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Focused Proof
In mathematical logic, focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and reductive logic. They form the most general definition of ''goal-directed'' proof-search—in which someone chooses a formula and performs hereditary reductions until the result meets some condition. The extremal case where reduction only terminates when axioms are reached forms the sub-family of ''uniform'' proofs. A sequent calculus is said to have the focusing property when focused proofs are complete for some terminating condition. For System LK, System LJ, and System LL, uniform proofs are focused proofs where all the atoms are assigned negative polarity. Many other sequent calculi has been shown to have the focusing property, notably the nested sequent calculi of both the classical and intuitionistic variants of the modal logics in the S5 cube. Uniform proofs In the sequent calculus for an intuition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Proof Theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also ... [...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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backtracking
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution. The classic textbook example of the use of backtracking is the eight queens puzzle, that asks for all arrangements of eight chess queens on a standard chessboard so that no queen attacks any other. In the common backtracking approach, the partial candidates are arrangements of ''k'' queens in the first ''k'' rows of the board, all in different rows and columns. Any partial solution that contains two mutually attacking queens can be abandoned. Backtracking can be applied only for problems which admit the concept of a "partial candidate solution" and a relatively quick test of whether it can possibly be completed to a valid solution. It is useless, for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduct
In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion". Definition Let ''A'' be an algebraic structure (in the sense of universal algebra) or a structure in the sense of model theory, organized as a set ''X'' together with an indexed family of operations and relations φ''i'' on that set, with index set ''I''. Then the reduct of ''A'' defined by a subset ''J'' of ''I'' is the structure consisting of the set ''X'' and ''J''-indexed family of operations and relations whose ''j''-th operation or relation for ''j'' ∈ ''J'' is the ''j''-th operation or relation of ''A''. That is, this reduct is the structure ''A'' with the omission of those operations and relations φ''i'' for which ''i'' is not in ''J''. A structure ''A'' is an expansion of ''B'' just when ''B'' is a reduct of ''A''. That is, reduct and expansion are mutual converses. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Logic Programming
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applying logical reasoning to that knowledge, to solve problems in the domain. Major logic programming language families include Prolog, Answer Set Programming (ASP) and Datalog. In all of these languages, rules are written in the form of ''clauses'': :A :- B1, ..., Bn. and are read as declarative sentences in logical form: :A if B1 and ... and Bn. A is called the ''head'' of the rule, B1, ..., Bn is called the ''body'', and the Bi are called '' literals'' or conditions. When n = 0, the rule is called a ''fact'' and is written in the simplified form: :A. Queries (or goals) have the same syntax as the bodies of rules and are commonly written in the form: :?- B1, ..., Bn. In the simplest case of Horn clauses (or "definite" clauses), all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Programming Language
A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually defined by a formal language. Languages usually provide features such as a type system, Variable (computer science), variables, and mechanisms for Exception handling (programming), error handling. An Programming language implementation, implementation of a programming language is required in order to Execution (computing), execute programs, namely an Interpreter (computing), interpreter or a compiler. An interpreter directly executes the source code, while a compiler produces an executable program. Computer architecture has strongly influenced the design of programming languages, with the most common type (imperative languages—which implement operations in a specified order) developed to perform well on the popular von Neumann architecture. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harrop Formula
In intuitionistic logic, the Harrop formulae, named after Ronald Harrop, are the class of formulae inductively defined as follows: * Atomic formulae are Harrop, including falsity (⊥); * A \wedge B is Harrop provided A and B are; * \neg F is Harrop for any well-formed formula F; * F \rightarrow A is Harrop provided A is, and F is any well-formed formula; * \forall x. A is Harrop provided A is. By excluding disjunction and existential quantification (except in the antecedent of implication), non-constructive predicates are avoided, which has benefits for computer implementation. Discussion Harrop formulae are "well-behaved" also in a constructive context. For example, in Heyting arithmetic , Harrop formulae satisfy a classical equivalence not generally satisfied in constructive logic: :\neg \neg A \leftrightarrow A. There are however \Pi_1-statements that are -independent, meaning these are simple \forall x. A statements for which excluded middle is not -provable. Ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modal Logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality, causation. For instance, in epistemic modal logic, the well-formed_formula, formula \Box P can be used to represent the statement that P is known. In deontic modal logic, that same formula can represent that P is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula \Box P \rightarrow P as a Tautology_(logic), tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operation, unary operators such as \Diamond and \Box, representing possibility and necessi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Proof
{{Short description, Fundamental theory of logical analysis In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and that does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided a proof of the theorem that was free from intuitions concerning lines crossing each other at a point, and so he felt happy calling it analytic (Bolzano 1817). Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter (Sebastik 2007). In proof theory, an analytic proof has come to mean a proof whose structure is simple in a special way, due to conditions on the kind of inferences that ensure none of them go beyond what is contained in the assumptions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
