P. Eklund, María Ángeles Galán García, J. Kortelainen, L. N. Stout
We will present three paradigms for non-classical substitution. Firstly, we have the classical substitution of variables with terms. This is written in a strict categorical form supporting presentation of the other two paradigms. The second paradigm is substitutions of variables with many-valued sets of terms. These two paradigms are based on functors and monads over the category of sets. The third paradigm is the substitution of many-valued sets of variables with terms over many-valued sets of variables. The latter is based on functors and monads over the category of many-valued sets. This provides a transparency of the underlying categories and also makes a clear distinction between set-theoretic operation in the meta language and operations on sets and many-valued sets as found within respective underlying categories.
{"title":"Paradigms for Non-classical Substitutions","authors":"P. Eklund, María Ángeles Galán García, J. Kortelainen, L. N. Stout","doi":"10.1109/ISMVL.2009.60","DOIUrl":"https://doi.org/10.1109/ISMVL.2009.60","url":null,"abstract":"We will present three paradigms for non-classical substitution. Firstly, we have the classical substitution of variables with terms. This is written in a strict categorical form supporting presentation of the other two paradigms. The second paradigm is substitutions of variables with many-valued sets of terms. These two paradigms are based on functors and monads over the category of sets. The third paradigm is the substitution of many-valued sets of variables with terms over many-valued sets of variables. The latter is based on functors and monads over the category of many-valued sets. This provides a transparency of the underlying categories and also makes a clear distinction between set-theoretic operation in the meta language and operations on sets and many-valued sets as found within respective underlying categories.","PeriodicalId":115178,"journal":{"name":"2009 39th International Symposium on Multiple-Valued Logic","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129189725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new way for synthesizing MVL functions by injecting pseudo minterm has been proposed in [13]. In this paper, we explore the idea of injecting multiple connected pseudo minterms as a means for reducing the number of implicants needed to synthesize an MVL function. A set of randomly generated 50000 2-variables 4-valued and 50000 2-variables 5-valued functions is used as benchmark. Comparisons made using the obtained results using the same benchmark show that the proposed algorithms improve the performance of technique proposed in [13] in terms of the average number of product terms needed to synthesize a given MVL function. The improvement in the number of PT is achieved at the expense of adding the MUX at the output which adds moderately to the cost of the circuit needed.
{"title":"The Use of Multiple Connected Pseudo Minterms in the Synthesis of MVL Functions","authors":"Bambang A. B. Sarif, M. Abd-El-Barr","doi":"10.1109/ISMVL.2009.56","DOIUrl":"https://doi.org/10.1109/ISMVL.2009.56","url":null,"abstract":"A new way for synthesizing MVL functions by injecting pseudo minterm has been proposed in [13]. In this paper, we explore the idea of injecting multiple connected pseudo minterms as a means for reducing the number of implicants needed to synthesize an MVL function. A set of randomly generated 50000 2-variables 4-valued and 50000 2-variables 5-valued functions is used as benchmark. Comparisons made using the obtained results using the same benchmark show that the proposed algorithms improve the performance of technique proposed in [13] in terms of the average number of product terms needed to synthesize a given MVL function. The improvement in the number of PT is achieved at the expense of adding the MUX at the output which adds moderately to the cost of the circuit needed.","PeriodicalId":115178,"journal":{"name":"2009 39th International Symposium on Multiple-Valued Logic","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126820595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Béjar, C. Fernández, Carles Mateu, Nuria Pascual
Edge matching puzzles have been, for a very long time, a common toy for children. Their simplicity hides a subtle and complex problem structure that results, in certain cases, in very hard problems. Those hard cases are being commercially exploited, capturing a wide attention due to generous prizes. Edge matching puzzles have been proven to be NP-Complete problems [1], and their phase transition has been recently located experimentally [2]. This work approaches the problem of defining and locating the phase transition for edge matching puzzles from an analytical point of view, defining statistical measures that; on one hand, upper bound the phase transition and prove to be good estimators for locating the hardest problems, and on the other hand, approaches the lower bound of the phase transition as a previous step to determine an exact asymptotic behavior.
{"title":"Bounding the Phase Transition on Edge Matching Puzzles","authors":"R. Béjar, C. Fernández, Carles Mateu, Nuria Pascual","doi":"10.1109/ISMVL.2009.55","DOIUrl":"https://doi.org/10.1109/ISMVL.2009.55","url":null,"abstract":"Edge matching puzzles have been, for a very long time, a common toy for children. Their simplicity hides a subtle and complex problem structure that results, in certain cases, in very hard problems. Those hard cases are being commercially exploited, capturing a wide attention due to generous prizes. Edge matching puzzles have been proven to be NP-Complete problems [1], and their phase transition has been recently located experimentally [2]. This work approaches the problem of defining and locating the phase transition for edge matching puzzles from an analytical point of view, defining statistical measures that; on one hand, upper bound the phase transition and prove to be good estimators for locating the hardest problems, and on the other hand, approaches the lower bound of the phase transition as a previous step to determine an exact asymptotic behavior.","PeriodicalId":115178,"journal":{"name":"2009 39th International Symposium on Multiple-Valued Logic","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122902064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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