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Refinement Method for Abstract State Machines Egon Börger Dipartimento di Informatica, Universita di Pisa http://www.di.unipi.it/~boerger

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“The intuition behind refinement” “The intuition behind refinement is just the following:     Principle of Substitutivity: it is acceptable to replace one program by another, provided it is impossible for a user of the programs to observe that the substitution has taken place.” [Derrick&Boiten 2001, pg.47] Why should “acceptable” refinements be restricted to those which guarantee that the substitution of one program by a refined one is not observable? e.g. imagine one wants to observe the desired improvement provided by a refinement (an executable instead of an abstract pgm, a faster or more general pgm serving also other purposes, a strengthening…) delimit the exact boundaries within which the refined program performs in the intended way

Characteristics of refinement notions in the literature Traditionally, refinement notions guided by the substitutivity principle come with additional restrictive assumptions: programs describe sequences of operations precluding parallelism of multiple simultaneous updates or iterative compositions of programs operations are global (binary) state relations yielding the frame problem for combinations of local effects observations are pairs of input/output sequences or of pre-post-states representing what is considered to be of interest before/after program execution making it difficult to look at arbitrary segments of computation

Role of syntactical issues in refinement notions in the literature numerous program refinement notions (e.g. for ADT, Z) are formulated for structurally equivalent programs with corresponding operations in the same places precluding the analysis of more complex relations bw operations invariants in refinements are often viewed as changing the state scheme or the operations, in terms of pre/post condition strengthenings or weakenings instead of analysing their effect as restricting the class of models

Role of syntactical issues in refinement notions in the literature most refinement notions are logic or proof-rule oriented, tailored to fit proof principles  [de Roever&Engelhardt] spec perceived as a (huge!) logical expression implementation understood as implication composition defined as conjunction thus possibly restricting the design space e.g. refinements should be pre-congruences: for every context C: x £ y  implies C[x] £ C[y]. This can be achieved for example by monotonicity of pgm constructors wrt refinement. commits to uniform context-independent “algebraic” refinements e.g. operation refinement by combining multiple operations “conjunctively” or “disjunctively” (“alphabet translation”)

Linking refinement and proof principles illustrated by B B links design & proofs by relating pgm constructs & proof principles  at the price of  restricting the design space Machine inclusion example (B-Book pg.317)) Let M include M’. Then “at most one operation of the included machine can be called from within an operation of the including machine. Otherwise we could break the invariant of the included machine.” Let M’ have the following operations, satisfying the invariant v £ w : increment  º  If v < w then v := v+1 decrement º  If v < w then w := w-1 Let M include M’ and contain the following operation: If v<w then increment                       decrement Then the invariant v £ w is broken by M for w = v+1 The ASM method allows parallel invocations of submachines at the price of having to care about the correctness proofs

Linking refinement and proof principles illustrated by CSP CSP links design & proofs by relating pgm constructs & proof principles  at the price of  restricting the design space Refining processes by adding assignment is restricted to certain assignments (Hoare CSP Book 1985, pg. 188)): When two processes P and Q are put into parallel, it is required that the variables P assigns to are disjoint from the variables of Q: Write(P) Ç  Var(Q) = Æ Otherwise the CSP laws would not work

Introducing refinement techniques into ASMs Refinements, one of the 3 building blocks of the ASM method, were introduced into ASMs in 1989 through Börger’s ASM models defining the ISO Prolog standard, triggered by the simple observation that    exploiting the freedom of abstraction ASMs offer, one can tailor ASM refinements to solve given design & analysis problems  also for complex real-life systems     as they occur in industrial practice Consequently, the  ASM refinement method is problem-oriented and its development was driven by practical refinement tasks, occurring in real-life system development the goal to support divide-and-conquer techniques for both design and verification without privileging one to the detriment of the other

Problem oriented tasks guiding the ASM refinement method In each case, “listen to the subject” to find/formulate an appropriate refinement /abstraction that faithfully reflects the intended  design decision (or reengineering idea) for the system under study can be justified to correctly implement the given model (or to abstract from the given code), namely through verification validation testing model-based runtime assertions to show by simulation that design assumptions hold in the implementation Effect (scaling to industrial-size systems): enhancement of communication of designs and system documentation (report of analysis) effective reuse  (exploiting orthogonalities, hierarchical layers) system maintenance based upon accurate, precise, richly indexed & easily searchable documentation                            E.Börger: High Level System Design and Analysis using ASMs                 LNCS 1012 (1999) 1-43

