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Dana Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His work on automata theory earned him the Turing Award in 1976, while his collaborative work with Christopher Strachey in the 1970s laid the foundations of modern approaches to the semantics of programming languages. He has worked also on modal logic, topology, and category theory. Early career He received his B.A. in Mathematics from the University of California, Berkeley, in 1954. He wrote his Ph.D. thesis on ''Convergent Sequences of Complete Theories'' under the supervision of Alonzo Church while at Princeton, and defended his thesis in 1958. Solomon Feferman (2005) writes of this period: After completing his Ph.D. studies, he moved to the University of Chicago, working as an instructor there until 1960. In 1959, he ...
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Berkeley, California
Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and Emeryville to the south and the city of Albany and the unincorporated community of Kensington to the north. Its eastern border with Contra Costa County generally follows the ridge of the Berkeley Hills. The 2020 census recorded a population of 124,321. Berkeley is home to the oldest campus in the University of California System, the University of California, Berkeley, and the Lawrence Berkeley National Laboratory, which is managed and operated by the university. It also has the Graduate Theological Union, one of the largest religious studies institutions in the world. Berkeley is considered one of the most socially progressive cities in the United States. History Indigenous history The site of today's City of Berkeley was the te ...
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Fred S
Fred may refer to: People * Fred (name), including a list of people and characters with the name Mononym * Fred (cartoonist) (1931–2013), pen name of Fred Othon Aristidès, French * Fred (footballer, born 1949) (1949–2022), Frederico Rodrigues de Oliveira, Brazilian * Fred (footballer, born 1979), Helbert Frederico Carreiro da Silva, Brazilian * Fred (footballer, born 1983), Frederico Chaves Guedes, Brazilian * Fred (footballer, born 1986), Frederico Burgel Xavier, Brazilian * Fred (footballer, born 1993), Frederico Rodrigues de Paula Santos, Brazilian * Fred Again (born 1993), British songwriter known as FRED Television and movies * '' Fred Claus'', a 2007 Christmas film * ''Fred'' (2014 film), a 2014 documentary film * Fred Figglehorn, a YouTube character created by Lucas Cruikshank ** ''Fred'' (franchise), a Nickelodeon media franchise ** '' Fred: The Movie'', a 2010 independent comedy film * ''Fred the Caveman'', French Teletoon production from 2002 * Fred Flint ...
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Scott's Trick
In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in ZF, Zermelo–Fraenkel set theory without the axiom of choice (Forster 2003:182). The method was introduced by . Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for cardinal numbers and more generally for isomorphism types, for example, order types of linearly ordered sets (Jech 2003:65). It is credited to be indispensable (even in the presence of the axiom of choice) when taking ultrapowers of proper classes in model theory. (Kanamori 1994:47) Application to cardinalities The use of Scott's trick for cardinal numbers shows how the method is typically employed. The ...
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Scott Continuity
In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset ''D'' of ''P'' with supremum in ''P'', its image has a supremum in ''Q'', and that supremum is the image of the supremum of ''D'', i.e. \sqcup f = f(\sqcup D), where \sqcup is the directed join. When Q is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets. A subset ''O'' of a partially ordered set ''P'' is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets ''D'' with supremum in ''O'' have non-empty intersection with ''O''. The Scott-open subsets of a partially ordered set ''P'' form a topology on ''P'', the Scott topology. A function b ...
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Scott Information System
In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains. Definition A Scott information system, ''A'', is an ordered triple (T, Con, \vdash) * T \mbox * Con \subseteq \mathcal_f(T) \mbox T * \subseteq (Con \setminus \lbrace \emptyset \rbrace)\times T satisfying # \mbox a \in X \in Con\mboxX \vdash a # \mbox X \vdash Y \mboxY \vdash a \mboxX \vdash a # \mboxX \vdash a \mbox X \cup \ \in Con # \forall a \in T : \ \in Con # \mboxX \in Con \mbox X^\prime\, \subseteq X \mboxX^\prime \in Con. Here X \vdash Y means \forall a \in Y, X \vdash a. Examples Natural numbers The return value of a partial recursive function, which either returns a natural number or goes into an infinite recursion, can be expressed as a simple Scott information system as follows: * T := \mathbb * Con := \ \cup \ * X \vdash a\iff a \in X. That is, the result can eit ...
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Mogensen–Scott Encoding
In computer science, Scott encoding is a way to represent (recursive) data types in the lambda calculus. Church encoding performs a similar function. The data and operators form a mathematical structure which is embedded in the lambda calculus. Whereas Church encoding starts with representations of the basic data types, and builds up from it, Scott encoding starts from the simplest method to compose algebraic data types. Mogensen–Scott encoding extends and slightly modifies Scott encoding by applying the encoding to Metaprogramming. This encoding allows the representation of lambda calculus terms, as data, to be operated on by a meta program. History Scott encoding appears first in a set of unpublished lecture notes by Dana Scott whose first citation occurs in the book ''Combinatorial Logic, Volume II''. Michel Parigot gave a logical interpretation of and strongly normalizing recursor for Scott-encoded numerals, referring to them as the "Stack type" representation of nu ...
