Diagnostic tests for discrete functions defined on rings.

By the assumption of the theorem and the definition of the direct sum of rings, among these rows there are HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msubsup><mi>l</mi><mi>B</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi></msubsup><mtext> rows </mtext><mo stretchy="false">(</mo><msubsup><mi>l</mi><mi>B</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi></msubsup></mtd></mtr></mtable></math> ht is the smallest possible number) that distinguish the elements ( I xb i SB 1 sb ), ..., ( I xb SB k sb i ) of ring I A i I B i , where I x i = I a i SB 1 sb , ..., I a SB n sb i , I x i I A i . Among these rows, by the hypothesis of the theorem and the definition of the direct sum of rings, there are HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msubsup><mi>l</mi><mi>A</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi></msubsup></mtd></mtr></mtable></math> ht rows (the minimal possible number) that distinguish these elements. Consider a source of faults HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msubsup><mi>U</mi><mi>n</mi><mo>,</mo><mi> </mi><mo stretchy="false">~</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">s</mi></msubsup></mtd></mtr></mtable></math> ht capable of acting on a Boolean function as follows. [Extracted from the article]