R. J. R. Back and K. Sere. Stepwise refinement of action systems. In Mathematics of Program Construction, pages 115--138, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....system and give the meaning of the parallel composition for several action systems. 2.1 Actions Action systems, originally proposed by Back and Kurki Suonio [2] are predicate transformer based systems. Action systems have been applied in modelling reactive and concurrent behaviour, see e.g.[3, 7]. Recently, the action systems framework has also been introduced for reasoning about discrete control systems [11] An action in action systems is any statement in Dijkstra s guarded command language [8] including the pure guarded commands. The actions are defined with the weakest precondition ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....system, with a larger number of (simpler) actions and an equivalent behavior with respect to input or output variables. Formal methods of concurrent programming become increasingly important in design of complex VLSI systems. In our earlier work, we have shown how the action systems formalism [1] can be applied in design of both self timed [9] and synchronous circuits [10, 11] In this paper, we continue our work on synchronous modeling by focusing on the formal aspects of clocked pipelining. The actions composing a synchronous action system are high level representations of circuit ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30,1991.

....a conflict with the classical axiom x = would arise in DRA. 2. 5 Modeling programs We want to use the refinement algebra to reason about loop programs in the style of Dijkstra s guarded commands [12] and also about parallel programs and distributed systems that have the form of action systems [3, 5], in the same way as KAT is used to reason about while programs in a partial correctness framework [15] Using DRA, conditionals and guarded loops are modeled as follows: if p then x else y fi = px u py do p x [ q y od = px u qy) here the guarded loop has two alternatives, but the same ....

R.J. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17--30, 1991.

....The transformer uses Dijkstra s idea of describing the effect of a given program by calculating the precondition which guarantees that a given postcondition holds. Action systems have been extensively used in sequential program refinement, and also in reactive and concurrent programs, see e.g. [4, 9]. Previously action systems have been extended with explicit clock variables when reasoning about real time systems, and with discrete approximations when modelling continuous phenomena in hybrid settings [13, 19] However, since there is no technical reason why predicates over program variables ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming,12, pp. 17-30, 1991.

....of these approaches. 2 Actions Action systems, originally proposed by Back and Kurki Suonio [2] are predicate transformer based systems that operate on some state base. Action systems have been applied in many non trivial applications when modeling reactive and concurrent behaviour, see e.g.[3, 7]. In this paper, we restrict ourselves to individual actions instead of studying the parallel behaviour several action systems. An action is any statement in an extended version of Dijkstra s guarded command language [8] where also the pure guarded commands are allowed as actions. The actions ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....is relevant for many frameworks of refinement, but we need a specific carrier to express our results. We choose the action system formalism of Back and Kurki Suonio [BKS83] as a paradigm for describing parallel systems. Refinement of action systems was studied by Back, Sere and Wright, in e.g. [BS91, Bac89, BvW94]. Our observations are directly useful to many other transition system based formalisms, e.g. AL91, KMP94, Jon91] 2 Preliminaries 2.1 Refinement calculus The basic domains of refinement calculus arise by pointwise extension from the boolean lattice. The truth values Bool = fT; Fg form a ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17--30, 1991.

....be extended to provide an adequate semantic treatment of promoted data types. Section 6 is a discussion of related work. That of Fischer, Smith, and Derrick [Fis97b, FS97, Fis98, Smi97, SD97] is particularly relevant. Earlier studies by Back, Sere, Hoare, He, Sanders, Woodcock, Morgan, and Butler [BS89, HHS87, WM90, But93] are also of interest. The paper ends with a brief summary and a list of references. 2 Notation 2.1 Abstract data types An abstract data type combines a notion of state with a collection of named operations, modelled as relations, that may involve input and output. Two operations are ....

.... op: OpsA n) P( StateA 0 ) 6 Discussion 6.1 Related work The problem of relating state based and behavioural models of the same component has attracted considerable theoretical and practical interest. A number of hybrid formalisms have been proposed, such as the Action Systems formalism [BS89, WM90, But93], but these are less suitable for addressing either of the two aspects of design in isolation. Accordingly, much of the recent work has concentrated upon identifying and exploiting links between complementary formalisms, each of which has proved successful in the description of a different aspect ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. In Proceedings of the International Conference on Mathematics of Program Construction, volume 375 of LNCS, pages 115 -- 138. Springer, 1989.

