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

Scientific Progress on the Semantic View : An Account of Scientific Progress as Objective Logical and Empirical Strength Increments

The aim of this master thesis is to make a convincing argument that scientific progress can be spoken of in objective terms. In order to make this argument I will propose a philosophical theory of scientific progress. Two concepts will be constructed with this aim in mind, both which are types of strength measures on scientific theories. The first concept, that of logical strength, pertains to the way a theory may exclude, or permit less, model classes compared to another theory. The second concept, that of empirical strength, pertains to an objective measure of the informational content of data models, defined in terms of Kolmogorov complexity. This latter idea stems from communication and computational theory. Scientific progress is then defined as the interaction, or the stepwise increases, of these two strength measures. Central for the conception of a scientific theory is the philosophical framework known as The Semantic View of Scientific Theories. This view can briefly be characterized as an empirical extension of Tarskian model-theory. Another central notion for this theory of scientific progress is the philosophical or metaphysical thesis called structural realism. Both will accordingly be explained and argued for. Finally, as a test on this proposed theory of scientific progress, it will be applied to two examples of theory transition from the history of physical theory. I conclude that the proposed theory handles these two cases well

Modular Logic Programming: Full Compositionality and Conflict Handling for Practical Reasoning

With the recent development of a new ubiquitous nature of data and the profusity of available knowledge, there is nowadays the need to reason from multiple sources of often incomplete and uncertain knowledge. Our goal was to provide a way to combine declarative knowledge bases – represented as logic programming modules under the answer set semantics – as well as the individual results one already inferred from them, without having to recalculate the results for their composition and without having to explicitly know the original logic programming encodings that produced such results. This posed us many challenges such as how to deal with fundamental problems of modular frameworks for logic programming, namely how to define a general compositional semantics that allows us to compose unrestricted modules. Building upon existing logic programming approaches, we devised a framework capable of composing generic logic programming modules while preserving the crucial property of compositionality, which informally means that the combination of models of individual modules are the models of the union of modules. We are also still able to reason in the presence of knowledge containing incoherencies, which is informally characterised by a logic program that does not have an answer set due to cyclic dependencies of an atom from its default negation. In this thesis we also discuss how the same approach can be extended to deal with probabilistic knowledge in a modular and compositional way. We depart from the Modular Logic Programming approach in Oikarinen & Janhunen (2008); Janhunen et al. (2009) which achieved a restricted form of compositionality of answer set programming modules. We aim at generalising this framework of modular logic programming and start by lifting restrictive conditions that were originally imposed, and use alternative ways of combining these (so called by us) Generalised Modular Logic Programs. We then deal with conflicts arising in generalised modular logic programming and provide modular justifications and debugging for the generalised modular logic programming setting, where justification models answer the question: Why is a given interpretation indeed an Answer Set? and Debugging models answer the question: Why is a given interpretation not an Answer Set? In summary, our research deals with the problematic of formally devising a generic modular logic programming framework, providing: operators for combining arbitrary modular logic programs together with a compositional semantics; We characterise conflicts that occur when composing access control policies, which are generalisable to our context of generalised modular logic programming, and ways of dealing with them syntactically: provided a unification for justification and debugging of logic programs; and semantically: provide a new semantics capable of dealing with incoherences. We also provide an extension of modular logic programming to a probabilistic setting. These goals are already covered with published work. A prototypical tool implementing the unification of justifications and debugging is available for download from http://cptkirk.sourceforge.net