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Refinement Calculus, 1998
Refinement Calculus, 1998
Refinement Calculus, 1998
In the previous chapter we showed that contract statements can be interpreted as (monotonic) pred... more In the previous chapter we showed that contract statements can be interpreted as (monotonic) predicate transformers. But predicate transformers are pure mathematical entities, functions from predicates (sets) to predicates, and as such have no direct computational interpretation. The way in which statements are assumed to be executed as programs has only been explained informally. Because both angelic and demonic nondeterminism is involved in statement execution, it is not obvious how statement execution should be defined formally. In this chapter, we therefore proceed to give an operational interpretation of contract statements as games We show that the predicate transformer semantics is an abstraction of the game semantics, in the sense that we can compute the former from the latter. Both correctness of statements and refinement can be explained operationally in terms of winning strategies in games.
Refinement Calculus, 1998
In this chapter we show how to formalize predicates in higher-order logic and how to reason about... more In this chapter we show how to formalize predicates in higher-order logic and how to reason about their properties in a general way. Sets are identified with predicates, so the formalization of predicates also gives us a formalization of set theory in higher-order logic. The inference rules for predicates and sets are also special cases of the inference rules for
Refinement Calculus, 1998
We now apply fixed-point theory to recursively defined statements, interpreting a recursive state... more We now apply fixed-point theory to recursively defined statements, interpreting a recursive statement as the least fixed point of a monotonic function on predicate transformers. We show how to construct recursive statements as limits of approximation chains and develop inference rules for introducing recursion as a refinement of a simpler nonrecursive statement. Finally, we show how to define recursive procedures
Refinement Calculus, 1998
In this chapter we show that the basic predicate transformers that we have introduced to model co... more In this chapter we show that the basic predicate transformers that we have introduced to model contracts (asserts, guards, functional updates, demonic and angelic updates) are all monotonic. In addition, composition, meet, and join preserve monotonicity. Conversely, any monotonic predicate transformer can be described in terms of these constructs, in a sense to be made more precise below. In fact,
Refinement Calculus, 1998
This chapter introduces the central mathematical structures that are needed to formalize the refi... more This chapter introduces the central mathematical structures that are needed to formalize the refinement calculus: partially ordered sets (posets), lattices, and categories. We identify the basic properties of posets and lattices, and use them for a classification of lattices. We also show how to construct new lattices out of old ones as Cartesian products and function spaces. We study structure-preserving mappings (homomorphisms) on lattices. Finally, we show how to form a certain kind of category out of these lattices. The simple notions identified in this chapter underlie the formal reasoning about properties of programs, specifications, and contracts in general.
Formal Aspects of Computing, 1990
A theory of commands with weakest precondition semantics is formalised using the HOL proof assist... more A theory of commands with weakest precondition semantics is formalised using the HOL proof assistant system. The concept of refinement between commands is formalised, a number of refinement rules are proved and it is shown how the formalisation can be used for proving refinements of actual program texts correct.
Refinement Calculus, 1998
Refinement Calculus, 1998
Refinement Calculus, 1998
In the previous chapter we showed that contract statements can be interpreted as (monotonic) pred... more In the previous chapter we showed that contract statements can be interpreted as (monotonic) predicate transformers. But predicate transformers are pure mathematical entities, functions from predicates (sets) to predicates, and as such have no direct computational interpretation. The way in which statements are assumed to be executed as programs has only been explained informally. Because both angelic and demonic nondeterminism is involved in statement execution, it is not obvious how statement execution should be defined formally. In this chapter, we therefore proceed to give an operational interpretation of contract statements as games We show that the predicate transformer semantics is an abstraction of the game semantics, in the sense that we can compute the former from the latter. Both correctness of statements and refinement can be explained operationally in terms of winning strategies in games.
Refinement Calculus, 1998
In this chapter we show how to formalize predicates in higher-order logic and how to reason about... more In this chapter we show how to formalize predicates in higher-order logic and how to reason about their properties in a general way. Sets are identified with predicates, so the formalization of predicates also gives us a formalization of set theory in higher-order logic. The inference rules for predicates and sets are also special cases of the inference rules for
Refinement Calculus, 1998
We now apply fixed-point theory to recursively defined statements, interpreting a recursive state... more We now apply fixed-point theory to recursively defined statements, interpreting a recursive statement as the least fixed point of a monotonic function on predicate transformers. We show how to construct recursive statements as limits of approximation chains and develop inference rules for introducing recursion as a refinement of a simpler nonrecursive statement. Finally, we show how to define recursive procedures
Refinement Calculus, 1998
In this chapter we show that the basic predicate transformers that we have introduced to model co... more In this chapter we show that the basic predicate transformers that we have introduced to model contracts (asserts, guards, functional updates, demonic and angelic updates) are all monotonic. In addition, composition, meet, and join preserve monotonicity. Conversely, any monotonic predicate transformer can be described in terms of these constructs, in a sense to be made more precise below. In fact,
Refinement Calculus, 1998
This chapter introduces the central mathematical structures that are needed to formalize the refi... more This chapter introduces the central mathematical structures that are needed to formalize the refinement calculus: partially ordered sets (posets), lattices, and categories. We identify the basic properties of posets and lattices, and use them for a classification of lattices. We also show how to construct new lattices out of old ones as Cartesian products and function spaces. We study structure-preserving mappings (homomorphisms) on lattices. Finally, we show how to form a certain kind of category out of these lattices. The simple notions identified in this chapter underlie the formal reasoning about properties of programs, specifications, and contracts in general.
Formal Aspects of Computing, 1990
A theory of commands with weakest precondition semantics is formalised using the HOL proof assist... more A theory of commands with weakest precondition semantics is formalised using the HOL proof assistant system. The concept of refinement between commands is formalised, a number of refinement rules are proved and it is shown how the formalisation can be used for proving refinements of actual program texts correct.