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Scott Domain
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete cpo. They are named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to algebraic lattices, being different only in possibly lacking a greatest element. They are also closely related to Scott information systems, which constitute a "syntactic" representation of Scott domains. While the term "Scott domain" is widely used with the above definition, the term "domain" does not have such a generally accepted meaning and different authors will use different definitions; Scott himself used "domain" for the structures now called "Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications. Originally, Dana Scott demanded a complete lattice, and the Russian mathematician Yuri Yershov constructed the isomorphic structure of cpo. B ...
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Nondeterministic Finite Automaton
In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if * each of its transitions is ''uniquely'' determined by its source state and input symbol, and * reading an input symbol is required for each state transition. A nondeterministic finite automaton (NFA), or nondeterministic finite-state machine, does not need to obey these restrictions. In particular, every DFA is also an NFA. Sometimes the term NFA is used in a narrower sense, referring to an NFA that is ''not'' a DFA, but not in this article. Using the subset construction algorithm, each NFA can be translated to an equivalent DFA; i.e., a DFA recognizing the same formal language. Like DFAs, NFAs only recognize regular languages. NFAs were introduced in 1959 by Michael O. Rabin and Dana Scott, who also showed their equivalence to DFAs. NFAs are used in the implementation of regular expressions: Thompson's construction is an algorithm for compiling a regular expression to an NFA ...
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Modal μ-calculus
In theoretical computer science, the modal μ-calculus (Lμ, Lμ, sometimes just μ-calculus, although this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic. The (propositional, modal) μ-calculus originates with Dana Scott and Jaco de Bakker, and was further developed by Dexter Kozen into the version most used nowadays. It is used to describe properties of labelled transition systems and for verifying these properties. Many temporal logics can be encoded in the μ-calculus, including CTL* and its widely used fragments— linear temporal logic and computational tree logic. An algebraic view is to see it as an algebra of monotonic functions over a complete lattice, with operators consisting of functional composition plus the least and greatest fixed point operators; from this viewpoint, the modal μ-calculus ...
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Semantics Of Programming Languages
In programming language theory, semantics is the rigorous mathematical study of the meaning of programming languages. Semantics assigns computational meaning to valid strings in a programming language syntax. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will be executed on a certain platform, hence creating a model of computation. History In 1967, Robert W. Floyd publishes the paper ''Assigning meanings to programs''; his chief aim is "a rigorous standard for proofs about computer programs, including proofs of correctness, equivalence, and termination". Floyd further writes: A semantic definition of a programming language, in our approach, is founded on a syntactic definition. It must specify which of the phrases in a syntactically correct program represent commands, and what condit ...
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Logic Of Computable Functions
Logic of Computable Functions (LCF) is a deductive system for computable functions proposed by Dana Scott in 1969 in a memorandum unpublished until 1993. It inspired: * Logic for Computable Functions (LCF), theorem proving logic by Robin Milner. * Programming Computable Functions In computer science, Programming Computable Functions (PCF) is a typed functional language introduced by Gordon Plotkin in 1977, based on previous unpublished material by Dana Scott. It can be considered to be an extended version of the typed lam ... (PCF), small theoretical programming language by Gordon Plotkin.{{ cite journal , first = Gordon D. , last = Plotkin , authorlink = Gordon Plotkin , title = LCF considered as a programming language , journal = Theoretical Computer Science , year = 1977 , pages = 223–255 , volume = 5 , issue = 3 , doi = 10.1016/0304-3975(77)90044-5 , url = http://homepages.inf.ed.ac.uk/gdp/publications/LCF.pdf , doi-access = free References ...
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Cartesian Monoid
A Cartesian monoid is a monoid, with additional structure of pairing and projection operators. It was first formulated by Dana Scott and Joachim Lambek independently.. Definition A Cartesian monoid is a structure with signature \langle *,e,(-,-),L,R\rangle where * and (-,-) are binary operations, L, R, and e are constants satisfying the following axioms for all x,y,z in its universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...: ; Monoid : * is a monoid with identity e ; Left Projection : L * (x,\,y) = x ; Right Projection :R * (x,\,y) = y ; Surjective Pairing : (L*x,\,R*x) = x ; Right Homogeneity : (x*z,\,y*z)=(x,\,y) * z The interpretation is that L and R are left and right projection functions respectively for the pairing function (-,-). References {{reflist M ...
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