On 0-Complete Partial Metric Spaces and Quantitative Fixed Point Techniques in Denotational Semantics
In 1994, Matthews introduced the notion of partial metric space with the aim of providing a quantitative mathematical model suitable for program verification. Concretely, Matthews proved a partial metric version of the celebrated Banach fixed point theorem which has become an appropriate quantitativ...
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| Vydané v: | Abstract and Applied Analysis Ročník 2013; číslo 2013; s. 692 - 702-1368 |
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| Médium: | Journal Article |
| Jazyk: | English |
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Cairo, Egypt
Hindawi Limiteds
01.01.2013
Hindawi Puplishing Corporation Hindawi Publishing Corporation John Wiley & Sons, Inc Wiley |
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| ISSN: | 1085-3375, 1687-0409 |
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| Abstract | In 1994, Matthews introduced the notion of partial metric space with the aim of providing a quantitative mathematical model suitable for program verification. Concretely, Matthews proved a partial metric version of the celebrated Banach fixed point theorem which has become an appropriate quantitative fixed point technique to capture the meaning of recursive denotational specifications in programming languages. In this paper we show that a few assumptions in statement of Matthews fixed point theorem can be relaxed in order to provide a quantitative fixed point technique useful to analyze the meaning of the aforementioned recursive denotational specifications in programming languages. In particular, we prove a new fixed point theorem for self-mappings between partial metric spaces in which the completeness has been replaced by 0-completeness and the contractive condition has been weakened in such a way that the new one best fits the requirements of practical problems in denotational semantics. Moreover, we provide examples that show that the hypothesis in the statement of our new result cannot be weakened. Finally, we show the potential applicability of the developed theory by means of analyzing a few concrete recursive denotational specifications, some of them admitting a unique meaning and others supporting multiple ones. |
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| AbstractList | In 1994, Matthews introduced the notion of partial metric space with the aim of providing a quantitative mathematical model suitable for program verification. Concretely, Matthews proved a partial metric version of the celebrated Banach fixed point theorem which has become an appropriate quantitative fixed point technique to capture the meaning of recursive denotational specifications in programming languages. In this paper we show that a few assumptions in statement of Matthews fixed point theorem can be relaxed in order to provide a quantitative fixed point technique useful to analyze the meaning of the aforementioned recursive denotational specifications in programming languages. In particular, we prove a new fixed point theorem for self-mappings between partial metric spaces in which the completeness has been replaced by 0-completeness and the contractive condition has been weakened in such a way that the new one best fits the requirements of practical problems in denotational semantics. Moreover, we provide examples that show that the hypothesis in the statement of our new result cannot be weakened. Finally, we show the potential applicability of the developed theory by means of analyzing a few concrete recursive denotational specifications, some of them admitting a unique meaning and others supporting multiple ones. In 1994, Matthews introduced the notion of partial metric space with the aim of providing a quantitative mathematical model suitable for program verification. Concretely, Matthews proved a partial metric version of the celebrated Banach fixed point theorem which has become an appropriate quantitative fixed point technique to capture the meaning of recursive denotational specifications in programming languages. In this paper we show that a few assumptions in statement of Matthews fixed point theorem can be relaxed in order to provide a quantitative fixed point technique useful to analyze the meaning of the aforementioned recursive denotational specifications in programming languages. In particular, we prove a new fixed point theorem for self-mappings between partial metric spaces in which the completeness has been replaced by 0-completeness and the contractive condition has been weakened in such a way that the new one best fits the requirements of practical problems in denotational semantics. Moreover, we provide examples that show that the hypothesis in the statement of our new result cannot be weakened. Finally, we show the potential applicability of the developed theory by means of analyzing a few concrete recursive denotational specifications, some of them admitting a unique meaning and others supporting multiple ones. |
| Audience | Academic |
| Author | O. Valero N. Shahzad |
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| Cites_doi | 10.1016/j.aml.2011.02.025 10.1016/S1571-0661(04)00029-5 10.1016/j.topol.2012.06.012 10.1186/1687-1812-2013-118 10.1016/0012-365X(91)90413-V 10.1155/2011/508730 10.1017/S0960129510000010 10.1016/j.mcm.2011.09.005 10.1007/BF02871458 10.1007/s00224-010-9310-7 10.1016/j.amc.2011.07.005 10.1007/s13398-011-0051-5 10.1186/1687-1812-2012-124 10.1093/comjnl/bxm108 10.1017/S0960129509007671 10.1016/j.topol.2005.01.023 10.1111/j.1749-6632.1994.tb44144.x 10.1016/S0304-3975(02)00705-3 10.4995/agt.2005.1957 10.1080/00207160.2011.559229 10.1016/j.tcs.2003.11.016 10.1016/S0166-8641(98)00102-3 10.1186/1687-1812-2012-40 10.1016/j.topol.2011.08.026 10.5772/36799 10.1186/1687-1812-2012-175 10.1007/3-540-63045-7_4 10.1111/j.1749-6632.1996.tb49177.x 10.1023/A:1008684018933 10.1016/0304-3975(95)00051-W 10.1016/j.topol.2012.11.004 10.1017/CBO9780511542725 |
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| Copyright | Copyright © 2013 N. Shahzad and O. Valero. COPYRIGHT 2014 John Wiley & Sons, Inc. Copyright © 2013 N. Shahzad and O. Valero. N. Shahzad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Copyright 2013 Hindawi Publishing Corporation |
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| SubjectTerms | Computer science Denotational semantics Fixed point theory Mathematical models Mathematical research Metric spaces Programming languages Recursive algorithms Semantics |
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| Title | On 0-Complete Partial Metric Spaces and Quantitative Fixed Point Techniques in Denotational Semantics |
| URI | https://www.airitilibrary.com/Article/Detail/P20160825001-201312-201609070065-201609070065-692-702-1368 https://search.emarefa.net/detail/BIM-513672 https://dx.doi.org/10.1155/2013/985095 https://www.proquest.com/docview/1609381348 http://projecteuclid.org/euclid.aaa/1393512237 https://doaj.org/article/31054b61482b4bbb9646ae69805eb55e |
| Volume | 2013 |
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