We give a sufficient condition for Kripke completeness of modal logics enriched with the transitive closure modality. More precisely, we show that if a logic admits what we call definable filtration (ADF), then such an expansion of the logic is complete; in addition, has the finite model property, and again ADF. This argument can be iterated, and as an application we obtain the finite model property for PDL-like expansions of logics that ADF.
The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Borel equivalence relations on that space. In the case of the Baire space, the two smooth relations are closed and the embedding is homeomorphic.
The present paper is devoted to study some completeness properties of transitive binary relational set, i.e., a set together with a transitive binary relation (so called t-set).
We study spherical completeness of ball spaces and its stability under expansions. We introduce the notion of an ultra-diameter, mimicking diameters in ultrametric spaces. We prove some positive results on preservation of spherical completeness involving ultra-diameters with values in narrow partially ordered sets. Finally, we show that in general, chain intersection closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.
It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra X+ over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q+, as well as the logic of a subalgebra of C+. This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R+. In th…
There has been a significant interest in extending various modal logics with intersection, the most prominent examples being epistemic and doxastic logics with distributed knowledge. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 like in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logics. A standard proof method for the S5 case was outlined in [8] and later explicated in more detail in [13], using an "unraveling-folding method" case to achieve a treelike model to deal with the …
Institute of Computer Science Czech Academy of Sciences（捷克科学院计算机科学研究所）；The Saul Kripke Center and Philosophy Program The Graduate Center, CUNY（纽约市立大学研究生中心苏尔·克里普克中心与哲学项目）
We give a constructive treatment of some basic concepts and results in semigroup theory. Focusing on semigroups equipped with an apartness relation, we give analogues, from the point of view of apartness, of several classical constructions and results, including transitive closure and congruence closure, free semigroups, periodicity, Rees factors, and Green's relations.
Most Properties are Undecidable for Transitive Tense Logics
The Tsinghua-UvA JRC for Logic Department of Philosophy Tsinghua University（清华大学哲学系清华-阿姆斯特丹逻辑联合研究中心）；Institute for Logic, Language and Computation University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）
The paper proves finite model property and decidability for a family of modal logics. A binary relation $R$ is called pretransitive, if $R^*=\cup_{i\leq m} R^i$ for some $m\geq 0$, where $R^*$ is the transitive reflexive closure of $R$. By the height of $(W,R)$ we mean the height of the preorder $(W,R^*)$. Special partitionings (filtrations) are described for pretransitive frames of finite height, which implies finite model property and decidability of logics of these frames.
We consider dynamic versions of epistemic logic as formulated in Baltag and Moss "Logics for epistemic programs" (2004). That paper proposed a logical language (actually families of languages parameterized by action signatures) for dynamic epistemic logic. It had been shown that validity in the language is Pi-1-1-complete, so there are no recursively axiomatized complete logical systems for it. In contrast, this paper proves a weak completeness result for the fragment without action iteration, and a strong completeness result for the fragment without action iteration and common knowledge. Our …
Institute of Computer Science University of Wrocław（弗罗茨瓦夫大学计算机科学研究所）；Technische Universität Dresden（德累斯顿工业大学）
The Triguarded Fragment (TGF) is among the most expressive decidable fragments of first-order logic, subsuming both its two-variable and guarded fragments without equality. We show that the TGF has the finite model property (providing a tight doubly exponential bound on the model size) and hence finite satisfiability coincides with satisfiability known to be N2ExpTime-complete. Using similar constructions, we also establish 2ExpTime-completeness for finite satisfiability of the constant-free (tri)guarded fragment with transitive guards.
