Word graphs: The third set

This is the third paper in a series of natural language processing in term of knowledge graphs. A word is a basic unit in natural language processing. This is why we study word graphs. Word graphs were already built for prepositions and adwords (including adjectives, adverbs and Chinese quantity words) in two other papers. In this paper, we propose the concept of the logic word and classify logic words into groups in terms of semantics and the way they are used in describing reasoning processes. A start is made with the building of the lexicon of logic words in terms of knowledge graphs

FO-Definability of Shrub-Depth

Shrub-depth is a graph invariant often considered as an extension of tree-depth to dense graphs. We show that the model-checking problem of monadic second-order logic on a class of graphs of bounded shrub-depth can be decided by AC^0-circuits after a precomputation on the formula. This generalizes a similar result on graphs of bounded tree-depth [Y. Chen and J. Flum, 2018]. At the core of our proof is the definability in first-order logic of tree-models for graphs of bounded shrub-depth

Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs

We show that the model-checking problem for successor-invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the exluded topological subgraph. Furthermore, we show that model-checking for order-invariant first-order logic is tractable on coloured posets of bounded width, parameterised by both the size of the input formula and the width of the poset. Our result for successor-invariant FO extends previous results for this logic on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors (Eickmeyer et al., LICS 2013), further narrowing the gap between what is known for FO and what is known for successor-invariant FO. The proof uses Grohe and Marx's structure theorem for graphs with excluded topological subgraphs. For order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO carries over to order-invariant FO

The hardness of the iconic must: Can Peirce’s existential graphs assist modal epistemology?

The current of development in 20th century logic bypassed Peirce’s existential graphs, but recently much good work has been done by formal logicians excavating the graphs from Peirce’s manuscripts, regularizing them and demonstrating the soundness and completeness of the alpha and beta systems (e.g. Roberts 1973, Hammer 1998, Shin 2002). However, given that Peirce himself considered the graphs to be his ‘chef d’oeuvre’ in logic, and explored the distinction between icons, indices and symbols in detail within the context of a much larger theory of signs, much about the graphs arguably remains to be thought through from the perspective of philosophical logic. For instance, are the graphs always merely of heuristic value or can they convey an ‘essential icon’ (analogous to the now standardly accepted ‘essential indexical’)? This paper claims they can and do, and suggests important consequences follow from this for the epistemology of modality. It is boldly suggested that structural articulation, which is characteristic of icons alone, is the source of all necessity. In other words, recognizing a statement as necessarily true consists only in an unavoidable recognition that a structure has the particular structure that it in fact has. (What else could it consist in?