1 / 33

Building a Model-Checker for Z

Z2. Building a Model-Checker for Z. John Derrick, Siobhán North and Anthony Simons Department of Computer Science University of Sheffield. Z2. Overview. Tool support for Z refinement Z2SAL translation strategy Basic types, free types, schemas The mathematical toolkit

coty
Télécharger la présentation

Building a Model-Checker for Z

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Z2 Building a Model-Checker for Z John Derrick, Siobhán North and Anthony Simons Department of Computer ScienceUniversity of Sheffield ABZ Conference, London 2008

  2. Z2 Overview • Tool support for Z refinement • Z2SAL translation strategy • Basic types, free types, schemas • The mathematical toolkit • Evaluation and performance translations andcircumlocutions! ABZ Conference, London 2008

  3. What Tools for Z? • CADiZ (Toyn & McDermid) • type checker, schema layout • CZT project (Miller, et al.) • Parser, checker for LaTeX-Z/ZML markup of ISO-Z • AST toolkit for java … more modules to follow • ProZ (Plagge & Leuschel) • validates Z, based on ProB tool (Leuschel & Butler) • Alloy and Z (Bolton) • recoding Z in Alloy, check refinements • SAL translation (Smith & Wildman; Derrick et al.) • leverage existing checkers, rich input language ABZ Conference, London 2008

  4. Advantages of • Symbolic Analysis Laboratory • from SRI (de Moura, et al.), good user base • rich input language has finite types, tuples, arrays, records, recursion (?), modules … • core engine based on BDD compilation, symbolic simulation using Bűchi automata • many tools: simulator, model-checker, bounded model-checker, … • checks both LTL and CTL properties • Z-to-SAL translation strategy • perhaps easier than building a native Z model-checker ABZ Conference, London 2008

  5. Z2 Issues • Bounding the infinite • Z supports infinite models, uninterpreted symbols • SAL needs finite models, concrete bounded ranges • Mismatched models • Z has separate operation schemas acting upon the data • SAL compiles all input, output, local vars into a single FSM with transitions representing the operations • Z functions are partial; SAL functions are total • Monolithic set types; vs. judgements on ordered variables • Non-constructive specifications • Z can express non-constructive specifications • SAL tools require a computable update step ABZ Conference, London 2008

  6. Z2 Translator • Bespoke Z parser/SAL generator • we use our own Java parser for LaTeX-Z • allows rapid prototyping, experimentation • much easier than (poorly documented ) CZT ASTs • Optimisation during analysis • model bounds set by range indicators found in the Z • early elimination of trivially satisfied predicates • Template-driven generation • Z structures map onto related SAL structures • we use our own SAL libraries for the Z math toolkit ABZ Conference, London 2008

  7. Z2 Strategy • Types, constants • translate unbounded Z types into bounded SAL types • translate uninterpreted Z constants into SAL variables, or simplify to constants (by symbolic reasoning) • State, operation schema variables • translate state schema vars into LOCAL vars • translate operation in?, out! to INPUT, OUTPUT vars • State init, operation schemas • translate all schemas into an executable MODULE • Z operations become the transitions of the FSM • Z pre-, post-conditions become SAL guarded commands ABZ Conference, London 2008

  8. Z Types • Built-in types ¥1ͥ͢NZNAT : TYPE = [1..3]; NAT : TYPE = [0..3]; INT : TYPE = [-1..3]; • Basic types [PERSON] PERSON : TYPE = {PERSON__1, PERSON__2, PERSON__3}; • Free types REPORT ::= REPORT : TYPE = DATATYPEok | ok,error «MESSAGE» error (message : MESSAGE) END; ABZ Conference, London 2008

  9. Z Constants • Uninterpreted • Either, pick a suitable constant value: max : ¥max : NAT = 3; • Or, treat as a local variable: max : ¥LOCAL max : NAT • Axiomatic Definitions • Treat as a constrained local variable: max : ¥ | max < 3 LOCAL max : NAT … DEFINITION invariant__ = … AND max < 3 … … more on invariant__ next … ABZ Conference, London 2008

  10. Z State Schema • Data declarations • treat as local vars in FSM module • State predicate • define var invariant__as an abbreviation State State : MODULE = BEGIN LOCAL level : NAT LOCAL invariant__ : BOOLEAN … DEFINITION invariant__ = (1 < level AND level <= max … ) …END; level : ¥ 1 < level  max ABZ Conference, London 2008

  11. Z Init Schema – 1 • Initialisation • set of assignments • Problems • SAL init, update is constructive  • Z may be equational State : MODULE = BEGIN LOCAL level : NAT … DEFINITION … INITIALIZATION level = 2; … …END; Init State level = 2 … how to handle? … ABZ Conference, London 2008

  12. Guarded Commands • SAL transition language • usual syntax: if guard then constructive assignments [ c1 AND c2 … AND cn --> v1’ = e1; v2’ = e2; … vn’ = en ] • Z2SAL translation idea : • move all update expressions back into the guard • enables equational reasoning for update expressions [ c1 AND c2 … AND cn AND v1’ = e1 AND v2’ = e2 … AND vn’ = en --> v1’ IN { x : NAT | TRUE }; v2’ IN { y : INT | TRUE }; … vn’ IN { z : NAT | TRUE }; ] sense: vars exist a posteriori ABZ Conference, London 2008

