Logical equations in monadic logic
type: Item
instance of: scholarly article
title: Logical equations in monadic logic
zbMATH Open document ID: 1190.03014
author: N. E. Zubov
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-01-15T00:00:00Z
review text: The authors regard a formula of logic, \(F(\mathbf{X,P})\), as an equation with unknowns \(\mathbf X\) and parameters \(\mathbf P\). Then, two questions arise naturally: under what conditions is it solvable, and what is the most general solution. [An example the authors list explains the situation. Let \(F(\mathbf {X,P})\) be \(\forall x({\mathbf P}x\to {\mathbf X}x)\&({\mathbf X}x\to {\mathbf Q}x)\). Then the solvability condition is \(\forall x({\mathbf P}x\to {\mathbf Q}x)\), and the general solution is \({\mathbf P}x\lor({\mathbf Q}x\& {\mathbf Y}x)\) with a fresh variable \(\mathbf Y\).] The authors answer these questions completely for monadic first-order formulas with equality, MFO. Based on the familiar procedure of quantifier elimination for MFO, they put the given formula in a special normal form that indicates how many elements there are in what classes. From this form and using \(\varepsilon\)-terms as witness constants, they give the solution in a detailed and easy manner.
Mathematics Subject Classification ID: 03B10
Mathematics Subject Classification ID: 03C10
zbMATH DE Number: 5659088
zbMATH Keywords: monadic formula
zbMATH Keywords: normal form
MaRDI profile type: Publication
full work available at URL: s10958-009-9400-8
OpenAlex ID: W2047303415
cites work: Solvable cases of the decision problem
cites work: Q4956005
cites work: Best solving modal equations
cites work: Q4756727
cites work: Q5582304
DOI: 10.1007/S10958-009-9400-8
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description: scientific article; zbMATH DE number 5659088