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Register a new userAlgebraic Logic, I Quantifier Theories and Completeness Theorems
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Added by
Zhaohua Luo
Publication date
2013
fields
Informatics Engineering
and research's language is
English
Authors
Zhaohua Luo
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Algebraic logic studies algebraic theories related to proposition and first-order logic. A new algebraic approach to first-order logic is sketched in this paper. We introduce the notion of a quantifier theory, which is a functor from the category of a monad of sets to the category of Boolean algebras, together with a uniquely determined system of quantifiers. A striking feature of this approach is that Cayleys Completeness Theorem and Godels Completeness Theorem can be stated and proved in a much simpler fashion for quantifier theories. Both theorems are due to Halmos for polyadic algebras. We also present a simple transparent treatment of ultraproducts of models of a quantifier theory.
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Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${mathcal{B}}^nto{mathcal{B}}$ for every n by an algebraic method.
The concept of a clone is central to many branches of mathematics, such as universal algebra, algebraic logic, and lambda calculus. Abstractly a clone is a category with two objects such that one is a countably infinite power of the other. Left and right algebras over a clone are covariant and contravariant functors from the category to that of sets respectively. In this paper we show that first-order logic can be studied effectively using the notions of right and left algebras over a clone. It is easy to translate the classical treatment of logic into our setting and prove all the fundamental theorems of first-order theory algebraically.
The nonstandard approach to program semantics has successfully resolved the completeness problem of Floyd-Hoare logic. The kno
Coalition logic is one of the most popular logics for multi-agent systems. While epistemic extensions of coalition logic have received much attention, existence of their complete axiomatisations has so far been an open problem. In this paper we settle several of those problems. We prove completeness for epistemic coalition logic with common knowledge, with distributed knowledge, and with both common and distributed knowledge, respectively.
Existing work on theorem proving for the assertion language of separation logic (SL) either focuses on abstract semantics which are not readily available in most applications of program verification, or on concrete models for which completeness is not possible. An important element in concrete SL is the points-to predicate which denotes a singleton heap. SL with the points-to predicate has been shown to be non-recursively enumerable. In this paper, we develop a first-order SL, called FOASL, with an abstracted version of the points-to predicate. We prove that FOASL is sound and complete with respect to an abstract semantics, of which the standard SL semantics is an instance. We also show that some reasoning principles involving the points-to predicate can be approximated as FOASL theories, thus allowing our logic to be used for reasoning about concrete program verification problems. We give some example theories that are sound with respect to different variants of separation logics from the literature, including those that are incompatible with Reynoldss semantics. In the experiment we demonstrate our FOASL based theorem prover which is able to handle a large fragment of separation logic with heap semantics as well as non-standard semantics.
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