Main usages of ASM refinements capture orthogonalities by modular machines (components) e.g. ASMs for sublanguages of Java and JVM instructions construct hierarchical levels for horizontal piecemeal extensions and adaptations (design for change) e.g. of  ISO Prolog model by constraints (Prolog III), polymorphism (Protos-L), narrowing (Babel), object-orientation (Müller), parallelism (Parlog, Concurrent Prolog etc), abstract execution strategy (Gödel) vertical stepwise detailing of models (design for reuse) in a proven to be correct way down to their implementation, e.g. model chains leading from Prolog to WAM Occam to Transputer Java to JVM ASMs to executable ASMs (Workbench, AsmGofer, AsmL, XASM) exploit reusable proof techniques for system properties e.g. reusing Prolog to WAM proof for CLP(R) to CLAM Protos-L to PAM using variety of logics for ASMs, KIV, PVS, Isabelle, model checkers

Examples of ASM Refinement & Verification Hierarchies Architectures: Pipelining of RISC DLX: model checking, PVS verification     Control Systems: Production Cell (model checked), Steam Boiler  (refinements to C++ code)   Light Control (executable requirements model)    Compiler correctness ISO Prolog to WAM: 12 refinement steps, KIV verified backtracking, structure of predicates, structure of clauses, structure of terms & substitution, optimizations Occam to Transputer :15 models  exhibiting channels, sequentialization of parallel procedures, pgm ctrl structure, env, transputer datapath and  workspace, relocatable code (relative instr addresses & resolving labels) Java to JVM: language and security driven decomposition into    5 horizontal sublanguage levels (imperative, modules, oo, exceptions, concurrency) and    4 vertical JVM levels for trustful execution, checking defensively at run time and  diligently at link time, loading (modular compositional structuring)

Illustrating Reusability of ASM Refinement Hierarchies

The ASM Refinement Scheme: Commuting Diagrams

Defining correctness of a refinement M* of M Fix any notions º of equivalence of states & of initial/final states Idea of correctness: refined runs simulate abstract ones Definition. M* is a correct refinement of M iff    every (infinite) refined run simulates an (infinite)    abstract run with equivalent corresponding states i.e. for each M*-run S*(0), S*(1),… there is an M-run S(0), S(1),… , either both terminating or both infinite, with infinite sequences i0< i1<…, j0< j1<… such that S(ik) º S*(jk) for each k, including the initial states (i0 = j0 =0)  and the final ones (if any) Wlog at final states, the state sequence becomes constant                 i.e. S(r) = S(r+k) for each final S(r) and each k, same for S*

Completeness condition for ASM refinements Completeness idea: abstract runs are simulated by (correspond to) refined ones, symmetrically to how for correctness refined runs simulate (correspond to) abstract ones Def. M* is a complete refinement of M                           iff M is a correct refinement of M* Related terminology: “bisimulation” or “interpreter equivalence” for correct and complete refinement (wrt terminating runs considering only the input/output behavior) “preservation of partial correctness” for correct refinement (wrt terminating runs) “preservation of total correctness” for complete refinement (adding to the correctness condition for terminating runs that every infinite refined run admits an infinite abstract run with an equivalent initial state)

Remarks on the correctness conditions for ASM refinements Corollary. Refinement correctness implies for terminating runs the equivalence of the input/output behavior of the abstract and the refined machine. S(ik), S*(jk) are the corresponding states (those of interest), end points of the corresponding computation segments (those of interest), for which the equivalence is defined in terms of a relation between their corresponding locations (those of interest). Wlog the sequences of corresponding states are minimal in the sense that between two sequence elements there are no other equivalent states i.e. there are no ik<i< ik+1  , jk<j< ik+1 with S(i) º S*(j)

Refinement notions in the literature as cases of ASM refinements Considering only the input/output behavior, restricting correctness (essentially) to terminating runs e.g. preservation of partial/total correctness (as used in compiler correctness verifications) or bisimulation Data refinement considering as initial/final the pre/post states of an operation (1,1)-refinements for corresponding operations (with unchanged signature, tailored to provide “unchanged” properties) forward simulation carries over º from pre-states to post-states backward simulation carries over º from post-states to pre-states see Hoare 1972, VDM, Z, B, de Roever & Engelhardt 1998 NB. Under a monolithic view (of each ASM as defining just one total operation on structures), ASM refinement becomes data refinement Non-atomic operation refinement (1,n)-refinements with fixed n (in Z, Object-Z, see Derrick & Boiten 2001) (1,1)-refinements for external operations with (1,0),(0,1)-refinements for finitely many invisible internal operations alphabet extension/translation,  I/O automata refinements,  etc.         see details in [Schellhorn2001]

