Closure properties for the class of behavioral models

Hidden k-logics can be considered as the underlying logics of program specification. They constitute natural generalizations of k-deductive systems and encompass deductive systems as well as hidden equational logics and inequational logics. In our abstract algebraic approach, the data structures are sorted algebras endowed with a designated subset of their visible parts, called filter, which represents a set of truth values. We present a hierarchy of classes of hidden k-logics. The hidden k-logics in each class are characterized by three different kinds of conditions, namely, properties of their Leibniz operators, closure properties of the class of their behavioral models, and properties of their equivalence systems. Using equivalence systems, we obtain a new and more complete analysis of the axiomatization of the behavioral models. This is achieved by means of the Leibniz operator and its combinatorial properties.

[1]  Joseph A. Goguen,et al.  A hidden agenda , 2000, Theor. Comput. Sci..

[2]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[3]  Josep Maria Font,et al.  A Survey of Abstract Algebraic Logic , 2003, Stud Logica.

[4]  T. Prucnal,et al.  An algebraic characterization of the notion of structural completeness , 1974 .

[5]  Karl Meinke,et al.  Universal algebra , 1993, LICS 1993.

[6]  Samuel R. Buss,et al.  Incompleteness of Behavioral Logics , 2000, CMCS.

[7]  Joseph A. Goguen,et al.  Types as theories , 1991 .

[8]  Michel Bidoit,et al.  Proving Behavioural Theorems with Standard First-Order Logic , 1994, ALP.

[9]  José Meseguer,et al.  General Logics , 2006 .

[10]  Grigore Rosu.  A Birkhoff-like Axiomatizability Result for Hidden Algebra and Coalgebra , 1998, CMCS.

[11]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[12]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[13]  Grant Malcolm,et al.  Behavioural Equivalence, Bisimulation, and Minimal Realisation , 1995, COMPASS/ADT.

[14]  B. Herrmann.  Equivalential and algebraizable logics , 1996 .

[15]  Michaël Rusinowitch,et al.  Observational proofs by rewriting , 2002, Theor. Comput. Sci..

[16]  Wolfgang Wechler,et al.  Universal Algebra for Computer Scientists , 1992, EATCS Monographs on Theoretical Computer Science.

[17]  Willem J. Blok,et al.  Protoalgebraic logics , 1986, Stud Logica.

[18]  Martin Hofmann,et al.  On Behavioural Abstraction and Behavioural Satisfaction in Higher-Order Logic , 1995, Theor. Comput. Sci..

[20]  W. Blok.  Algebraic Semantics for Universal Horn Logic Without Equality , 1992 .

[21]  Michel Bidoit,et al.  Observational Logic , 1998, AMAST.

[22]  J. Czelakowski.  Equivalential logics (II) , 1981 .

[23]  Marcos A. D. Martins,et al.  Behavioral Reasoning in generalized Hidden Logics , 2004 .

[24]  Michel Bidoit,et al.  Observer Complete Definitions are Behaviourally Coherent , 1999 .

[25]  Oliver Schoett.  Two impossibility theorems on behaviour specification of abstract data types , 2005, Acta Informatica.

[26]  Peter Padawitz,et al.  Swinging types=functions+relations+transition systems , 2000, Theor. Comput. Sci..

[27]  Don Pigozzi,et al.  Algebraic logic , 1985, Problem books in mathematics.

[28]  G. Leavens,et al.  Equational Reasoning with Subtypes , 2002 .

[29]  José Luiz Fiadeiro,et al.  Structuring Theories on Consequence , 1988, ADT.

[30]  Grigore Rosu,et al.  Hiding More of Hidden Algebra , 1999, World Congress on Formal Methods.

[31]  Manuel A. Martins.  Behavioral Institutions and Refinements in Generalized Hidden Logics , 2006, J. Univers. Comput. Sci..

[32]  Martin Wirsing,et al.  Behavioural and Abstractor Specifications , 1995, Sci. Comput. Program..

[33]  Don Pigozzi.  Equality-Test and If-Then-Else Algebras: Axiomatization and Specification , 1991, SIAM J. Comput..

[34]  Michel Bidoit,et al.  Behavioural Theories and the Proof of Behavioural Properties , 1996, Theor. Comput. Sci..

[35]  Grigore Rosu,et al.  Conditional Circular Coinductive Rewriting with Case Analysis , 2002, WADT.

[36]  Razvan Diaconescu,et al.  Behavioural Coherence in Object-Oriented Algebraic Specification , 2000, J. Univers. Comput. Sci..