Non-deterministic multi-valued matrices (Nmatrices) are a new, fruitful and quickly expanding field of research first introduced a few years ago. Since then it has been rapidly developing towards a foundational logical theory and has found numerous applications. The novelty of Nmatrices is in extending the usual many-valued algebraic semantics of logical systems by importing the idea of non-deterministic computations, and allowing the truth-value of a formula to be chosen non-deterministically out of a given set of options. Nmatrices have proved to be a powerful tool, the use of which preserves all the advantages of ordinary many-valued matrices, but is applicable to a much wider range of logics. Indeed, there are many useful (propositional) non-classical logics, which have no finite multi-valued characteristic matrices, but {do} have finite Nmatrices, and thus are decidable. In this tutorial we introduce the reader to the concept of Nmatrices, and demonstrate their usefulness by providing modular non-deterministic semantics for a well-known family of logics for reasoning under uncertainty.
{"title":"Non-deterministic Multi-valued Logics--A Tutorial","authors":"A. Avron, A. Zamansky","doi":"10.1109/ISMVL.2010.18","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.18","url":null,"abstract":"Non-deterministic multi-valued matrices (Nmatrices) are a new, fruitful and quickly expanding field of research first introduced a few years ago. Since then it has been rapidly developing towards a foundational logical theory and has found numerous applications. The novelty of Nmatrices is in extending the usual many-valued algebraic semantics of logical systems by importing the idea of non-deterministic computations, and allowing the truth-value of a formula to be chosen non-deterministically out of a given set of options. Nmatrices have proved to be a powerful tool, the use of which preserves all the advantages of ordinary many-valued matrices, but is applicable to a much wider range of logics. Indeed, there are many useful (propositional) non-classical logics, which have no finite multi-valued characteristic matrices, but {do} have finite Nmatrices, and thus are decidable. In this tutorial we introduce the reader to the concept of Nmatrices, and demonstrate their usefulness by providing modular non-deterministic semantics for a well-known family of logics for reasoning under uncertainty.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130715958","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}
All maximal partial clones on 4-element, 5-element, and 6-element sets have been found and are compared to the case of maximal clones of all total functions. Due to the large numbers of maximal partial clones other criteria to check for generating systems of all partial functions are analyzed.
{"title":"Number of Maximal Partial Clones","authors":"Karsten Schölzel","doi":"10.1109/ISMVL.2010.60","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.60","url":null,"abstract":"All maximal partial clones on 4-element, 5-element, and 6-element sets have been found and are compared to the case of maximal clones of all total functions. Due to the large numbers of maximal partial clones other criteria to check for generating systems of all partial functions are analyzed.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131677734","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}
Multiway Decision Graphs (MDGs) subsume Binary Decision Diagrams(BDDs) and extend them by a first-order formulae suitable for model checking of data path circuits. In this paper, we propose a reduction technique to improve MDGs model checking. We use a reduction platform based on combining MDGs together with the rewriting engine in the HOL theorem prover. The idea is to prune the transition relation of the circuits using pre-proved theorems and lemmas from the specification given at system level. Then, the actual proof of temporal MDG formulae will be achieved by the MDGs model checker. We support our reduction technique by experimental results executed on benchmark properties.
{"title":"MDGs Reduction Technique Based on the HOL Theorem Prover","authors":"Sa'ed Abed, O. Mohamed","doi":"10.1109/ISMVL.2010.12","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.12","url":null,"abstract":"Multiway Decision Graphs (MDGs) subsume Binary Decision Diagrams(BDDs) and extend them by a first-order formulae suitable for model checking of data path circuits. In this paper, we propose a reduction technique to improve MDGs model checking. We use a reduction platform based on combining MDGs together with the rewriting engine in the HOL theorem prover. The idea is to prune the transition relation of the circuits using pre-proved theorems and lemmas from the specification given at system level. Then, the actual proof of temporal MDG formulae will be achieved by the MDGs model checker. We support our reduction technique by experimental results executed on benchmark properties.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116432730","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}
Boolean logic interpretations have been recently proposed for analogical proportions (i.e. statements of the form ''$a$ is to $b$ as $c$ is to $d$"), and for two other related formal proportions named reverse analogy (''what $a$ is to $b$ is the reverse of what $c$ is to $d$"), and paralogy (''what $a$ and $b$ have in common $c$ and $d$ have it also"). These proportions relate items $a$, $b$, $c$, and $d$ in different ways, on the basis of their differences, or of their similarities. This paper investigates multiple-valued models for these proportions. These extensions may serve different purposes, such as taking into account graded features, or handling crisp ones in non binary attribute domains in a compact way. After summarizing the main results in the binary case, we discuss what multiple valued patterns make sense for the different proportions, starting with tri-valued interpretations, and then considering $[0,1]$-valued interpretations. It appears that Lukasiewicz implication-based interpretation fits well with the intended meaning of analogy and reverse analogy, while a minimum-based interpretation is more suitable for paralogy in case of graded features. Besides, the interest of an interpretation based on Post algebra in case of non binary attribute domains is briefly outlined. Similarities and differences with the standard Boolean setting are highlighted.