Conservative ASM refinement: incrementally adding machines Adding an entire machine M - not limited to a single “operation” - to another machine     Exl. Adding a bytecode verifier to the Java interpreter in JVM

Procedural refinements & their specialization to sequential submachine refinements of ctl state ASMs Procedural refinement: replacing a machine by another (usually more complex) machine Specialization for control state ASMs: replacing control state transitions (machines at nodes) by    submachine diagrams with entry/exit nodes The Scheme:

Illustrating sequential submachine refinements refining the control state ASM model for a debugger

Sequential submachine refinement of machine onStart into a sequence of three submachines

Parallel and sequential refinement of callback(LoadModule) callback(LoadModule (proc,mod)) =      displayMessage("Loaded module: " ++ mod.name())       record mod in shell        forall bp in shell.BPs               bind all breakpoints to the mod (in any order)           bp.bind(mod)  seq          mod.enableClassLoadCallbacks()   proc.resume()                                    continue via external call                                                                                Analogously for UnloadModule

(1,1)-refinements of ASMs allow parallelism replacing an action - part of a parallel step, not limited to a single “operation” - by multiple parallel actions  (not viewed as a new “operation”, but as part of a new parallel step)  e. g. rule  by   rule1  … rulen Exl. Defining submachine execJava of execJavaThread         by parallel submachines separating semantics of thread execution from thread scheduling

(1,1)-refinement of execJava as parallel composition of language driven submachines execJava = execJavaI           imperative control constructs         execJavaO                     oo features execJavaE                  exception handling execJavaT                  concurrent threads where each  execJavasub =                                   execJavaExpsub      expression evaluation                                   execJavaStmsub               statement execution allowing semantics to be defined instructionwise

Backtracking Machine (for Tree Computations) If mode = ramify then         Let k = |alternatives (Params)|         Let o1 ,..., ok =new (NODE)               candidates (currnode) := { o1 ,..., ok }               forall 1 £ i £ k  do                         parent (oi) := currnode                         env (oi) := i-th (alternatives (Params))          mode := select If mode = select then         If candidates (currnode) = Æ            then backtrack            else try-next-candidate                    mode := execute

Backtracking Machine backtrack º   if parent (currnode) = root                                   then mode := Stop                                   else currnode := parent (currnode) try-next-candidate º                depth-first tree traversal     currnode:= next (candidates(currnode))     delete next (candidates(currnode)) from candidates (currnode) The fctn next is a choice fct, possibly dynamic, which determines the order for trying out the alternatives. The fct alternatives, possibly dynamic and coming with parameters, determines the solution space. The execution machine may update mode again to ramify (in case of successful exec) or to select  (for failed exec)

Backtracking Machine: logic instantiation Prolog  Börger/Rosenzweig Science of Computer Programming 24 (1995) alternatives = procdef (act,pgm), yielding a sequence of clauses in pgm, to be tried out in this order to execute the current statement (“goal”) act procdef (act,constr,pgm) in CLAM with constraints for indexing mechanism                  Börger/Salamone OUP 1995 next = first-of-sequence (depth-first left-to-right tree traversal) execute mode resolves act against the head of the next candidate, if possible, replacing act by that clauses’ body & proceeding in mode ramify, otherwise it deletes that candidate & switches to mode select

Backtracking Machine: functioal progg instantiation Babel           Börger et al. IFIP 13 World Computer Congress 1994, Vol.I alternatives = fundef (currexp,pgm), yielding the list of defining rules provided in pgm for the outer fct of currexp next = first-of-sequence execute applies the defining rules in the given order to reduce currexp to normal form (using narrowing, a combination of unification and reduction)

Backtracking Machine: context free grammar instantiation Generating leftmost derivations of cf grammars G alternatives (currnode,G), yields sequence of symbols Y1...Yk of the  conclusion of a G-rule with premisse X labeling currnode. Includes a choice bw different rules X®w env yields the label of a node: variable X or terminal letter a next = first-of-sequence (depth-first left-to-right tree traversal) execute mode for nodes labeled by a variable triggers tree expansion for terminal nodes extracts the yield, concatenating terminal word to output, continues derivation at parent node in mode select