  13. Z Init Schema – 2 • Non-constructive • if the assignment holds • then the initial state isvalid (empty consequent) • A bonus for Z  • can assert the invariantin the initial state • abbreviates a largeconjunction of terms State : MODULE = BEGIN LOCAL level : NAT … DEFINITION … INITIALIZATION [ level = 2 AND invariant__ --> ] …END; ABZ Conference, London 2008

  14. Z Operation Schema – 1 • Input, output vars • exist in one SAL scope • must rename uniquely State : MODULE = BEGIN LOCAL level : NAT INPUT Inc__in? : NAT OUTPUT Inc__out_ : REPORT … DEFINITION invariant__ = … INITIALIZATION … AND invariant__ … TRANSITION … AND invariant__’ … …END; Inc Δ Statein? : ¥out! : REPORT level + in?  maxlevel’ = level + in?out! = ok assert invariant__’ after each step… ABZ Conference, London 2008

  15. Z Operation Schema – 2 State : MODULE = BEGIN LOCAL level : NAT INPUT Inc__in? : NAT OUTPUT Inc__out_ : REPORT … TRANSITION [ Inc : level + Inc__in? <= max AND level’ = level + Inc__in? AND Inc__out_’ = ok AND invariant__’ --> level’ IN { x : NAT | TRUE} [] … ]END; • Pre-, post-conditions • guarded commands • update FSM vars Inc Δ Statein? : ¥out! : REPORT level + in?  maxlevel’ = level + in?out! = ok ABZ Conference, London 2008

  16. Z2 MathToolkit • General strategy • define Z math data types in separate SAL text-units • parameterized CONTEXTs reused with different types • Sets, relations • translate sets, relations into Bryant’s ordered propositions • specific problems with cardinality, product-types • Functions, sequences, bags • translate partial Z functions into total SAL functions • requires bottom elements, rules for bottom • still working on sequences, bags ABZ Conference, London 2008

  17. Set Context set {T : TYPE;} : CONTEXT = BEGIN Set : TYPE = [T -> BOOLEAN]; empty : Set = LAMBDA (elem : T) : FALSE; … contains? (set : Set, elem : T) : BOOLEAN = set(elem); union (setA : Set, setB : Set) : Set = LAMBDA (elem : T) : setA(elem) OR setB(elem); …END LOCAL members : set{PERSON;} ! SetINITIALIZATION members = set{PERSON;} ! empty set definition; and usage ABZ Conference, London 2008

  18. Ordered Propositions • Bryant’s encoding for sets • conversion into ordered propositions over elements • monolithic “set” has no direct representation • set type is a function from element  boolean • polylithic judgements over ordered elements • Pros and Cons • highly efficient compilation into BDDs  • set-operations  boolean graphs for each element • difficult to count the elements in a set  • but Z needs a cardinality operation ABZ Conference, London 2008

  19. Brute Force Counting count3 {T : TYPE; e1, e2, e3 : T} : CONTEXT = BEGIN Set : TYPE = [T -> BOOLEAN]; size? (set : Set) : NATURAL = IF set(e1) THEN 1 ELSE 0 ENDIF + IF set(e2) THEN 1 ELSE 0 ENDIF + IF set(e3) THEN 1 ELSE 0 ENDIF;END LOCAL num : NATLOCAL friends : set{PERSON;} ! Set… num = count3{PERSON; PERSON__1, PERSON__2, PERSON__3} ! size?(friends) … Z2SAL generates counting contexts, as required count3 definition; and usage ABZ Conference, London 2008

  20. Relation Encoding • Follows set encoding • set type  ordered propositions over elements • relation type  set of pairs  ordered props. over pairs • Encoding choices • re-implement all set-ops in the relation context • provide only the rel-ops in the relation context, and re-use all set-ops from the set context  • SAL typing issue • set{…}!Set ≠ relation{…}!Relation  because type names specific to their local context • solution: pick only one local context to export “public” names by which types are known in main context ABZ Conference, London 2008

  21. Relation Context relation {X, Y : TYPE;} : CONTEXT = BEGINXY : TYPE = [X, Y]; Domain : TYPE = [X -> BOOLEAN]; Relation : TYPE = [XY -> BOOLEAN]; … domain (rel : Relation) : Domain = LAMBDA (x : X) : EXISTS (y : Y) : LET (pair : XY) = (x, y) IN rel(pair); …END PERSON__X__NAT : TYPE = [PERSON, NAT];LOCAL phonebook : set{PERSON__X__NAT;} ! Set… friends = relation{PERSON, NAT;} ! domain(phonebook) … SAL bug : type subst. expects single symbol; can’t subst. structure relation definition; and usage ABZ Conference, London 2008

