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Realizability
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another. Realizability can be seen as a formalization of the Brouwer–Heyting–Kolmogorov (BHK) interpretation of intuitionistic logic. In realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Markov's Principle
Markov's principle (also known as the Leningrad principle), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function. In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. These statements are provable in cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Effective Topos
In mathematics, the effective topos introduced by captures the mathematical idea of effectivity within the category theoretical framework. Definition Preliminaries Kleene realizability The topos is based on the partial combinatory algebra given by Kleene's first algebra _1. In Kleene's notion of recursive realizability, any predicate is assigned realizing numbers, i.e. a subset of . The extremal propositions are \top and \bot, realized by and \. However in general, this process assigns more data to a proposition than just a binary truth value. A formula with k free variables will give rise to a map in (\mathcal P)^ the values of which is the subset of corresponding realizers. Realizability topoi is a prime example of a realizability topos. These are a class of elementary topoi with an intuitionistic internal logic and fulfilling a form of dependent choice. They are generally not Grothendieck topoi. In particular, the effective topos is (_1). Other realizability topos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Heyting Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. Axiomatization Heyting arithmetic can be characterized just like the first-order theory of Peano arithmetic , except that it uses the intuitionistic predicate calculus for inference. In particular, this means that the double-negation elimination principle, as well as the principle of the excluded middle , do not hold. Note that to say does not hold exactly means that the excluded middle statement is not automatically provable for all propositions—indeed many such statements are still provable in and the negation of any such disjunction is inconsistent. is strictly stronger than in the sense that all -theorems are also -theorems. Heyting arithmetic comprises the axioms of Peano arithmetic and the intended model is the collection of natural numbers . The signature includes zero "0" and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Brouwer–Heyting–Kolmogorov Interpretation
In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, is an explanation of the meaning of proof in intuitionistic logic, proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene. It is the standard explanation of intuitionistic logic. The interpretation The interpretation states what is intended to be a proof of a given Formula (mathematical logic), formula. This is specified by induction on the structure of that formula: *A proof of P \wedge Q is a pair \langle a, b \rangle where a is a proof of P and b is a proof of Q. *A proof of P \vee Q is either \langle 0, a \rangle where a is a proof of P or \langle 1, b\rangle where b is a proof of Q. *A proof of P \to Q is a construction (see ) that converts a (hypothetical) proof of P into a proof of Q. *A proof of (\exists x S) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ludics
In proof theory, ludics is an analysis of the principles governing inference rules of mathematical logic. Key features of ludics include notion of compound connectives, using a technique known as ''focusing'' or ''focalisation'' (invented by the computer scientist Jean-Marc Andreoli), and its use of ''locations'' or ''loci'' over a base instead of propositions. More precisely, ludics tries to retrieve known logical connectives and proof behaviours by following the paradigm of interactive computation, similarly to what is done in game semantics to which it is closely related. By abstracting the notion of formulae and focusing on their concrete uses—that is distinct occurrences—it provides an abstract syntax for computer science, as loci can be seen as pointers on memory. The primary achievement of ludics is the discovery of a relationship between two natural, but distinct notions of type, or proposition. The first view, which might be termed the proof-theoretic or Gentzen-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometry Of Interaction
In proof theory, the Geometry of Interaction (GoI) was introduced by Jean-Yves Girard shortly after his work on linear logic. In linear logic, proofs can be seen as various kinds of networks as opposed to the flat tree structures of sequent calculus. To distinguish the real proof nets from all the possible networks, Girard devised a criterion involving trips in the network. Trips can in fact be seen as some kind of operator acting on the proof. Drawing from this observation, Girard described directly this operator from the proof and has given a formula, the so-called ''execution formula'', encoding the process of cut elimination at the level of operators. Subsequent constructions by Girard proposed variants in which proofs are represented as flows, or operators in von Neumann algebras. Those models were later generalised by Seiller's Interaction Graphs models. One of the first significant applications of GoI was a better analysis of Lamping's algorithm for optimal reduction for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Disjunction And Existence Properties
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). Definitions * The disjunction property is satisfied by a theory if, whenever a sentence ''A'' ∨ ''B'' is a theorem, then either ''A'' is a theorem, or ''B'' is a theorem. * The existence property or witness property is satisfied by a theory if, whenever a sentence is a theorem, where ''A''(''x'') has no other free variables, then there is some term ''t'' such that the theory proves . Related properties Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: * The numerical existence property (NEP) states that if the theory proves (\exists x \in \mathbb)\varphi(x), where ''φ'' has no other free variables, then the theory proves \varphi(\bar) for some n \in \mathbb\text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stephen Cole Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism. Biography Kleene was awarded a bachelor's degree from Amherst College in 1930. He was awarded a Ph.D. in mathematics from Princeton University in 1934, where his thesis, entitled ''A Theory of Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Logic
Linear logic is a substructural logic proposed by French logician 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 pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logical Methods In Computer Science
''Logical Methods in Computer Science'' (LMCS) is a peer-reviewed open access scientific journal covering theoretical computer science and applied logic. It opened to submissions on September 1, 2004. The editor-in-chief is Stefan Milius ( Friedrich-Alexander Universität Erlangen-Nürnberg). History The journal was initially published by the International Federation for Computational Logic, and then by a dedicated non-profit. It moved to the . platform in 2017. The first editor-in-chief was Dana Scott. In its first year, the journal received 75 submissions. Abstracting and indexing The journal is abstracted and indexed in Current Contents/Engineering, Computing & Technology, Mathematical Reviews, Science Citation Index Expanded, Scopus, and Zentralblatt MATH. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computable
Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power. Other forms of computability are studied as well: computability notions weaker than Turing machines are studied in automata theory, while computability notions stronger than Turing machines are studied in the field of hypercomputation. Problems A central idea in computability is that of a (computational) computational problem, problem, which is a task whose computability can be explored. There are two key types of problems: * A decision problem fixes a set ''S'', which may be a set of string ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
