and I have removed some personal references. The 11th century Arabic-Persian logician Ibn Sīnā (Avicenna) in section 9.6 of his book Qiyās gives what appears to be a proof search algorithm for syllogisms. We confirm that it is indeed a proof search State Machine from Ibn Sīnā’s text. The paper also contains a translation of the passage from Ibn Sina’s Arabic, and some notes on the text and translation. 1

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for

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Dynamic graphs are locally finite, infinite graphs consisting of an infinite number of representations of a finite graph. The nature of these repetitions determines the dimension of a dynamic graph. This regular structure leads us to study different problems like the planarity with or without vertices and edges accumulation. In this paper, we obtain a complete scheme (results and efficient algorithms) to test whether a half-, one- or two-dimensional dynamic graph is planar, VAP-free or EAP-free planar. 1 Introduction The easiest way to get an infinite graph is to copy infinitely many times a finite graph. This is a dynamic graph so it is an infinite locally finite graph with a regular structure. They appear naturally in many problems as in transportation planning, communications, and operations management that, as it was pointed out by Orlin [21], can be modeled by one-dimensional dynamic graphs, also what we call here half-dimensional dynamic graphs can appear in such modeling. Obviou...

Problems that are today considered to be part of modern graph theory originally appeared in a variety of different connections and contexts. Some of these original questions appear little more than games or puzzles. In the instance of the ‘Icosian Game’, this observation seems quite literally true. Yet for the game’s inventor, the Icosian Game encapsulated deep mathematical ideas which

The earliest origins of graph theory can be found in puzzles and game, including Euler’s Königsberg Bridge Problem and Hamilton’s Icosian Game. A second important branch of mathematics that grew out of these same humble beginnings was the study of position (“analysis situs”), known today as topology 1. In this project, we examine some important connections between algebra, topology

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Learners ’ conceptions in different class situations around Königsberg’s bridges problem