{"title":"标记[公式省略]-自体的可诊断性","authors":"","doi":"10.1016/j.tcs.2024.114743","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we formulate a notion of diagnosability for labeled weighted automata over a class of dioids which admit both positive and negative numbers as well as vectors. The weights can represent diverse physical meanings such as time elapsing and position deviations. We also develop an original tool called concurrent composition to verify diagnosability for such automata. These results are fundamentally new compared with the existing ones in the literature.</p><p>In a little more detail, <em>diagnosability</em> is characterized for a labeled weighted automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math></span> over a special dioid <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> called <em>progressive</em>, which can represent diverse physical meanings such as time elapsing and position deviations. In a progressive dioid, the canonical order is total, there is at least one <em>eventually dominant</em> element, there is no zero divisor, and the cancellative law is satisfied, where the functionality of an eventually dominant element <em>t</em> is to make every nonzero element <em>a</em> arbitrarily large by multiplying <em>a</em> by <em>t</em> for sufficiently many times. A notion of diagnosability is formulated for <span><math><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math></span>. By developing a notion of <em>concurrent composition</em>, a necessary and sufficient condition is given for diagnosability of automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math></span>. It is proven that the problem of computing the concurrent composition for an automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><munder><mrow><mi>Q</mi></mrow><mo>_</mo></munder></mrow></msup></math></span> is <span><math><mi>NP</mi></math></span>-complete, then the problem of verifying diagnosability of <span><math><msup><mrow><mi>A</mi></mrow><mrow><munder><mrow><mi>Q</mi></mrow><mo>_</mo></munder></mrow></msup></math></span> is proven to be <span><math><mi>coNP</mi></math></span>-complete, where the <span><math><mi>NP</mi></math></span>-hardness and <span><math><mi>coNP</mi></math></span>-hardness results even hold for deterministic, deadlock-free, and divergence-free automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><munder><mrow><mi>N</mi></mrow><mo>_</mo></munder></mrow></msup></math></span>, where <span><math><munder><mrow><mi>Q</mi></mrow><mo>_</mo></munder></math></span> and <span><math><munder><mrow><mi>N</mi></mrow><mo>_</mo></munder></math></span> are the max-plus dioids having elements in <span><math><mi>Q</mi><mo>∪</mo><mo>{</mo><mo>−</mo><mo>∞</mo><mo>}</mo></math></span> and <span><math><mi>N</mi><mo>∪</mo><mo>{</mo><mo>−</mo><mo>∞</mo><mo>}</mo></math></span>, respectively. Several extensions of the main results have also been obtained.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Diagnosability of labeled Dp-automata\",\"authors\":\"\",\"doi\":\"10.1016/j.tcs.2024.114743\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we formulate a notion of diagnosability for labeled weighted automata over a class of dioids which admit both positive and negative numbers as well as vectors. The weights can represent diverse physical meanings such as time elapsing and position deviations. We also develop an original tool called concurrent composition to verify diagnosability for such automata. These results are fundamentally new compared with the existing ones in the literature.</p><p>In a little more detail, <em>diagnosability</em> is characterized for a labeled weighted automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math></span> over a special dioid <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> called <em>progressive</em>, which can represent diverse physical meanings such as time elapsing and position deviations. In a progressive dioid, the canonical order is total, there is at least one <em>eventually dominant</em> element, there is no zero divisor, and the cancellative law is satisfied, where the functionality of an eventually dominant element <em>t</em> is to make every nonzero element <em>a</em> arbitrarily large by multiplying <em>a</em> by <em>t</em> for sufficiently many times. A notion of diagnosability is formulated for <span><math><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math></span>. By developing a notion of <em>concurrent composition</em>, a necessary and sufficient condition is given for diagnosability of automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math></span>. It is proven that the problem of computing the concurrent composition for an automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><munder><mrow><mi>Q</mi></mrow><mo>_</mo></munder></mrow></msup></math></span> is <span><math><mi>NP</mi></math></span>-complete, then the problem of verifying diagnosability of <span><math><msup><mrow><mi>A</mi></mrow><mrow><munder><mrow><mi>Q</mi></mrow><mo>_</mo></munder></mrow></msup></math></span> is proven to be <span><math><mi>coNP</mi></math></span>-complete, where the <span><math><mi>NP</mi></math></span>-hardness and <span><math><mi>coNP</mi></math></span>-hardness results even hold for deterministic, deadlock-free, and divergence-free automaton <span><math><msup><mrow><mi>A</mi></mrow><mrow><munder><mrow><mi>N</mi></mrow><mo>_</mo></munder></mrow></msup></math></span>, where <span><math><munder><mrow><mi>Q</mi></mrow><mo>_</mo></munder></math></span> and <span><math><munder><mrow><mi>N</mi></mrow><mo>_</mo></munder></math></span> are the max-plus dioids having elements in <span><math><mi>Q</mi><mo>∪</mo><mo>{</mo><mo>−</mo><mo>∞</mo><mo>}</mo></math></span> and <span><math><mi>N</mi><mo>∪</mo><mo>{</mo><mo>−</mo><mo>∞</mo><mo>}</mo></math></span>, respectively. Several extensions of the main results have also been obtained.</p></div>\",\"PeriodicalId\":49438,\"journal\":{\"name\":\"Theoretical Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-08-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304397524003608\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003608","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
In this paper, we formulate a notion of diagnosability for labeled weighted automata over a class of dioids which admit both positive and negative numbers as well as vectors. The weights can represent diverse physical meanings such as time elapsing and position deviations. We also develop an original tool called concurrent composition to verify diagnosability for such automata. These results are fundamentally new compared with the existing ones in the literature.
In a little more detail, diagnosability is characterized for a labeled weighted automaton over a special dioid called progressive, which can represent diverse physical meanings such as time elapsing and position deviations. In a progressive dioid, the canonical order is total, there is at least one eventually dominant element, there is no zero divisor, and the cancellative law is satisfied, where the functionality of an eventually dominant element t is to make every nonzero element a arbitrarily large by multiplying a by t for sufficiently many times. A notion of diagnosability is formulated for . By developing a notion of concurrent composition, a necessary and sufficient condition is given for diagnosability of automaton . It is proven that the problem of computing the concurrent composition for an automaton is -complete, then the problem of verifying diagnosability of is proven to be -complete, where the -hardness and -hardness results even hold for deterministic, deadlock-free, and divergence-free automaton , where and are the max-plus dioids having elements in and , respectively. Several extensions of the main results have also been obtained.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.