## Transactions on Computational Science II, Volume 2The denotational and expressive needs in cognitive informatics, computational intelligence, software engineering, and knowledge engineering have led to the development of new forms of mathematics collectively known as denotational mathematics. Denotational mathematics is a category of mathematical structures that formalize rigorous expressions and long-chain inferences of system compositions and behaviors with abstract concepts, complex relations, and dynamic processes. Typical paradigms of denotational mathematics are concept algebra, system algebra, Real-Time Process Algebra (RTPA), Visual Semantic Algebra (VSA), fuzzy logic, and rough sets. A wide range of applications of denotational mathematics have been identified in many modern science and engineering disciplines that deal with complex and intricate mathematical entities and structures beyond numbers, Boolean variables, and traditional sets. This issue of Springer’s Transactions on Computational Science on Denotational Mathematics for Computational Intelligence presents a snapshot of current research on denotational mathematics and its engineering applications. The volume includes selected and extended papers from two international conferences, namely IEEE ICCI 2006 (on Cognitive Informatics) and RSKT 2006 (on Rough Sets and Knowledge Technology), as well as new contributions. The following four important areas in denotational mathem- ics and its applications are covered: Foundations and applications of denotational mathematics, focusing on: a) c- temporary denotational mathematics for computational intelligence; b) deno- tional mathematical laws of software; c) a comparative study of STOPA and RTPA; and d) a denotational mathematical model of abstract games. |

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### Contents

New Means of Thought | 1 |

On Contemporary Denotational Mathematics for Computational Intelligence | 6 |

Mereological Theories of Concepts in Granular Computing | 30 |

On Mathematical Laws of Software | 46 |

Rough Logic and Its Reasoning | 84 |

On Reduct Construction Algorithms | 100 |

Attribute Set Dependence in Reduct Computation | 118 |

A General Model for Transforming Vague Sets into Fuzzy Sets | 133 |

Quantifying Knowledge Base Inconsistency Via Fixpoint Semantics | 145 |

Rank and Statistical Independence in a Contigency Table | 161 |

Applying Rough Sets to Information Tables Containing Possibilistic Values | 180 |

Toward a Generic Mathematical Model of Abstract Game Theories | 205 |

A Comparative Study of STOPA and RTPA | 224 |

246 | |

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### Common terms and phrases

abstract game algorithms Apr(UB attribute set behaviors big-R notation Boolean Cognitive Informatics computational intelligence Computer Science concept algebra contingency table Data decision deﬁned deﬁnition deletion denotational mathematics dispatch E(UA element equivalence classes Example expBL ﬁtness fixpoint formal function fuzzy sets game theories Granular Computing granule Heidelberg IEEE inconsistency indiscernibility information system information table isequal knowledge Kryszkiewicz laws of software Lemma LNCS LNAI mathematical entities mathematical structures matrix maximum expected utility memory meta-processes method object obtained operational semantics operations players Polkowski possibilistic possibilistic degree possible table Process Algebra process relations Processes Law properties recursive reduct construction rough inclusion rough logical formula rough set theory RTPA rules semantics sets of matches Skowron software engineering software system Springer statistical independence strategy subset system algebra system architectures t-norm Theorem U/IND(Uai upper approximations vague value variables voting Wang weighted equivalence classes