Advancements in CASL: A Comprehensive Study of the Common Algebraic Specification Language

1. CASL-Common Algebraic Specification Language Anis Yousefi Ph.D. Candidate Department of Computing & Software McMaster University yousea2@mcmaster.ca

2. Outline • Introduction • CASL’s Basic Specifications • CASL’s Structured Specifications • CASL’s Architectural Specifications • Tools & Case Study

3. The Common Algebraic Specification Language • Common • CoFI: the international common framework initiative • Designed to replace existing algebraic languages and provide a standard • Algebraic Specification • Programs as algebraic structures • Data Values + Functions • Specification in terms of axiom

4. CASL Layers • Basic (Unstructured) Specifications • Specifications in terms of signatures and sentences • Structured Specifications • Naming, Parameterization, etc. • Architectural Specifications • Reusable implementation units • Libraries of named specifications

5. Basic Specifications • Declaration of symbols + set of axioms and constraints (restricting the interpretations of the declared symbols) Σ Symbols Class of Σ-Models  Interpretations of Σ which satisfy the axioms and constraints of the specifications • M ╞═ SP : “Model M satisfies specification SP” or “M is a model of SP”

6. Specifications and Institutions • To develop concepts of a spec language independent of the underlying logical system • I = (Sign, Sen, Mod,╞═) • Sign: category of signatures • Sen: set of sentences for each signature • Senσ : sentence translation map • Mod: category of models for each signature • Modσ : model reduction functor • ╞═: satisfaction relations for each signature (whether a sentence holds in a model or not)

7. Notations • Σ = (S,F) • σ : Σ Σ’ • Sign: category of signatures • Sen(Σ): set of sentences for signature Σ • Senσ: Sen(Σ)  Sen(Σ’) • Mod(Σ): set of models for signature Σ • Modσ:Mod(Σ’)  Mod(Σ) • ╞═Σ |Mod(Σ)| Sen(Σ)

8. Discussion on signatures • Signatures define (non-logical) symbols used in sentences & interpreted in models • Signature Morphisms allow to extend signatures, change notations,… • Signature Morphisms lead to translation of sentences & models such that satisfaction is preserved

9. Logic • Institutions + Entailment System • Extending institutions with proof-theoretic entailment relations compatible with semantic entailment • LOG = (Sign, Sen, Mod,╞═,├─) ├─ Σ P(Sen(Σ)) Sen(Σ) (derivation rules) • Soundness: if Γ├─Σφ then Γ╞═Σφ • Completeness: if Γ╞═Σφ then Γ├─Σφ

10. CASL’s Basic Specifications • Adds sub-sorting, partiality, first order logic & induction • To add sub-sorts • Many-sorted institutions with partial functions, FOL, sort generation constraints • Constructing sub-sorted institutions from many-sorted institutions

11. Many-Sorted Institutions • I = (Sign, Sen, Mod,╞═) • Sign: category of many-sortedsignatures • Sen: set of many-sorted sentences for each signature • Senσ : sentence translation map • Mod: category of many-sorted models for each signature • Modσ : model reduction functor • ╞═: satisfaction relation for each signature

12. Many-Sorted Signature • Σ = (S, TF, PF, P) • S: set of sorts • TF: total function symbols • PF: partial function symbols • P: predicates • Signature Morphism

13. Many-Sorted Σ-Models • M = (MS, TFM, PFM, PM) • MS: a family of non-empty carrier sets indexed by sort s in S, for each sort in Σ • TFM: Mw Ms , for each TF in Σ Mw: Cartesian product of Ms for sorts in the domain of TF • PFM: Mw? Ms , for each PF in Σ • PM Mw , for each P in Σ

14. Many-Sorted Σ-Sentences • Closed many-sorted first-order formulae (no variables) • Sort generation constraint over Σ • Σ X  T  Many-Sorted Atomic Σ-Formulae  Many-Sorted First-Order Formulae • Variables X over Σ • Terms T: X + applications of Σ-functions over X

15. Many-Sorted First-Order Formulae • Many-Sorted Atomic Σ-Formulae • Application of a predicate symbol to terms • Existential equation • Strong equation • Assertion def t: defining value of term t • Many-Sorted First-Order Formulae • Add formula “false” and closing under implication and universal quantification

16. Sort Generation Constraint Over Σ • A given set of sorts is generated by a given set of functions • Along with a signature morphism that allows translation • ({nat}, {0; suc}, id)

17. ╞═ ╞═ Satisfaction of sentences • P(t1,…,tn) is satisfied if the value of t1 to tn is defined and give a tuple belonging to PM • “def t” is satisfied if the value of t is defined • is satisfied if values of t1 and t2 are defined and equal • is satisfied if values of t1 and t2 are defined and equal or both values are undefined • A sort-generation constraint is satisfied if the carriers of the sorts in are generated by the function symbols in from the values in the carriers of sorts not in . Then

18. Entailment System • Rules of derivation

19. Soundness and Completeness • Soundness: • CASL institutions equipped with the provided entailment system is sound • Completeness: • it is complete if sort generation constraints are not used.

