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Hypersequent
In mathematical logic, the hypersequent framework is an extension of the proof-theoretical framework of sequent calculi used in structural proof theory to provide analytic calculi for logics that are not captured in the sequent framework. A hypersequent is usually taken to be a finite multiset of ordinary sequents, written : \Gamma_1 \Rightarrow \Delta_1 \mid \cdots \mid \Gamma_n \Rightarrow \Delta_n The sequents making up a hypersequent are called components. The added expressivity of the hypersequent framework is provided by rules manipulating different components, such as the communication rule for the intermediate logic LC ( Gödel–Dummett logic) : \frac or the modal splitting rule for the modal logic S5: : \frac Hypersequent calculi have been used to treat modal logics, intermediate logics, and substructural logics. Hypersequents usually have a formula interpretation, i.e., are interpreted by a formula in the object language, nearly always as some kind of disjunction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Structural Proof Theory
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide decision procedures, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to model theory. Analytic proof The notion of analytic proof was introduced into proof theory by Gerhard Gentzen for the sequent calculus; the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as was shown by Dag Prawitz; the definition is slightly more complex—the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Structural Rule
In the logical discipline of proof theory, a structural rule is an inference rule of a sequent calculus that does not refer to any logical connective but instead operates on the sequents directly. Structural rules often mimic the intended meta-theoretic properties of the logic. Logics that deny one or more of the structural rules are classified as substructural logics. Common structural rules Three common structural rules are: * , where the hypotheses or conclusion of a sequence may be extended with additional members. In symbolic form weakening rules can be written as \frac on the left of the turnstile, and \frac on the right. Known as monotonicity of entailment in classical logic. * , where two equal (or unifiable) members on the same side of a sequent may be replaced by a single member (or common instance). Symbolically: \frac and \frac. Also known as factoring in automated theorem proving systems using resolution. Known as idempotency of entailment in classical logic. * Exc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cut Elimination
The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in part I of his landmark 1935 paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus making use of the cut rule also possesses a cut-free proof, that is, a proof that does not make use of the cut rule. The Natural Deduction version of cut-elimination, known as ''normalization theorem'', has been first proved for a variety of logics by Dag Prawitz in 1965 (a similar but less general proof was given the same year by Andrès Raggio). The cut rule A sequent is a logical expression relating multiple formulas, in the form , which is to be read as "If all of hold, then at least one of must hold", or (as Gentzen glossed): "If (A_1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
International Conference On Automated Reasoning With Analytic Tableaux And Related Methods
The International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX) is an annual international academic conference that deals with all aspects of automated reasoning with analytic tableaux. Periodically, it joins with CADE and TPHOLs into the International Joint Conference on Automated Reasoning (IJCAR). The first table convened in 1992. Since 1995, the proceedings of this conference have been published by Springer's LNAI series. In August 2006 TABLEAUX was part of the Federated Logic Conference in Seattle, USA. The following TABLEAUX were held in 2007 in Aix en Provence, France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan ..., as part of IJCAR 2008, in Sydney, Australia, as TABLEAUX 2009, in Oslo, Norway, as part of IJCAR 2010, Edinburgh, U ... [...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] |
Information Processing Letters
''Information Processing Letters'' is a peer review, peer-reviewed scientific journal in the field of computer science, published by Elsevier. The aim of the journal is to enable fast dissemination of results in the field of Data processing, information processing in the form of short papers. Submissions are limited to nine double-spaced pages. The scope of IPL covers fundamental aspects of information processing and computing. This naturally covers topics in the broadly understood field of theoretical computer science, including algorithms, formal languages and automata, computational complexity, computational logic, distributed and parallel algorithms, computational geometry, learning theory, computational number theory, computational biology, coding theory, theoretical cryptography, and applied discrete mathematics. Generally, submissions in all areas of scientific inquiry are considered, provided that they describe research contributions credibly motivated by applications to com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Symposium On Logic In Computer Science
The ACM–IEEE Symposium on Logic in Computer Science (LICS) is an annual academic conference on the theory and practice of computer science in relation to mathematical logic. Extended versions of selected papers of each year's conference appear in renowned international journals such as Logical Methods in Computer Science and ACM Transactions on Computational Logic. History LICS was originally sponsored solely by the IEEE, but as of the 2014 founding of the ACM Special Interest Group on Logic and Computation LICS has become the flagship conference of SIGLOG, under the joint sponsorship of ACM and IEEE. From the third installment in 1988 until 2013, the cover page of the conference proceedings has featured an artwork entitled ''Irrational Tiling by Logical Quantifiers'', by Alvy Ray Smith. Since 1995, each year the '' Kleene award'' is given to the best student paper. In addition, since 2006, the ''LICS Test-of-Time Award'' is given annually to one among the twenty-year-old L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is ''expressively complete'' if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cut Rule
In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. Its meaning is that, if a formula ''A'' appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula ''A'' does not appear can be deduced. This applies to cases of modus ponens, such as how instances of ''man'' are eliminated from ''Every man is mortal, Socrates is a man'' to deduce ''Socrates is mortal''. Formal notation It is normally written in formal notation in sequent calculus notation as : : : \begin\Gamma \vdash A, \Delta \quad \Gamma', A \vdash \Delta' \\ \hline \Gamma, \Gamma' \vdash \Delta, \Delta'\end cut Elimination The cut rule is the subject of an important theorem, the cut-elimination theorem The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
