Abstract:We give an algebraic model of the designs of UTP based on a variant of modal semirings, hence generalising the original relational model. This is intended to exhibit more clearly the algebraic principles behind UTP and to provide deeper insight into the general properties of designs, the program and specification operators, and refinement. Moreover, we set up a formal connection with general and total correctness of programs as discussed by a number of authors. Finally we show that the designs form a left semi… Show more
“…Since we are going to import some results from the third framework, we will show some connections between variants 1 and 3 (we do not need variant 2 in the present paper, but the treatment for it would be symmetrical). We only give a summary of the necessary theory; a more thorough treatment can be found in [4].…”
Section: The Algebra Of Prescriptionsmentioning
confidence: 99%
“…The identity in algebraic structure is used in the companion paper [4] to give a uniform treatment of normal designs t a and normal prescriptions t a as pairs (a, t) consisting of a transition part a and a termination condition part t.…”
Abstract. We show that the well-known algebra of matrices over a semiring can be used to reason conveniently about predicates as used in the Unifying Theories of Programming (UTP). This allows a simplified treatment of the designs of Hoare and He and the prescriptions of Dunne. In addition we connect the matrix approach with the theory of test and condition semirings and the modal operators diamond and box. This allows direct re-use of the results and proof techniques of Kleene algebra with tests for UTP as well as a connection to traditional wp/wlp semantics. Finally, we show that matrices of predicate transformers allow an even more streamlined treatment and removal of a restricting assumption on the underlying semirings.
“…Since we are going to import some results from the third framework, we will show some connections between variants 1 and 3 (we do not need variant 2 in the present paper, but the treatment for it would be symmetrical). We only give a summary of the necessary theory; a more thorough treatment can be found in [4].…”
Section: The Algebra Of Prescriptionsmentioning
confidence: 99%
“…The identity in algebraic structure is used in the companion paper [4] to give a uniform treatment of normal designs t a and normal prescriptions t a as pairs (a, t) consisting of a transition part a and a termination condition part t.…”
Abstract. We show that the well-known algebra of matrices over a semiring can be used to reason conveniently about predicates as used in the Unifying Theories of Programming (UTP). This allows a simplified treatment of the designs of Hoare and He and the prescriptions of Dunne. In addition we connect the matrix approach with the theory of test and condition semirings and the modal operators diamond and box. This allows direct re-use of the results and proof techniques of Kleene algebra with tests for UTP as well as a connection to traditional wp/wlp semantics. Finally, we show that matrices of predicate transformers allow an even more streamlined treatment and removal of a restricting assumption on the underlying semirings.
“…In [9] prescriptions were introduced as the general-correctness counterparts of Hoare and He's total-correctness designs, and their properties have since been further explored in [6] and [13]. Let v be the list of state variables of the state space and ok be an additional auxiliary boolean variable with the same interpretation as that already described in Section 3 for designs.…”
Section: A General-correctness Program Calculusmentioning
Abstract. We introduce a calculus for reasoning about programs in total correctness which blends UTP designs with von Wright's notion of a demonic refinement algebra. We demonstrate its utility in verifying the familiar loop-invariant rule for refining a total-correctness specification by a while loop. Total correctness equates non-termination with completely chaotic behaviour, with the consequence that any situation which admits non-termination must also admit arbitrary terminating behaviour. General correctness is more discriminating in allowing nontermination to be specified together with more particular terminating behaviour. We therefore introduce an analogous calculus for reasoning about programs in general correctness which blends UTP prescriptions with a demonic refinement algebra. We formulate a loop-invariant rule for refining a general-correctness specification by a while loop, and we use our general-correctness calculus to verify the new rule.
“…The equivalence classes correspond to the designs of the Unifying Theories of Programming of [9] and hence represent a total correctness view. It has been shown in [7] (in the setting of condition semirings that is isomorphic to that of test semirings) that the set of these classes forms again a left semiring and can be made into a weak Kleene and omega algebra by using exactly the same definitions as above (as class representatives). Now top-left-strictness holds, since chaos ≡ loop and loop is a left zero by the definition of command composition.…”
Section: The Demonic Refinement Algebra Of Commandsmentioning
confidence: 99%
“…So these model programs where no miraculous termination can occur; they correspond to the feasible designs of [9]. In [7] it is shown that the set F(S) classes of feasible commands can isomorphically be represented by simple semiring elements. The mediating functions are…”
Section: The Demonic Refinement Algebra Of Commandsmentioning
Abstract. Weak omega algebra and demonic refinement algebra are two ways of describing systems with finite and infinite iteration. We show that these independently introduced kinds of algebras can actually be defined in terms of each other. By defining modal operators on the underlying weak semiring, that result directly gives a demonic refinement algebra of commands. This yields models in which extensionality does not hold. Since in predicate-transformer models extensionality always holds, this means that the axioms of demonic refinement algebra do not characterise predicate-transformer models uniquely. The omega and the demonic refinement algebra of commands both utilise the convergence operator that is analogous to the halting predicate of modal µ-calculus. We show that the convergence operator can be defined explicitly in terms of infinite iteration and domain if and only if domain coinduction for infinite iteration holds.
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