....on an extended version of Dijkstra s language of guarded commands [8] This language includes assignment, sequential composition, conditional choice, and iteration, and is defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be 1 found elsewhere [1, 2]. Actions. An action A is defined, for example, by A : abort (abortion, nontermination) j skip (empty statement) j x : x 0 :R (nondeterministic assignment) j [P ] assumption) j A 1 [ A 2 (nondeterministic choice) where P and R are predicates (boolean conditions) x a variable or a ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....A j are disjoint if they have no state variables in common: vA i vA j = Actually, we relax this condition by permitting the actions to have read read conflict on the state variables. If several disjoint actions are simultaneously enabled then they can be implemented in parallel. Back et al. [1, 4, 5, 6] gave a thorough treatment of the action systems where an action body is a standard program, i.e. statement of language (1) They used a backward, predicate transformer model of the programs, assuming, however, that the body of each action A in an action system must satisfy two conditions: 1) It ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....guarded command language of Dijkstra [7] The statements of this language include assignment, sequential composition, assertion, conditional choice and iteration, and are defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be found for example in [2, 3]. The action systems framework in asynchronous design is treated in [13, 14, 15] Actions An action is a guarded command of the form g S where g, the guard, is a boolean condition, and S, the body, is any statement in our language. The action A is said to be enabled when the guard is true, ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....Action Systems, an invariant verification algorithm of linear complexity and a reachability algorithm of squared complexity. Keywords: action systems, linear hybrid systems, model checking, invariants TUCS Research Group Programming Methodology Research Group 1 Introduction Action systems [3, 4, 5] are a predicate transformer based systems used especially in the development of reactive and concurrent systems. Recently, Ronkko and Ravn [15] extended Action Systems with a new action called the differential action. It allows the use of differential equations in capturing continuous behaviour. ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming,12, pp. 17-30, 1991.

....on an extended version of Dijkstra s language of guarded commands [7] This language includes assignment, sequential composition, conditional choice, and iteration, and is defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be found elsewhere [1, 4]. Actions. An action A is defined (for example) by A : skip x : x 0 :R A1 [ A2 A1 # A2 P A b = P ]# A j[ var x : x0# A ]j (empty statement) nondeterministic assignment) nondeterministic choice) sequential composition) guarded action) block with local variables) where ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....two main approaches [8] post design verification and correctness preserving transformations. This study presents features of the action systems framework, formalism that uses correctness preserving transformations in order to derive system implementation from higher level specifications [2]. The majority of modern digital hardware uses synchronous techniques, although the asynchronous model gains again, more and more attention, especially because of the potential for low power systems. The basic reason for this is that the design experience base and CAD tool support for synchronous ....

....based on an extended version of Dijkstra s language of guarded commands [5] This language includes assignment, sequential composition, conditional choice, and iteration, and is defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be found elsewhere [1, 2]. Actions. An action A is defined, for example, by A : skip (empty statement) j x : x 0 :R (nondeterministic assignment) j A 1 [ A 2 (nondeterministic choice) j A 1 ; A 2 (sequential composition) j P A (guarded action) where P and R are predicates, x a variable or a list of ....

[Article contains additional citation context not shown here]

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991. 13

....two main approaches [8] post design verification and . correctness preserving transformations. This study is based on features of the action systems framework, formalism that uses correctness preserving transformations in order to derive system implementation from higher level specifications [2]. The majority of modern digital hardware uses synchronous techniques, although the asynchronous model gains again, more and more attention, especially because of the potential for low power systems. The basic reason for this is that the design experience base and CAD tool support for synchronous ....

....based on an extended version of Dijkstra s language of guarded commands [5] This language includes assignment, sequential composition, conditional choice, and iteration, and is defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be found elsewhere [1,2]. An action A is defined, for example, by A: skip empty statement x : x .R nondeterministic assignment A 1 [ A 2 nondeterministic choice A 1 ; A 2 sequential composition [ var x: x 0 ; A] block with local variables P A guarded action where P and R are predicates, x a variable or a ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12, pp. 17-30, 1991.

....on an extended version of Dijkstra s language of guarded commands [14] This language includes assignment, sequential composition, conditional choice, and iteration, and is defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be found elsewhere [3, 4]. Actions An action A is defined by A : abort (abortion, nontermination) j skip (empty statement) j x : x 0 :R (nondeterministic assignment) j [P ] assumption) j A 1 [ A 2 (nondeterministic choice) j A 1 ; A 2 (sequential composition) j j[ var x : x0; A ]j (block with local ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....based on an extended version of Dijkstra s language of guarded commands [7] This language includes assignment, sequential composition, conditional choice, and iteration, and is defined using weakest precondition predicate transformers. Comprehensive study on this formalism can be found elsewhere [1, 4]. Actions An action A is defined by A : abort (abortion, nontermination) j skip (empty statement) j x : x 0 :R (nondeterministic assignment) j [P ] assumption) j A 1 [ A 2 (nondeterministic choice) j A 1 ; A 2 (sequential composition) j j[ var x : x0; A ]j (block with local ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17-30, 1991.

