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Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
The Algebra of 3Graphs
 Proc. Steklov Inst. Math. 221
, 1998
"... We introduce and study the structure of an algebra in the linear space spanned by all regular 3valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular gra ..."
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We introduce and study the structure of an algebra in the linear space spanned by all regular 3valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular graphs of degree 3, i. e. graphs in which every vertex is incident with exactly three edges, often occur in mathematics. Apart from graph theory proper, where such graphs are referred to as `cubic', they appear in a natural way in the topology of 3manifolds, in the Vassiliev knot invariant theory and in connection with the four colour theorem. It turns out that in all these applications 3valent graphs are endowed with a natural structure that consists in fixing, at every vertex of the graph, one of the two possible cyclic orders in the set of three edges issuing from this vertex. In the theory of Vassiliev invariants 3valent graphs can be viewed as elements of the primitive subspace P of th...
A short survey of automated reasoning
"... 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 f ..."
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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
The Block Connectivity of Random Trees
"... Let r, m, and n be positive integers such that rm = n. For each i ∈ {1,..., m} let Bi = {r(i − 1) + 1,..., ri}. The rblock connectivity of a tree on n labelled vertices is the vertex connectivity of the graph obtained by collapsing the vertices in Bi, for each i, to a single (pseudo)vertex vi. In ..."
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Let r, m, and n be positive integers such that rm = n. For each i ∈ {1,..., m} let Bi = {r(i − 1) + 1,..., ri}. The rblock connectivity of a tree on n labelled vertices is the vertex connectivity of the graph obtained by collapsing the vertices in Bi, for each i, to a single (pseudo)vertex vi. In this paper we prove that, for fixed values of r, with r ≥ 2, the rblock connectivity of a random tree on n vertices, for large values of n, is likely to be either r − 1 or r, and furthermore that r − 1 is the right answer for a constant fraction of all trees on n vertices. 1
Complexity Results for the Empire Problem in Collection of Stars
"... Abstract. In this paper, we study the Empire Problem, a generalization of the coloring problem to maps on twodimensional compact surface whose genus is positive. Given a planar graph with a certain partition of the vertices into blocks of size r, for a given integer r, the problem consists of decid ..."
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Abstract. In this paper, we study the Empire Problem, a generalization of the coloring problem to maps on twodimensional compact surface whose genus is positive. Given a planar graph with a certain partition of the vertices into blocks of size r, for a given integer r, the problem consists of deciding if s colors are sufficient to color the vertices of the graph such that vertices of the same block have the same color and vertices of two adjacent blocks have different colors. In this paper, we prove that given a 5regular graph, deciding if there exists a 4coloration is NPcomplete. Also, we propose conditional NPcompleteness results for the Empire Problem when the graph is a collection of stars. A star is a graph isomorphic to K1,q for some q ≥ 1. More exactly, we prove that for r ≥ 2, if the (2r − 1)coloring problem in 2rregular connected graphs is NPcomplete, then the Empire Problem for blocks of size r +1and s =2r − 1 is NPcomplete for forests of K1,r. Moreover, we prove that this result holds for r =2.Alsofor r ≥ 3, ifthercoloring problem in (r +1)regular graphs is NPcomplete, then the Empire Problem for blocks of size r +1and s = r is NPcomplete for forests of K1,1 = K2, i.e., forest of edges. Additionally, we prove that this result is valid for r =2and r =3. Finally, we prove that these results are the best possible, that is for smallest value of s or r, the Empire Problem in these classes of graphs becomes polynomial.