{"title":"Multiple-Valued Logic Interpretations of Analogical, Reverse Analogical, and Paralogical Proportions","authors":"H. Prade, G. Richard","doi":"10.1109/ISMVL.2010.55","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.55","url":null,"abstract":"Boolean logic interpretations have been recently proposed for analogical proportions (i.e. statements of the form ''$a$ is to $b$ as $c$ is to $d$\"), and for two other related formal proportions named reverse analogy (''what $a$ is to $b$ is the reverse of what $c$ is to $d$\"), and paralogy (''what $a$ and $b$ have in common $c$ and $d$ have it also\"). These proportions relate items $a$, $b$, $c$, and $d$ in different ways, on the basis of their differences, or of their similarities. This paper investigates multiple-valued models for these proportions. These extensions may serve different purposes, such as taking into account graded features, or handling crisp ones in non binary attribute domains in a compact way. After summarizing the main results in the binary case, we discuss what multiple valued patterns make sense for the different proportions, starting with tri-valued interpretations, and then considering $[0,1]$-valued interpretations. It appears that Lukasiewicz implication-based interpretation fits well with the intended meaning of analogy and reverse analogy, while a minimum-based interpretation is more suitable for paralogy in case of graded features. Besides, the interest of an interpretation based on Post algebra in case of non binary attribute domains is briefly outlined. Similarities and differences with the standard Boolean setting are highlighted.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129909091","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}
In this paper we present some applications of the reticulation of a residuated lattice, in the form of a transfer of properties between the category of bounded distributive lattices and that of residuated lattices through the reticulation functor. The results we are presenting are related to co-Stone algebras; among other applications, we transfer a known characterization of $m$-co-Stone bounded distributive lattices to residuated lattices and we prove that the reticulation functor for residuated lattices preserves the strongly co-Stone hull.
{"title":"Co-stone Residuated Lattices","authors":"C. Mureşan","doi":"10.1109/ISMVL.2010.27","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.27","url":null,"abstract":"In this paper we present some applications of the reticulation of a residuated lattice, in the form of a transfer of properties between the category of bounded distributive lattices and that of residuated lattices through the reticulation functor. The results we are presenting are related to co-Stone algebras; among other applications, we transfer a known characterization of $m$-co-Stone bounded distributive lattices to residuated lattices and we prove that the reticulation functor for residuated lattices preserves the strongly co-Stone hull.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2010-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124097388","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}
Let $A$ be a finite non-singleton set. For $|A|=2$ we show that the partial clone consisting of all self-dual monotonic partial functions on $A$ is not finitely generated, while it is the intersection of two finitely generated maximal partial clones on $A$. Moreover, for $|A| ge 3$ we show that there are pairs of finitely generated maximal partial clones whose intersection is a not finitely generated partial clone on $A$.
设$A$是一个有限非单元素集合。对于$|A|=2$，我们证明了$A$上由所有自对偶单调偏函数组成的部分克隆不是有限生成的，而它是$A$上两个有限生成的极大部分克隆的交集。此外，对于$|A| ge 3$，我们证明了存在一对有限生成的极大部分克隆，它们的交集是$A$上的非有限生成的部分克隆。
{"title":"Finitely Generated Maximal Partial Clones and Their Intersections","authors":"Miguel Couceiro, L. Haddad","doi":"10.1109/ISMVL.2010.31","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.31","url":null,"abstract":"Let $A$ be a finite non-singleton set. For $|A|=2$ we show that the partial clone consisting of all self-dual monotonic partial functions on $A$ is not finitely generated, while it is the intersection of two finitely generated maximal partial clones on $A$. Moreover, for $|A| ge 3$ we show that there are pairs of finitely generated maximal partial clones whose intersection is a not finitely generated partial clone on $A$.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130746399","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}
We describe the classes of operations closed under permutation of variables, addition of dummy variables and composition in terms of a preservation relation between operations and certain systems of multisets.
利用运算与若干多集系统之间的保存关系，描述了在变量置换、虚拟变量的加法和复合下闭闭的运算类。
{"title":"Classes of Operations Closed under Permutation, Cylindrification and Composition","authors":"Miguel Couceiro, Erkko Lehtonen","doi":"10.1109/ISMVL.2010.30","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.30","url":null,"abstract":"We describe the classes of operations closed under permutation of variables, addition of dummy variables and composition in terms of a preservation relation between operations and certain systems of multisets.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115875714","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}
Let $A$ and $B$ be arbitrary sets with at least two elements. The arity gap of a function $fcolon A^nto B$ is the minimum decrease in its essential arity when essential arguments of $f$ are identified. In this paper we study the arity gap of polynomial functions over bounded distributive lattices and present a complete classification of such functions in terms of their arity gap. To this extent, we present a characterization of the essential arguments of polynomial functions, which we then use to show that almost all lattice polynomial functions have arity gap 1, with the exception of truncated median functions, whose arity gap is 2.
设$A$和$B$是至少有两个元素的任意集合。函数$f: a ^n到B$的密度差是当$f$的基本参数被确定时其基本密度的最小减少。本文研究了有界分布格上多项式函数的度差问题，并根据函数的度差给出了这类函数的完全分类。在这种程度上，我们提出了多项式函数的基本参数的表征，然后我们用它来证明几乎所有的格多项式函数都有一个密度间隙1，除了截断的中位数函数，它的密度间隙为2。
{"title":"The Arity Gap of Polynomial Functions over Bounded Distributive Lattices","authors":"Miguel Couceiro, Erkko Lehtonen","doi":"10.1109/ISMVL.2010.29","DOIUrl":"https://doi.org/10.1109/ISMVL.2010.29","url":null,"abstract":"Let $A$ and $B$ be arbitrary sets with at least two elements. The arity gap of a function $fcolon A^nto B$ is the minimum decrease in its essential arity when essential arguments of $f$ are identified. In this paper we study the arity gap of polynomial functions over bounded distributive lattices and present a complete classification of such functions in terms of their arity gap. To this extent, we present a characterization of the essential arguments of polynomial functions, which we then use to show that almost all lattice polynomial functions have arity gap 1, with the exception of truncated median functions, whose arity gap is 2.","PeriodicalId":447743,"journal":{"name":"2010 40th IEEE International Symposium on Multiple-Valued Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2009-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126638905","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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