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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