20. Sub-Sorting in CASL • Injective embedding between carriers (not necessarily as inclusions) • Allows for more general models in which values of a sub-sort and super-sort are represented differently • Integers (represented as 32-bit words) and Reals (represented using floating-point representation)

21. Sub-sorted Institutions • A category of sub-sorted signatures is defined • A functor from this category into the category of many-sorted signatures is defined • The notations of models, sentences, and satisfaction is borrowed from the many-sorted institutions via this functor

22. Sub-sorted Signature • Σ = (S, TF, PF, P, ≤) • ≤: reflexive and transitive relation on set of sorts • Sub-sorted signature morphism: many-sorted signature morphism that preserves “≤” and “overloading relations” between functions • Overloading relation • Shared sub-sort in their domain • Shared super-sort in their range

23. em Constructing Sub-Sorted Signatures from Many-Sorted Signatures s s’ • em: embedding • pr: projection • in: membership predicate for each pair of sorts s ≤ s’ • Construction ^ is a functor from category of sub-sorted signatures into the category of many-sorted signatures pr

24. Sub-Sorted Models • Many-sorted models satisfying some axioms • Embedding functions are injective. • The embedding of a sort into itself is the identity function. • All compositions of embedding functions between the same two sorts are equal functions. • Projection functions are injective when defined. • Embedding followed by projection is identity. • Membership in a sub-sort holds just when the projection to the sub-sort is defined. • Embedding is compatible with those functions and predicates that are in the overloading relations.

25. Sub-Sorted Sentences • Sub-sorted Σ-sentences are ordinary many-sorted sentences for • Translation of sentences along a sub-sorted signature morphism σ is ordinary many-sorted translation along

26. Satisfaction and proofs • Reuse satisfaction conditions from many-sorted • Proof calculus can borrowed from the many-sorted case Φ├─ΣφΦ UAx(Σ)├─Σφ • Soundness and completeness follow from the many-sorted case ^

27. Structured Specifications

28. Structured Specifications • SPEC1 and SPEC2 (Union): Combines specifications (re-use) • SPEC1 with Symbol Mapping (Translation) : Renaming of symbols • SPEC1 hide Symbol List (Reduction): Hiding symbols • SPEC1 Then SPEC2 (Extension): Enriching models by declaring new symbols and asserting their properties and/or specializing the interpretation of already declared symbols • Free {SPEC}: Restricting to free data-types

29. Naming, Parameterization, & Views • Name to refer to the specification • Generic specifications: parameters • Instantiation: providing an argumentfor each parameter + a fitting morphismfrom the parameter to the argument • Fitting may also be achieved by use of named views between the parameter and argument specifications • view VN : SP to SP’ = Symbol Mapping

30. Architectural Specifications • Provides a means for stating how implementation units are used as building blocks for larger components • Consist of declaration and/or definition of units together with a way of assembling them

31. Architectural Specifications

32. Architectural Specifications Keywords • given: imported units • with: renaming (mapping of symbols) • hide/reveal: unit reduction (hide some symbols in unit) • and: amalgamation of units • fit: fitting an argument to the corresponding formal argument for the generic unit, via a signature morphismin the same

33. Specification Refinement • Fixes some expected properties but says nothing about implementation • Model class becomes smaller and smaller • Techniques: • Views: model class inclusion • Refinement R1 = SP1 refined to SP2 • Refinement R1 = SP1 refined to arch spec ASP • Reducing to implementation of smaller specifications via an architectural specification

34. Tools • Hets: The Heterogeneous Tool Set • Support for all layers of CASL + CASL sub-languages and extensions • Parsing, analysis, proof • Supporting multiple logics • Hets web-based interface: http://www.informatik.uni-bremen.de/cgi-bin/cgiwrap/maeder/hets.cgi • Other tools: CASL consistency checker, CASLtoPVS, CATS, HOL-CASL

35. Case Study: Warehouse System

36. Case Study: Warehouse System

37. Case Study: Warehouse System

38. Conclusion • CASL is a complex specification language that provides formal semantics and proof calculus for all its constructs • Orthogonal layers • Basic specs: writing theories in a specific logic • Structured and architectural specs: logic independent semantics • Tool support

39. References • T. Mossakowski, A. Haxthausen, D. Sannella and A. Tarlecki. CASL -- the Common Algebraic Specification Language: semantics and proof theory. Computing and Informatics 22:285-321 (2003). An extended version of this paper appeared in the book Logics of Specification Languages, 241--298, D. Bjørner and M. Henson eds., Springer (2008). • P.D. Mosses and M. Bidoit. Casl – The Common Algebraic Specification Language: User Manual, volume 2900 of Lecture Notes in Computer Science, Springer, 2004. • P.D. Mosses (ed.). Casl – The Common Algebraic Specification Language: Reference Manual, volume 2960 of Lecture Notes in Computer Science, Springer, 2004. • Hets is available from http://www.informatik.uni-bremen.de/agbkb/forschung/formal_methods/CoFI/hets/ . • R. Khedri, Formal Methods for Software Specification and Development, McMaster University.