A Reasoning System for Satisfiability of Diagrammatic Specifications

Diagrammatic modelling is the foundation of many popular knowledge representation and software development techniques. In Model Driven Software Engineering, domain specific modelling languages are represented as metamodels and domain specific specifications are represented as models. The (meta-)models are represented by graphs and (models) instances are represented by graphs typed by the (meta)model. The typing relation is formalised by graph homomorphisms. Constraints are used to further specify the semantics of models. The state of the art modelling techniques of today have limited support for expressing and reasoning about diagrammatic constraints; constraints are usually expressed in an external textual language, not fully integrated in the metamodelling process. The diagram predicate framework, DPF is a fully diagrammatic meta modelling technique where one can express arbitrary diagrammatic constraints in the form of predicates on graphs. In this paper we build on ideas, successfuly exploited in a variety of logical systems by Orlowska and collaborators, to build a logical reasoning system for diagrammatic specifications. We enrich the expressiveness of DPF specifications to include semantic dependencies between DPF constraints and present a sound and complete reasoning system using a dual tableaux deduction system to determine if DPF specifications are satisfiable. We briefly discuss an extension of the reasoning system which uses the relational approach to encode the existence of certain graph homomorphisms and provide deduction rules to account for necessary properties of these homomorphisms.

[1]  M. Makkai.  Generalized sketches as a framework for completeness theorems , 1997 .

[2]  Wendy MacCaull,et al.  A Kripke Semantics for the Logic of Gelfand Quantales , 2001, Stud Logica.

[3]  Adrian Rutle,et al.  A Diagrammatic Formalisation of MOF-Based Modelling Languages , 2009, TOOLS.

[4]  Ewa Orłowska,et al.  Dual Tableaux: Foundations, Methodology, Case Studies , 2010 .

[5]  Wendy MacCaull,et al.  A Proof System for Dependencies for Information Relations , 2000, Fundam. Informaticae.

[6]  Zinovy Diskin,et al.  A Diagrammatic Logic for Object-Oriented Visual Modeling , 2008, Electron. Notes Theor. Comput. Sci..

[7]  Adrian Rutle,et al.  A formalisation of the copy-modify-merge approach to version control in MDE , 2010, J. Log. Algebraic Methods Program..

[8]  Colin Atkinson,et al.  The Essence of Multilevel Metamodeling , 2001, UML.

[9]  Adrian Rutle,et al.  A User-friendly Tool for Model Checking Healthcare Workflows , 2013, EUSPN/ICTH.

[10]  Xiaoliang Wang,et al.  Verification of Graph-based Model Transformations Using Alloy , 2014, Electron. Commun. Eur. Assoc. Softw. Sci. Technol..

[11]  Daniel Jackson,et al.  Some Shortcomings of OCL, the Object Constraint Language of UML , 2000, TOOLS.

[12]  Ewa Orlowska,et al.  Relational Proof Systems for Modal Logics , 1996 .

[13]  Adrian Rutle,et al.  DPF Workbench: A Diagrammatic Multi-Layer Domain Specific (Meta-)Modelling Environment , 2012 .

[14]  Ewa Orlowska,et al.  Relational Semantics for Nonclassical Logics: Formulas are Relations , 1994 .

[15]  Ewa Orlowska,et al.  A Logic of Type Relations and its Applications to Relational Databases , 2006, J. Log. Comput..

[16]  Fernando Orejas,et al.  Symbolic graphs for attributed graph constraints , 2011, J. Symb. Comput..

[17]  Wendy MacCaull,et al.  Relational semantics and a relational proof system for full Lambek calculus , 1998, The Journal of Symbolic Logic.

[18]  Lars Michael Kristensen,et al.  A Diagrammatic Approach to Model Completion , 2015, AMT@MoDELS.

[19]  Frank Budinsky,et al.  EMF: Eclipse Modeling Framework 2.0 , 2009 .

[20]  Fernando Orejas,et al.  Tableau-Based Reasoning for Graph Properties , 2014, ICGT.

[21]  Hartmut Ehrig,et al.  Reasoning with graph constraints , 2009, Formal Aspects of Computing.

[22]  Yngve Lamo,et al.  A Bottom Up Approach to Model Based Program Validation , 2016, FlexMDE@MoDELS.

[23]  Yngve Lamo,et al.  Towards a categorical approach for meta-modelling epistemic game theory , 2016, MoDELS.

[24]  Adrian Rutle,et al.  Towards User-Friendly and Efficient Analysis with Alloy , 2015, MoDeVVa@MoDELS.

[25]  Arend Rensink,et al.  Representing First-Order Logic Using Graphs , 2004, ICGT.

[26]  Ivar Jacobson,et al.  The unified modeling language reference manual , 2010 .

[27]  Peter P. Chen.  The entity-relationship model: toward a unified view of data , 1975, VLDB '75.

[28]  Wendy MacCaull.  Relational Proof System for Linear and Other Substructural Logics , 1997, Log. J. IGPL.

[29]  Beata Konikowska,et al.  Reasoning with first order nondeterministic specifications , 1999, Acta Informatica.

[30]  Bruno Courcelle,et al.  Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach , 2012, Encyclopedia of mathematics and its applications.

[31]  Lars Michael Kristensen,et al.  Diagrammatic Development of Domain Specific Modelling Languages with WebDPF , 2016, Int. J. Inf. Syst. Model. Des..

[32]  Yngve Lamo,et al.  Quantifier-free logic for nondeterministic theories , 2006, Theor. Comput. Sci..

[33]  R. Meyer,et al.  The semantics of entailment — III , 1973 .

[34]  Ewa Orlowska,et al.  Relational proof system for relevant logics , 1992, Journal of Symbolic Logic.

[35]  Hao Wang,et al.  A metamodelling approach to behavioural modelling , 2012, BM-FA '12.

[36]  Ewa Orlowska,et al.  Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus , 2002, Stud Logica.

[37]  Hartmut Ehrig,et al.  Fundamentals of Algebraic Graph Transformation (Monographs in Theoretical Computer Science. An EATCS Series) , 1992 .