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First-order Syntactical Anti-unification
Anti-unification is the process of constructing a generalization common to two given symbolic expressions. As in unification, several frameworks are distinguished depending on which expressions (also called terms) are allowed, and which expressions are considered equal. If variables representing functions are allowed in an expression, the process is called "higher-order anti-unification", otherwise "first-order anti-unification". If the generalization is required to have an instance literally equal to each input expression, the process is called "syntactical anti-unification", otherwise "E-anti-unification", or "anti-unification modulo theory". An anti-unification algorithm should compute for given expressions a complete and minimal generalization set, that is, a set covering all generalizations and containing no redundant members, respectively. Depending on the framework, a complete and minimal generalization set may have one, finitely many, or possibly infinitely many members, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unification (computer Science)
In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expression (mathematics), expressions, each of the form ''Left-hand side = Right-hand side''. For example, using ''x'',''y'',''z'' as variables, and taking ''f'' to be an uninterpreted function, the Singleton (mathematics), singleton equation set is a syntactic first-order unification problem that has the substitution as its only solution. Conventions differ on what values variables may assume and which expressions are considered equivalent. In first-order syntactic unification, variables range over first-order terms and equivalence is syntactic. This version of unification has a unique "best" answer and is used in logic programming and programming language type system implementation, especially in Hindley–Milner based type inference algorithms. In higher-order unification, possibly restricted to higher-order pattern unification, terms may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Mapping
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Theorem Proving
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major motivating factor for the development of computer science. Logical foundations While the roots of formalized Logicism, logic go back to Aristotelian logic, Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalized mathematics. Gottlob Frege, Frege's ''Begriffsschrift'' (1879) introduced both a complete propositional logic, propositional calculus and what is essentially modern predicate logic. His ''The Foundations of Arithmetic, Foundations of Arithmetic'', published in 1884, expressed (parts of) mathematics in formal logic. This approach was continued by Bertrand Russell, Russell and Alfred North Whitehead, Whitehead in their influential ''Principia Mathematica'', first published 1910� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ACM Transactions On Programming Languages And Systems
The ''ACM Transactions on Programming Languages and Systems'' (''TOPLAS'') is a bimonthly, open access, peer-reviewed scientific journal on the topic of programming languages published by the Association for Computing Machinery. Background Published since 1979, the journal's scope includes programming language design, implementation, and semantics of programming languages, compilers and interpreters, run-time systems, storage allocation and garbage collection, and formal specification, testing, and verification of software. It is indexed in Scopus and SCImago. The editor-in-chief is Andrew Myers (Cornell University). According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.410. References External links * TOPLASat ACM Digital Library TOPLASat DBLP DBLP is a computer science bibliography website. Starting in 1993 at Universität Trier in Germany, it grew from a small collection of HTML files and became an organization hosting a datab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SRI International
SRI International (SRI) is a nonprofit organization, nonprofit scientific research, scientific research institute and organization headquartered in Menlo Park, California, United States. It was established in 1946 by trustees of Stanford University to serve as a center of innovation to support economic development in the region. The organization was founded as the Stanford Research Institute. SRI formally separated from Stanford University in 1970 and became known as SRI International in 1977. SRI performs client-sponsored research and development for government agencies, commercial businesses, and private foundations. It also licenses its technologies, forms strategic partnerships, sells products, and creates Research spin-off, spin-off companies. SRI's headquarters are located near the Stanford University campus. SRI's annual revenue in 2014 was approximately $540 million, which tripled from 1998 under the leadership of Curtis Carlson. In 1998, the organization was on the ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equational Theories
Equational may refer to: * Equative (other), a construction in linguistics * something pertaining to equations, in mathematics * something pertaining to equality Equality generally refers to the fact of being equal, of having the same value. In specific contexts, equality may refer to: Society * Egalitarianism, a trend of thought that favors equality for all people ** Political egalitarianism, in which ..., in logic See also * * Equation (other) * Equality (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inductive Logic Programming
Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "''inductive''" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. * Schema: ''positive examples'' + ''negative examples'' + ''background knowledge'' ⇒ ''hypothesis''. Inductive logic programming is particularly useful in bioinformatics and natural language processing. History Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golem (ILP)
Golem is an inductive logic programming algorithm developed by Stephen Muggleton and Cao Feng in 1990. It uses the technique of Inductive logic programming#Least general generalisation, relative least general generalisation proposed by Gordon Plotkin, leading to a bottom-up search through the theta-subsumption, subsumption lattice. In 1992, shortly after its introduction, Golem was considered the only inductive logic programming system capable of scaling to tens of thousands of examples. Description Golem takes as input a Definite clause, definite program as background knowledge together with sets of positive and negative examples, denoted E^ and E^ respectively. The overall idea is to construct the least general generalisation of E^ with respect to the background knowledge. However, if is not merely a finite set of Ground expression, ground Atomic formula, atoms, then this relative least general generalisation may not exist. Therefore, rather than using directly, Golem uses t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inductive Logic Programming
Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "''inductive''" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. * Schema: ''positive examples'' + ''negative examples'' + ''background knowledge'' ⇒ ''hypothesis''. Inductive logic programming is particularly useful in bioinformatics and natural language processing. History Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hash Table
In computer science, a hash table is a data structure that implements an associative array, also called a dictionary or simply map; an associative array is an abstract data type that maps Unique key, keys to Value (computer science), values. A hash table uses a hash function to compute an ''index'', also called a ''hash code'', into an array of ''buckets'' or ''slots'', from which the desired value can be found. During lookup, the key is hashed and the resulting hash indicates where the corresponding value is stored. A map implemented by a hash table is called a hash map. Most hash table designs employ an Perfect hash function, imperfect hash function. Hash collision, Hash collisions, where the hash function generates the same index for more than one key, therefore typically must be accommodated in some way. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored in the table. Many hash table designs also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton (mathematics)
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and \ are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
