkes. At the same university, he received his Ph.D. in 1990 under the supervision of Horst Luckhardt and passed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Buss
Samuel R. (Sam) Buss is an American computer scientist and mathematician who has made major contributions to the fields of mathematical logic, complexity theory and proof complexity. He is currently a professor at the University of California, San Diego, Department of Computer Science and Department of Mathematics. Biography Buss received his bachelor's degree in 1979 from the Emory University, and his master's degree and Ph.D. from Princeton University, respectively in 1983 and 1985. He joined the University of California, Berkeley, mathematics department in 1986 as an Lecturer, and stayed there until 1988. Buss joined the faculty of University of California, San DiegoComputer ScienceanMathematicsDepartments in 1988 as an assistant professor, where he was promoted to Professor in 1993. In 2019, Buss gave the Gödel Lecture titled ''Totality, provability and feasibility.'' Research Buss is considered one of the forefathers of bounded arithmetic and proof complexity. During ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon Feferman
Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to the United States after World War I and had met and married in New York. Neither parent had any advanced education. The family moved to Los Angeles, where Feferman graduated from high school at age 16. He received his B.S. from the California Institute of Technology in 1948, and in 1957 his Ph.D. in mathematics from the University of California, Berkeley, under Alfred Tarski, after having been drafted and having served in the U.S. Army from 1953 to 1955. In 1956 he was appointed to the Departments of Mathematics and Philosophy at Stanford University, where he later became the Patrick Suppes Professor of Humanities and Sciences. Feferman died on 26 July 2016 at his home in Stanford, following an illness that lasted three months and a stroke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeremy Avigad
Jeremy Avigad is a professor of philosophy at Carnegie Mellon University. He received a B.A. in mathematics from Harvard University in 1989, and a Ph.D. in mathematics from the University of California at Berkeley in 1995 under the supervision of Jack Silver. He has contributed to the areas of mathematical logic and foundations, formal verification and interactive theorem proving, and the philosophy and history of mathematics. He became Director of the Hoskinson Center for Formal Mathematics at Carnegie Mellon University after Charles Hoskinson Charles Hoskinson is an American entrepreneur who is a co-founder of the blockchain engineering company Input Output Global, Inc. (formerly IOHK), and the Cardano blockchain platform, and was a co-founder of the Ethereum blockchain platform. E ... donated $20 Million in September 2021 to establish it. References 20th-century American mathematicians 21st-century American mathematicians American logicians Living people ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak König's Lemma
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 978 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Logic
Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also be characterized as linear logic with weakening. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. Jean-Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterizes linear logic in terms of linear algebra; here he alludes to affine transformations on vector spaces. Affine logic predated linear logic. V. N. Grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom.Cf. Frederic Fitch's demonstrably consistent set theory Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989). Affine logic can be embedded into linear logic by rewriting th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valeria De Paiva
Valeria Correa Vaz de Paiva is a Brazilian mathematician, logician, and computer scientist. Her work includes research on logical approaches to computation, especially using category theory, knowledge representation and natural language semantics, and functional programming with a focus on foundations and type theories... Education De Paiva earned a bachelor's degree in mathematics in 1982, a master's degree in 1984 (on pure algebra) and completed a doctorate at the University of Cambridge in 1988, under the supervision of Martin Hyland. UCAM-CL-TR-213 Her thesis introduced Dialectica spaces, a categorical way of constructing models of linear logic, based on Kurt Gödel's Dialectica interpretation . Career and research She worked for nine years at PARC in Palo Alto, California, and also worked at Rearden Commerce and Cuil before joining Nuance. She is an honorary research fellow in computer science at the University of Birmingham. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dialectica Spaces
Dialectica spaces are a categorical way of constructing models of linear logic. They were introduced by Valeria de Paiva, Martin Hyland's student, in her doctoral thesis, as a way of modeling both linear logic and Gödel's Dialectica interpretation—hence the name. Given a category ''C'' and a specific object ''K'' of ''C'' with certain (logical) properties, one can construct the category of Dialectica spaces over ''C'', whose objects are pairs of objects of ''C'', related by a ''C''-morphism into ''K''. Morphisms of Dialectica spaces are similar to Chu space morphisms, but instead of an equality condition, they have an inequality condition, which is read as a logical implication Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...: the first object implies the second. Refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of persistent "truths", ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |