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Register a new userExpressive power of infinitary [0, 1]-valued logics
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Authors
Christopher J. Eagle
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We consider model-theoretic properties related to the expressive power of three analogues of $L_{omega_1, omega}$ for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for complete separable metric structures.
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We describe an infinitary logic for metric structures which is analogous to $L_{omega_1, omega}$. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.
This paper shows how to transform explosive many-valued systems into paraconsistent logics. We investigate especially the case of three-valued systems showing how paraconsistent three-valued logics can be obtained from them.
The logics RL, RP, and RG have been obtained by expanding Lukasiewicz logic L, product logic P, and Godel--Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in L, P, and G. Namely, RL is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG we provide a base of admissible rules. We leave the problem open whether the variety of rational Godel algebras is Q-universal.
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Eric Alsmann (School of Electrical Engineering
, Computer Science
, n University of Kassel
2021
We investigate the expressive power of the two main kinds of program logics for complex, non-regular program properties found in the literature: those extending propositional dynamic logic (PDL), and those extending the modal mu-calculus. This is inspired by the recent discovery of a decidable program logic called Visibly Pushdown Fixpoint Logic with Chop which extends both the modal mu-calculus and PDL over visibly pushdown languages, which, so far, constituted the ends of two pillars of decidable program logics. Here we show that this logic is not only more expressive than either of its two fragments, but in fact even more expressive than their union. Hence, the decidability border amongst program logics has been properly pushed up. We complete the picture by providing results separating all the PDL-based and modal fixpoint logics with regular, visibly pushdown and arbitrary context-free constructions.
Data streams occur widely in various real world applications. The research on streaming data mainly focuses on the data management, query evaluation and optimization on these data, however the work on reasoning procedures for streaming knowledge bases on both the assertional and terminological levels is very limited. Typically reasoning services on large knowledge bases are very expensive, and need to be applied continuously when the data is received as a stream. Hence new techniques for optimizing this continuous process is needed for developing efficient reasoners on streaming data. In this paper, we survey the related research on reasoning on expressive logics that can be applied to this setting, and point to further research directions in this area.
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