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Łukasiewicz normal forms and toric desingularizations. (English) Zbl 0865.03014
Hodges, Wilfrid (ed.) et al., Logic: from foundations to applications. European logic colloquium, Keele, UK, July 20–29, 1993. Oxford: Clarendon Press. 401-423 (1996).
As a main result, this paper presents a coding of nonsingular toric varieties by tautologies. The simplicial machinery used throughout the paper is introduced in a proof of the presentation of piecewise linear functions \(f_\tau\) over the cube \([0,1]^n\) as linear polynomials \(p(x_1,\dots,x_n)=a_1x_1+\dots+a_n x_n+b\) with integer coefficients. \(f_\tau\) is the associated function of a formula \(\tau\) in the infinite valued sentential calculus of Łukasiewicz. For \(U\) a unimodular simplicial complex over the unit square \([0,1]^2\) (the paper only deals with formulas with two variables), the notion of admissible simplex is considered. Through the notion of Schauder hat, a formal basis \(B\) of \(U\) is defined as a set of formulas that represents the Schauder basis of \(U\). If \(\Theta\) is admissible for \(U\), the transformation \(U \to U\Theta\) naturally induces a transformation of a formal basis \(B\) into a formal basis \(B\Theta\). It is shown that for every unimodular simplicial complex \(U\) over \([0,1]^2\) there is a sequence \(\Theta_1,\dots,\Theta_n\) of simplexes such that \(B_0\Theta_1\dots\Theta_n\) is a formal basis of \(U\). Different complexes give rise to different formal bases. As a corollary it follows that for a finite set of formulas \(\alpha_i\), there is a sequence of simplexes \(\Theta_1,\dots,\Theta_n\) and a formal Schauder basis \(B_0\Theta_1\dots\Theta_n\) such that each \(\alpha_i\) is equivalent to a disjunction of basic formulas. As the main theorem, it follows that a toric variety can be coded by a tautology obtained as a disjunction of formulas from a formal basis. The paper concludes with an application of the infinite-valued calculus of Łukasiewicz to the general solution of Ulam’s “Twenty Questions game” with two lies.
For the entire collection see [Zbl 0851.00045].
Reviewer: A.Hoogewijs (Gent)

MSC:
03B50 Many-valued logic
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