Behavioral Extensions of Institutions

We show that any institution ${\mathcal I}$ satisfying some reasonable conditions can be transformed into another institution, ${\mathcal I}_{beh}$, which captures formally and abstractly the intuitions of adding support for behavioral equivalence and reasoning to an existing, particular algebraic framework. We call our transformation an “extension” because ${\mathcal I}_{beh}$ has the same sentences as ${\mathcal I}$ and because its entailment relation includes that of ${\mathcal I}$. Many properties of behavioral equivalence in concrete hidden logics follow as special cases of corresponding institutional results. As expected, the presented constructions and results can be instantiated to other logics satisfying our requirements as well, thus leading to novel behavioral logics, such as partial or infinitary ones, that have the desired properties.

[1]  R. Azvan Diaconescu,et al.  Grothendieck Institutions , 2002 .

[2]  Robin Milner,et al.  Theories for the Global Ubiquitous Computer , 2004, FoSSaCS.

[3]  Andrzej Tarlecki,et al.  Bits and Pieces of the Theory of Institutions , 1985, ADT.

[4]  Razvan Diaconescu,et al.  Institution-independent Ultraproducts , 2002, Fundam. Informaticae.

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

[6]  A. W. Roscoe.  A classical mind: essays in honour of C. A. R. Hoare , 1994 .

[7]  Martin Hofmann,et al.  On Behavioral Abstraction and Behavioural Satisfaction in Higher-Order Logic , 1995, TAPSOFT.

[8]  Răzvan Diaconescu,et al.  Hiding and behaviour: an institutional approach , 1994 .

[9]  Samson Abramsky,et al.  Category Theory and Computer Programming , 1986, Lecture Notes in Computer Science.

[10]  Gernot Salzer,et al.  Automated Deduction in Classical and Non-Classical Logics , 2002, Lecture Notes in Computer Science.

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

[12]  Razvan Diaconescu.  An Institution-independent Proof of Craig Interpolation Theorem , 2004, Stud Logica.

[13]  Grigore Rosu,et al.  Hidden Congruent Deduction , 1998, FTP.

[14]  S. Anderson,et al.  Secure Synthesis of Code: A Process Improvement Experiment , 1999, World Congress on Formal Methods.

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

[16]  Tom Lyche,et al.  From Object-Orientation to Formal Methods , 2004, Lecture Notes in Computer Science.

[17]  José Meseguer,et al.  Membership algebra as a logical framework for equational specification , 1997, WADT.

[18]  José Meseguer,et al.  Conditioned Rewriting Logic as a United Model of Concurrency , 1992, Theor. Comput. Sci..

[19]  Horst Herrlich,et al.  Abstract and concrete categories , 1990 .

[20]  Hartmut Ehrig,et al.  Recent trends in data type specification : 9th Workshop on Specification of Abstract Data Types, joint with the 4th COMPASS Workshop, Caldes de Malavella, Spain, October, 26-30, 1992 : selected papers , 1994 .

[21]  Michel Bidoit,et al.  On the Integration of Observability and Reachability Concepts , 2002, FoSSaCS.

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

[23]  Helene Kirchner.  Trees in Algebra and Programming — CAAP '96 , 1996, Lecture Notes in Computer Science.

[24]  Michel Bidoit,et al.  Behavioural Satisfaction and Equivalence in Concrete Model Categories , 1996, CAAP.

[25]  Grigore Rosu,et al.  Composing Hidden Information Modules over Inclusive Institutions , 2004, Essays in Memory of Ole-Johan Dahl.

[26]  Virgil Emil Cazanescu,et al.  Weak Inclusion Systems , 1997, Math. Struct. Comput. Sci..

[27]  H. Keisler.  Model theory for infinitary logic , 1971 .

[28]  Joseph A. Goguen,et al.  Towards an Algebraic Semantics for the Object Paradigm , 1992, COMPASS/ADT.

[29]  José Meseguer,et al.  Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations , 1992, Theor. Comput. Sci..

[30]  Alexander Kurz,et al.  Observational logic, constructor-based logic, and their duality , 2003, Theor. Comput. Sci..

[31]  Joseph A. Goguen,et al.  Institutions: abstract model theory for specification and programming , 1992, JACM.

[32]  Petros S. Stefaneas,et al.  Logical support for modularisation , 1993 .

[33]  Donald Sannella,et al.  On Observational Equivalence and Algebraic Specification , 1985, TAPSOFT, Vol.1.

[34]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[35]  Michael Makkai,et al.  Accessible categories: The foundations of categorical model theory, , 2007 .

[36]  Jean Benabou,et al.  Fibered categories and the foundations of naive category theory , 1985, Journal of Symbolic Logic.

[37]  Razvan Diaconescu.  Elementary Diagrams in Institutions , 2004, J. Log. Comput..