811 research outputs found
Relational extensions to feature logic: applications to constraint based grammars
This thesis investigates the logical and computational foundations of unification-based
or more appropriately constraint based grammars. The thesis explores extensions to
feature logics (which provide the basic knowledge representation services to constraint
based grammars) with multi-valued or relational features. These extensions are useful
for knowledge representation tasks that cannot be expressed within current feature
logics.The approach bridges the gap between concept languages (such as KL-ONE), which
are the mainstay of knowledge representation languages in AI, and feature logics. Va¬
rious constraints on relational attributes are considered such as existential membership,
universal membership, set descriptions, transitive relations and linear precedence con¬
straints.The specific contributions of this thesis can be summarised as follows:
1. Development of an integrated feature/concept logic
2. Development of a constraint logic for so called partial set descriptions
3. Development of a constraint logic for expressing linear precedence constraints
4. The design of a constraint language CL-ONE that incorporates the central ideas
provided by the above study
5. A study of the application of CL-ONE for constraint based grammarsThe thesis takes into account current insights in the areas of constraint logic programming, object-oriented languages, computational linguistics and knowledge representation
Convolution algebras: Relational convolution, generalised modalities and incidence algebras
Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of relational convolution leads to generalised binary and unary modal operators for qualitative and quantitative models, and to more conventional variants, when ternary relations arise from identities over partial semigroups. Convolution-based semantics for fragments of categorial, linear and incidence (segment or interval) logics are provided as qualitative applications. Quantitative examples include algebras of durations and mean values in the duration calculus
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Semantical Investigations on Non-classical Logics with Recovery Operators: Negation
We investigate mathematical structures that provide a natural semantics for
families of (quantified) non-classical logics featuring special unary
connectives, called recovery operators, that allow us to 'recover' the
properties of classical logic in a controlled fashion. These structures are
called topological Boolean algebras. They are Boolean algebras extended with
additional unary operations, called operators, such that they satisfy
particular conditions of a topological nature. In the present work we focus on
the paradigmatic case of negation. We show how these algebras are well-suited
to provide a semantics for some families of paraconsistent Logics of Formal
Inconsistency and paracomplete Logics of Formal Undeterminedness, which feature
recovery operators used to earmark propositions that behave 'classically' in
interaction with non-classical negations. In contrast to traditional semantical
investigations, carried out in natural language (extended with mathematical
shorthand), our formal meta-language is a system of higher-order logic (HOL)
for which automated reasoning tools exist. In our approach, topological Boolean
algebras become encoded as algebras of sets via their Stone-type
representation. We employ our higher-order meta-logic to define and interrelate
several transformations on unary set operations (operators), which naturally
give rise to a topological cube of opposition. Furthermore, our approach allows
for a uniform characterization of propositional, first-order and higher-order
quantification (also restricted to constant and varying domains). With this
work we want to make a case for the utilization of automated theorem proving
technology for doing computer-supported research in non-classical logics. All
presented results have been formally verified (and in many cases obtained)
using the Isabelle/HOL proof assistant
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