389 research outputs found
Discovering Implicational Knowledge in Wikidata
Knowledge graphs have recently become the state-of-the-art tool for
representing the diverse and complex knowledge of the world. Examples include
the proprietary knowledge graphs of companies such as Google, Facebook, IBM, or
Microsoft, but also freely available ones such as YAGO, DBpedia, and Wikidata.
A distinguishing feature of Wikidata is that the knowledge is collaboratively
edited and curated. While this greatly enhances the scope of Wikidata, it also
makes it impossible for a single individual to grasp complex connections
between properties or understand the global impact of edits in the graph. We
apply Formal Concept Analysis to efficiently identify comprehensible
implications that are implicitly present in the data. Although the complex
structure of data modelling in Wikidata is not amenable to a direct approach,
we overcome this limitation by extracting contextual representations of parts
of Wikidata in a systematic fashion. We demonstrate the practical feasibility
of our approach through several experiments and show that the results may lead
to the discovery of interesting implicational knowledge. Besides providing a
method for obtaining large real-world data sets for FCA, we sketch potential
applications in offering semantic assistance for editing and curating Wikidata
The D-Completeness of T→
A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T→, and any logic of which this is a sublogic, is D-complete
Goal-directed proof theory
This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape
The model checking problem for intuitionistic propositional logic with one variable is AC1-complete
We show that the model checking problem for intuitionistic propositional
logic with one variable is complete for logspace-uniform AC1. As basic tool we
use the connection between intuitionistic logic and Heyting algebra, and
investigate its complexity theoretical aspects. For superintuitionistic logics
with one variable, we obtain NC1-completeness for the model checking problem.Comment: A preliminary version of this work was presented at STACS 2011. 19
pages, 3 figure
Relevance Logic: Problems Open and Closed
I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way
On the construction and algebraic semantics of relevance logic
Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Joan Gispert Brasó[en] The truth-functional interpretation of classical implication gives rise to relevance paradoxes, since it doesn't adequately model our usual understanding of a valid implication, which assumes the antecedent is relevant to the truth of the consequent. This work gives an overview of the system of relevance logic, which aims to avoid said paradoxes. We present the logic with a Hilbert calculus and then prove the Variable-sharing Theorem. We also give an equivalent algebraic semantics for and a semantics for its first-degree entailment fragment
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
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