  22. Partitioning Z MathOps • Multiple contexts • relation type also defined as set of pairs • allows reuse of set-ops for relations • specific rel-ops provided by relation context • can we extend this idea? • Partitioning criteria • package mathops by the number of type params • eg: give separate contexts for closure{X;}, relation{X,Y;} and compose{X,Y,Z;} ABZ Conference, London 2008

  23. Function Encoding • Follow relation encoding? • function type  set of pairs  similar to relation  impose extra restrictions on range • supports empty, partial, mutable functions  • slower execution, bigger search space  • Follow native SAL encoding? • function type  ordered mappings over elements • only supports total functions  • faster execution, close to BDD encoding  • Totalising strategy • by extending partial types with bottom values ABZ Conference, London 2008

  24. Function Context function {X, Y : TYPE; xb : X, yb : Y} : CONTEXT = BEGIN Function : TYPE = [X -> Y]; Domain : TYPE = [X -> BOOLEAN]; … domain (fun : Function) : Domain = LAMBDA (x : X) : fun(x) /= yb; …END NAT : TYPE = [0..4];PERSON : TYPE = {PERSON__1, PERSON__2, … PERSON__B};LOCAL passport : [PERSON -> NAT]LOCAL citizens : set{PERSON;} ! Set … citizens = function{PERSON, NAT; PERSON__B, 4} ! domain(passport) … xb, yb are formal params for bottom sentinel value, or explicit bottom ABZ Conference, London 2008

  25. Extended Invariant NAT : TYPE = [0..4];PERSON : TYPE = {PERSON__1, PERSON__2, … PERSON__B};LOCAL passport : [PERSON -> NAT]INPUT Apply__citizen? : PERSONOUTPUT Apply__passid_ : NATDEFINITION invariant__ = ( … AND Apply__citizen? /= PERSON__B AND Apply__passid_ /= 4 … AND passport(PERSON__B) = 4 …) All Z inputs, outputs must be well-defined f(xb) = yb asserted globally for each fn shorter fn defns ABZ Conference, London 2008

  26. Function Types • Z function typology • partial or total, combined with • unmarked, surjective, injective, bijective • Encode as distinct SAL types? • would require duplicated function contexts  • provide semantic predicates, rather than extra syntax surjective? (fun : Function) : BOOLEAN = FORALL (y : Y) : EXISTS (x : X) : fun(x) = y; ABZ Conference, London 2008

  27. Translation Templates • Set insertion • friends’ = friends U {pers?} • Function insertion • passport’ = passport Å {citizen? a passid!} …where the literal SAL would be very inefficient… insert (set : Set, new : T) : Set = LAMBDA (elem : T) : elem = new OR set(elem); insert (fun : Function, pair : XY) : Function = LAMBDA (x : X) : IF pair.1 = x THEN pair.2 ELSE fun(x) ENDIF; Z2SAL identifies cases with singleton sets … likewise for singleton set, function subtraction… ABZ Conference, London 2008

  28. Z2 Evaluation • Parser evaluation strategy • analyze diverse handwritten LaTeX-Z specs • extend parser to recognise alternative LaTeX forms • inspect SAL output, simulate SAL output • Math toolkit evaluation strategy • create a CONTEXT for a given Z math data type • simulate with many Z specs using this data type • test using counter-theorems …see next slide… • Example findings • can shrink state-space by clamping initial outputs • semantic function properties that don’t apply to Æ ABZ Conference, London 2008

  29. Counter-Theorem LOCAL members : set{PERSON;} ! SetLOCAL rented : set{PERSON__X__TITLE;} ! SetLOCAL stockLevel : [TITLE -> NAT]INITIALIZATION [ …]… th1 : THEOREM State |– G( set{PERSON__X__TITLE;} ! empty? (rented) ); Counter-theorem says: “the State module allows us to derive that the relation rented is always empty”, expected not to hold. …all vars initialised to empty sets/relations/functions… ABZ Conference, London 2008

  30. Proof Trace After 3 steps, the counter-theorem is disproved; so the negationholds; i.e. it is possible for a person to rent at least one video ABZ Conference, London 2008

  31. Z2 Performance • Video shop example in the paper • 3 base types: PERSON, TITLE, NAT • 4 constructed types: pair, set, relation, function • 3 local vars (of set, relation, function types) • 8 input/output vars (of basic types) • 5 ops (rentVideo, addTitle, delTitle, addMember, copiesOut) • SAL compilation and execution times • about 6-7 seconds to compile to BDDs, Bűchi automata • counter-theorems disproved in 1-2 seconds • 11,664 initial states  61,568,640 states after 5 steps ABZ Conference, London 2008

  32. Z2 What Next? • Sequences and bags • experimenting with SAL records for sequences • extra field stores length of the sequence • issues in preserving the order of a sequence • Porting to CZT • MSc team project adapted CZT parser, AST • used Visitor-pattern to generate similar SAL output • poorly-documented AST is fairly hard to use • PhD project • provable refinement of Z-specs by model-checking SAL translation ABZ Conference, London 2008

  33. Z2 Thank You! John Derrick, Siobhán North and Anthony Simons Department of Computer ScienceUniversity of Sheffield ABZ Conference, London 2008

More Related