Backtracking Machine: instantiation  for attribute grammars Synthesis of node attribute from children’s attributes via backtrack º   if parent (currnode) = root  then mode := Stop                                 else currnode := parent (currnode)                                         X.a := f(Y1.a1, ..., Yk.ak) where X = env(parent(currnode)), Yi =env(oi ) for children nodes Inheriting attribute from parent and siblings included in update of env (e.g. upon node creation) generalized to update also node attributes Attribute conditions for grammar rules included in execute-rules as additional guard to yielding output

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Looking for invariants to prove ASM refinement correctness Idea: find commuting diagrams with end points s, s* which satisfy an invariant » implying the to be established equivalence º Realization: for each pair of corresponding states - not both final -  satisfying », follow the two runs to find a successor pair s’, s*’ (of corresponding states satisfying ») Two cases are possible for such run extensions: only one of the two runs can be extended the abstract one, producing an (m,0)-diagram the refined one, producing a (0,n)-diagram both runs can be extended

Extending runs by triangles and trapezoids

Definition of the forward simulation condition FSC(s,s*) If s » s* and not both s,s* are final states, then either the abstract run can be extended               by an (m,0)-triangle               leading in m>0 steps to an s’»s* with (s’,s*) <m0 (s,s*) or the refined run can be extended               by a (0,n)-triangle               leading in n>0 steps to an s*’»s with (s,s*’) <0n (s,s*) or both runs can be extended               by an (m,n)-trapezoid leading                    in m>0 abstract steps to an s’                    in n>0 refined steps to an s*’               such that s’»s*’ NB. A minor modification covers also nondeterministic ASMs

Schellhorn’s coupling invariant for correct ASM refinements Theorem. M* is a correct refinement of M     wrt an equivalence notion º and a notion of initial/final states    if there is a relation » such that the coupling invariant » implies equivalence  º each refined initial state s* is coupled by the invariant to an abstract initial state s»s* the forward simulation condition FSC holds for every pair (s,s*) of abstract and refined states

Exercise Prove that in the correctness definition of ASM refinements one can assume without loss of generality that the sequences of corresponding states are minimal, in the sense that between two sequence elements there are no other equivalent states i.e. there are no ik<i< ik+1  , jk<j< ik+1 with S(i) º S*(j)

"References" References AsmBook   E. Börger, R. Stärk: Abstract State Machines.  A Method for High-Level System Design and Analysis Springer-Verlag  2003, see  http://www.di.unipi.it/~AsmBook ASM Refinement Case Study Book   R. Stärk, J. Schmid, E. Börger Java and the Java Virtual Machine: Definition, Verification, Validation  Springer 2001, see http://www.inf.ethz.ch/~jbook ASM Refinement Analysis   G. Schellhorn Verification of ASM Refinements Using Generalized Forward Simulation   J. Universal Computer Science 7 (11) 2001 ASM Survey  E. Börger High Level System Design and Analysis using ASMs  LNCS Vol. 1012 (1999),  pp. 1-43 ASM History  E. Börger The Origins and the Development of the ASM Method for High Level System Design and Analysis   JUCS 8 (1) 2002

References on Backtracking Machine E.Börger and D. Rosenzweig: Mathematical Definition of Full Prolog In: Science of Computer Programming 24 (1995) 249-286 E.Börger and R.F.Salamone: CLAM Specification for Provably Correct Compilation of CLP ( R ) Programs In: E.Börger (Ed.) Specification and Validation Methods. Oxford University Press, 1995, 97-130 E.Börger, F.J.Lopez-Fraguas, M.Rodrigues-Artalejo: A Model for Mathematical Analysis of Functional Programs and their Implementations In: B.Pehrson and I.Simon (Eds.): IFIP 13 World Computer Congress 1994, Vol.I: Technology/Foundations, 410-415 D. Johnson and L. Moss: Grammar Formalisms Viewed als Evolving Algebras Linguistics and Philosophy 17 (1994) 537-560

"References" References Three Books on Refinement Methods J. Derrick, E. Boiten  Refinement in Z and Object-Z    Springer-Verlag 2001 W. de Roever, K. Engelhardt  Data Refinement: Model-Oriented Proof Methods and their Comparison  Cambridge University Press 1998 J. C. P. Woodcock, J. Davies Using Z: Specification, Refinement, and Proof Prentice-Hall 1996