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MINLOG
MINLOG is a proof assistant developed at the University of Munich by the team of Helmut Schwichtenberg. MINLOG is based on First-order logic, first order natural deduction calculus. It is intended to reason about computable functionals, using Minimal logic, minimal rather than Classical logic, classical or intuitionistic logic. The primary motivation behind MINLOG is to exploit the proofs-as-programs paradigm for program development and program verification. Proofs are, in fact, treated as first-class objects, which can be normalized. If a formula is existential, then its proof can be used for reading off an instance of it or changed appropriately for program development by proof transformation. To this end, MINLOG is equipped with tools to extract functional programs directly from proof terms. This also applies to non-constructive proofs, using a refined A-translation. The system is supported by automatic proof search and normalization by evaluation as an efficient term rewriting de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normalization By Evaluation
In programming language semantics, normalisation by evaluation (NBE) is a style of obtaining the normal form of terms in the λ-calculus by appealing to their denotational semantics. A term is first ''interpreted'' into a denotational model of the λ-term structure, and then a canonical (β-normal and η-long) representative is extracted by ''reifying'' the denotation. Such an essentially semantic, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms. NBE was first described for the simply typed lambda calculus. It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory. Outline Consider the simply typed lambda calculus, where types τ can be basic types (α), function types (→ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Schwichtenberg
Helmut Schwichtenberg (born 5 April 1942 in Żagań) is a German mathematical logician. Schwichtenberg studied mathematics from 1961 at the FU Berlin and from 1964 at the University of Münster, where he received his doctorate in 1968 from Dieter Rödding. He then worked as an assistant and then as a professor in Münster, and since 1978 has been professor of mathematical logic at the Ludwig-Maximilians-Universität Munich (successor of Kurt Schütte). Schwichtenberg deals with, among other things, proof theory, theory of computability, lambda calculus and applications of logic in computer science. He is a member of the Bavarian Academy of Sciences The Bavarian Academy of Sciences and Humanities (german: Bayerische Akademie der Wissenschaften) is an independent public institution, located in Munich. It appoints scholars whose research has contributed considerably to the increase of knowledg .... Selected publications * * (2nd edition 2000: ) * * References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists''"'' is a quantifier, while ''x'' is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of ax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning. Motivation Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise ''Principia Mathematica''. Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz that advocated a more natural treatment of logic, Jaśkowski made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935. His proposals led to diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Functionals
Computability is the ability to solve a problem in an effective manner. 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) 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 strings, natural numbers, or o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Logic
Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded middle as well as the principle of explosion (''ex falso quodlibet''), and therefore holding neither of the following two derivations as valid: :\vdash (B \lor \neg B) :(A \land \neg A) \vdash where A and B are any propositions. Most constructive logics only reject the former, the law of excluded middle. In classical logic, the ''ex falso'' laws :(A \land \neg A) \to B, :\neg(A \lor \neg A) \to B, :\neg A \to (A \to B), as well as their variants with A and \neg A switched, are equivalent to each other and valid. Minimal logic also rejects those principles. Axiomatization Minimal logic is axiomatized over the positive fragment of intuitionistic logic. These logics may be formulated in the language using implication \to, conjunction \land and disjunction \lor as the basic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclop ... [...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 the 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 interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
