The \u03f5-logic (which is called \u03f5E-logic in this paper) of Kuyper and Terwijn is a variant of first order logic with the same syntax, in which the models are equipped with probability measures and in which the \u2200x quantifier is interpreted as “there exists a set A of measure \u22651\u2212\u03f5 such that for each x\u2208A, ….” Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational \u03f5\u2208(0,1), respectively \u03a311-complete and \u03a011-hard, and ii) for \u03f5=0, respectively decidable and \u03a301-complete. The adjective “general” here means “uniformly over all languages.”
\nWe extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability by and validity over finite models in \u03f5E-logic are, i) for rational \u03f5\u2208(0,1), respectively \u03a301- and \u03a001-complete, and ii) for \u03f5=0, respectively decidable and \u03a001-complete. Although partial results toward the countable case are also achieved, the computability of \u03f5E-logic over countable models still remains largely unsolved. In addition, most of the results, of this paper and of Kuyper and Terwijn, do not apply to individual languages with a finite number of unary predicates. Reducing this requirement continues to be a major point of research.
\nOn the positive side, we derive the decidability of the corresponding problems for monadic relational languages — equality- and function-free languages with finitely many unary and zero other predicates. This result holds for all three of the unrestricted, the countable, and the finite model cases.
\nApplications in computational learning theory, weighted graphs, and neural networks are discussed in the context of these decidability and undecidability results.<\/p>\n","protected":false},"excerpt":{"rendered":"
The \u03f5-logic (which is called \u03f5E-logic in this paper) of Kuyper and Terwijn is a variant of first order logic with the same syntax, in which the models are equipped with probability measures and in which the \u2200x quantifier is interpreted as “there exists a set A of measure \u22651\u2212\u03f5 such that for each x\u2208A, […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"gregyang","user_id":"37245"}],"msr_publishername":"Taylor and Francis","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Journal of Applied Non-Classical Logics","msr_editors":"","msr_how_published":"","msr_isbn":"1166-3081","msr_issue":"","msr_journal":"Journal of Applied Non-Classical 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