Algebraic Semantics of Imperative Programs

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The trick with denotational semantics is to give a mathematical interpretation that lets you avoid this universal quantification -- you say that the meaning of a piece of code is some mathematical value, and you compare them by checking to see if they're mathematically equal or not. This is local ie, compositional , and does not involve quantification over all possible clients. You do need to show that the denotational semantics implies contextual equivalence for it to be sound, of course. When it is complete -- when denotational equality is exactly the same as contextual equivalence, we say the semantics is "fully abstract".

But means that you need to ensure that the denotational semantics validates those equivalences. So for this example, if you wanted to give a denotational semantics for this Java-like language, you need to ensure not just that calling new takes a heap and gives you back a new heap with the newly-created object, but that the meaning of the program is invariant same under all permutations of the input heap.

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This can involve quite complex mathematical structures eg, in this case working in a category which ensures everything works modulo a suitable permutation group. There are of course ways of dealing with effects in denotational semantics.

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For example, we can use Eugenio Moggi's idea that computational effects are monads this idea has also been used in the design of Haskell. One of the problems with this is that monads are hard to combine. For a comprehensive treatment, see Matija Pretnar's thesis. I should also mention the possible worlds semantics for local state, developed by Frank Oles and John Reynolds sorry, can't find a better link, this stuff is from , which predates Moggi's monads.

They used categories of presheaves to provide a semantics of an algol-like language which correctly modeled many aspects of local state but not all of them, I think the model allowed snapback, but maybe my memory serves me wrong. Matthias Felleisen presented a compelling solution to the side effects problem in semantics in his series on "Syntactic Theories of Control and State. That line of work resulted in the CESK machine, a simple abstract machine framework capable of concisely modeling functional, object-oriented, imperative and even logic languages.

The CESK framework handles not just side effects, but also "complex" control constructs like exceptions, continuations, laziness and even threads. The CESK machine, and small-step operational semantics more broadly, have been the de facto standard in programming language theory for about two decades.

Algebraic Semantics of Imperative Programs

In short, a CESK machine is a small-step machine with four components to describe every machine state: the control string a generalization of the program counter , the environment, a store also called the heap and the current continuation. It also makes it easy to model pointers and dynamic allocation: just make store addresses first-class values. It doesn't necessarily make it difficult, but it does impose restrictions on the way the semantics of larger expressions can be constructed out of smaller ones.

It can interact very badly with certain other programming constructs, for instance, if one wants to give a Scott-style denotational semantics for a language allowing assignment of higher-order functions to global references. It is not simply side effects like state that cause the trouble.


Simple imperative languages such as Dijkstra's guarded command language have these kinds of side effects, and have nice semantics. Trouble arises with extensions of the lambda-calculus with the kind of operational semantics expected of programming languages even in the absence of side effects: the earliest, Plotkin's PCF, was given denotational models relatively early, but the semantics were not fully abstract, meaning that the denotational semantics was overly general, not exactly corresponding to their operational semantics. PCF finally recieved a fully abstract denotational semantics in the late s with game semantics, which is not at all like Scott's order-theoretic semantics.

Concurrency still has not received a fully adequate denotational treatment. Many question the importance of this kind of semantics. We can always provide some kind of operational semantics, even if that "semantics" is just the program source and the names of some machines that have compiled and run the program: for this reason Strachey condemned operational semantics. But Plotkin's structural operation semantics has shown how operational semantics can be separated from machine models, and Pitt's work has shown how such semantics can support similar reasoning about programs and programming languages to denotational semantics.

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Thus operational semantics is a viable alternative to denotational semantics, and have been applied with success to a substantial number of programming languages such as Standard ML. To some extent, difficulties providing semantics correspond to the difficulty of providing powerful programming languages that behave in the way one would expect.

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