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Some Applications of Lawvere’s fixpoint Theorem

Frontiers of Philosophy in China 2019年第3期14卷 490-510页

作者： LI Xi Department of Philosophy Central South UniversityChangsha 410083China

The famous diagonal argument plays a prominent role in set the ory as well as in the proof of undecidability results in computability the ory and incompleteness results in ***(1969)brings to light the common schema among them through a pretty neat fixpoint the orem which generalizes the diagonal argument behind Cantor's theorem and characterizes self-reference explicitly in category *** until Yanofsky(2003)rephrases Lawvere’s fixpoint theorem using sets and functions,Lawvere's work has been overlooked by *** paper will continue Yanofsky's work,and show more applications of Lawvere's fixpoint theorem to demonstrate the ubiquity of the *** example,this paper will use it to construct uncomputable real number,unnameable real number,partial re cursive but not potentially recursive function,Berry paradox,and fast growing Busy Beaver *** interesting lambda fixpoint combinators can also be fitted into this *** Curry's Y combinator and Turing's combinator follow from Lawvere's theorem,as well as their call-by-value *** last,it can be shown that the lambda calculus version of the fixpoint lemma also fits Lawvere’s schema.

关键词： paradox fixpoint diagonalization combinator

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Declarative semantics of programming in residuated lattice-valued logic

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Science China(Technological Sciences) 2000年第5期43卷 481-494页

作者：应明生 State Key Laboratory of Intelligent Technology and Systems Department of Computer Science and Technology Tsinghua University Beijing 100084 China

We give two generalizations of Tarski’s fixpoint theorem in the setting of residuated lattices and use them to establish van Emdem-Kowalski’s least fixpoint semantics for residuated lattice-valued logic programs.

关键词： logic programming declarative semantics residuated lattice fixpoint theorem completion of program.

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A Constructor-Based EI-Model Semantics ofEI-CTRS

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Journal of Computer Science & Technology 1995年第1期10卷 85-95页

作者：王怀民陈火旺 Department of Computer Science National University of Defence Technology Changsha

This paper investigates the semantics of conditional term rewriting systemswith negation (denoted by EI-CTRS), called constructor-based EI-model se-mantics. The introduction of '≠' in EI-CTRS make EI-CTRS more difficult tostudy. This is in part because of a failure of EI-CTRS to guarantee that thereexist least Herbrand models in classical logical point of views. The key idea ofEI-model is to explain that 't ≠ s' means that the two concepts representedby t and s respectively actually belong to distinguished basic concepts repre-sented by two constructor-ground terms. We define the concept of EI-model,and show that there exist least Herbrand ELmodels for EI-satisfiable *** algebraic and logic point of view, we show that there are very strong rea-sons for regarding the least Herbrand EI-models as the intended semantics ofEI-CTRS. According to fixpoint theory, we develop a method to construct leastHerbrand EI-models in a bottom-up manner. Moreover, we discuss soundnessand completeness of EI-rewrite for EI-model semantics.

关键词： Conditional term rewriting model semantics negation fixpoint constructor

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Specification and Verification for Semi-Structured Data

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Wuhan University Journal of Natural Sciences 2006年第1期11卷 107-112页

作者： CHEN Tao-lue HAN Ting-ting LU Jian State Key Laboratory of Novel Software Technology Nanjing University Nanjing 210093 Jiangsu China

Tree logic, inherited from ambient logic, is introduced as the formal foundation of related programming language and type systems, In this paper, we introduce recursion into such logic system, which can describe the tree data more dearly and concisely. By making a distinction between proposition and predicate, a concise semantics interpretation for our modal logic is given. We also develop a model checking algorithm for the logic without △ operator. The correctness of the algorithm is shown. Such work can be seen as the basis of the semi-structured data processing language and more flexible type system.

关键词： semi structured data tree logic fixpoint model checking algorithm

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