Demonic semantics: using monotypes and residuals

Abstract

Relations and relational operators can be used to define the semantics of programming languages. The operations ∨ and • serve to give angelic semantics by defining a program to go right when there is a possibility to go right. On the other hand, the demonic operations and do the opposite: if there is a possibility to go wrong, a program whose semantics is given by these operators will go wrong; it is the demonic semantics. This type of semantics is known at least since Dijkstra's introduction of the language of guarded commands. Recently, there has been a growing interest in demonic relational semantics of sequential programs. Usually, a construct is given an ad hoc semantic definition based on an intuitive understanding of its behavior. In this note, we show how the notion of relational flow diagram (essentially a matrix whose entries are relations on the set of states of the program), introduced by Schmidt, can be used to give a single demonic definition for a wide range of programming constructs. This research had originally been carried out by J. Desharnais and F. Tchier (1996) in the same framework of the binary homogeneous relations. We show that all the results can be generalized by using the monotypes and the residuals introduced by Desharnais et al. (2000).2000 Mathematics Subject Classification: 18C10, 18C50, 68Q55, 68Q65, 03B70, 06B35.1. Introduction. The approaches to semantics are categorized as operational, axiomatic, or denotational. We will be concerned with the operational and the denotational semantics of nondeterministic programs. The operational semantics is described by the relation between the initial and final states. In our case, we consider the worst execution of the program; that is, we suppose that the program behaves as badly as possible according to the demonic relational semantics . Usually this last one is given an ad hoc semantics definition based on an intuitive understanding of the behavior of the program. Denotational semantics has been introduced by Scott and Strachey. To give the denotational semantics, we associate to a program a mathematical object. In our case, this object is a flow diagram which is a graph whose arrows are weighted by the different steps of the program. The operations are "the demonic choice" and "demonic composition." In this note, we show how the notion of the flow diagram can be exploited to give single demonic operational semantics (with only demonic operators) for a wide range of programming constructs.

Nondeterministic semantics of compound diagrams

“…To prove Theorem 6.1, we needed many intermediate results. Some of them have been demonstrated by the generalisation of Mills' theorem [18,46,47,48,49,50,51,52]. In Section 7, we have applied Theorem 6.1 to a compound diagram corresponding to a particular case (sequence) of language of guarded commands.…”

“…program not to terminate normally, then it will not terminate normally (like a demon it looks for the worst case). The nondeterminism is important because, for a computation to be successful all possible executions must lead to a successful result [2,3,6,9,46,47,48,49,50,51,52].…”

Relational Demonic Fuzzy Refinement

“…Actually, once problems in these fields are formalized in terms of relational calculus, these problems can be considered by using formulae of relations; that is, we need only calculus of relations in order to solve the problems. In the context of software development, one important approach is that of developing programs from specifications by stepwise refinement; see, for example, [16][17][18][19][20]. One point of view is that a specification is a relation constraining the input-output (resp., argument-result) behaviour of programs.…”

“…These operations are, respectively, illustrated by Figures 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21, and 22.…”

“…The operator (⊓) is called fuzzy demonic intersection. This definition is equivalent to the definition given in ( [18,19,36,43,[48][49][50]). In other words,…”