....concrete or refine the way the component moves in the network, and still preserve the functionality of the original computation. If we refine the state variables and leave the location variable unchanged during the refinement, we follow the rules for data refinement of topological action systems [2, 20]. This basically means that, having the topological action systems C and C 0 and an abstraction relation R, C 0 is a data refinement of C using R, C vR C 0 , if the initialisation, each exported procedure and each action of C is data refined using R by the corresponding initialisation, ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, Vol. 12, pp. 17-30, 1991.

....introduced and applied during the development of a real time, multi tasking system. A development procedure consisting of six major steps is outlined. Many smaller refinement steps may be applied at each level, of course, using the standard refinement rules established for action systems [BS91] For each step a generic description is given and then illustrated with a concrete example based on the well known mineshaft pump controller case study [BW90] Figure 1 shows the environment in which the pump controller is intended to operate. The overall goal is to keep the depth of ....

R.-J. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12(1):17--30, 1991.

....is to respond to short fluctuations such as that between times e and f . The heater may or may not be switched on at some time(s) between e and h, but it must definitely be off after time h. 2 2. 2 A State Machine Model The following state machine solution is expressed as an action system [3], adopting Abadi and Lamport s approach for modelling concurrent, real time state machines [1] Variables Firstly, we consider the three observable system properties. Let input variable thermometer , of type Temperature, denote the observed temperature at the current moment in time. Let output ....

....biased against the state machine model, an action as complex as UpdateTemp would still be needed in an open specification, where thermometer is a true input, in order to update history markers fell and rose. Concurrency State machine specifications simulate concurrency by interleaving actions [3]. However interleaving is inappropriate for modelling parallel time consuming actions because actions that overlap consume less time than the same actions performed in sequence. The actions in Section 2.2 are not time consuming they occur in under a single time unit and thus do not illustrate ....

[Article contains additional citation context not shown here]

R.-J. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12(1):17--30, 1991.

....instructions implementing S 1 , and addresses y , such that ff :2 4 y OE , are available for instructions implementing S 2 . Obligations (1) and (2) are maintained and each component S i can now be refined independently. Proof. To prove this law we must change a sequence of actions into a loop [1]. A v by R3 con Sigma = fff; ff :2g [ M ffl fpc = ffg ; S 1 ; S 2 ; pc 2 M] v by R1 con Sigma = fff; ff :2g [ M ffl fpc = ffg ; S 1 ; pc = ff :2] S 2 ; pc 2 M] v by R5 con Sigma = fff; ff :2g [ M ffl fpc = ffg ; do pc = ff S 1 ; pc = ff :2] pc ....

R.-J. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12(1):17--30, 1991.

....logic, and proved using associated proof methods. The standard notion of refinement for nondeterministic systems is based on behaviour inclusion; a system A is refined by another system A 0 if and only if every behaviour of A 0 is also a possible behaviour of A. In the action system formalism [2, 3], a nondeterministic system is described as a composition of action systems working on a common global state. An action system is characterised by an initialisation predicate which describes how the local state of the component is initialised, an action that is executed repeatedly (atomically) ....

....consequence of the following basic property which is easily verified: Su[p] tfqg = Stfqg)u[p] Thus, rather that considering action contracts we consider quadruples of the form (p; T; q a ; q d ) resulting from identifying a coalition. Action contracts generalise the traditional action system [2] which is identified by an initialisation and a collection of actions that are repeatedly executed until no action is enabled. In action systems, the choice of action is (demonically) nondeterministic, and no angelic nondeterminism is permitted. Semantically, such an action system can be treated ....

R.J.R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17--30, 1991.

....point of view as it allows the system to be developed and verified in manageable tasks, it still lacks e.g. good tool support. In this paper we concentrate on formal specification and derivation of systems within so called action systems. Action systems [2] and the associated refinement calculus [4], which provides a mathematical reasoning basis for the stepwise development of action systems, have shown their value in the design of reliable and correct systems in many ways [21] The formal design methods supporting action systems and reasoning about them are heavily influenced by approaches ....

R. J. R. Back and K. Sere. Stepwise refinement of action systems. Structured Programming, 12:17--30, 1991.

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R. J. R. Back and K. Sere. Stepwise refinement of action systems. In Mathematics of Program Construction, pages 115--138, 1989.

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Back, R. J. & Sere, K. (1989), Stepwise refinement of action systems, in `International Conference on Mathematics of Program Construction, 375th Anniversary of the Groningen University', pp. 115--138.

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R. J. R. Back and K. Sere. Stepwise refinement of action systems. In Mathematics of Program Construction, volume 375 of LNCS, pages 115--138. Springer-Verlag, 1989.

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