Proposed Thesis Title: Higher-order embeddings of program logics

Once we have formal operational semantics for a programming language, it is in theory possible to prove any valid statement about a program meeting its specification. However, in applications, a more abstract mechanism is desirable; an axiomatic style of reasoning. Perhaps the most well known axiomatic semantics is Hoare Logic[5]. Hoare Logic is both elegant and simple though the underlying programming language is, by modern standards, limiting. Recent developments in axiomatic semantics have given more elaborate logics, for example the Logic of Objects (AL) by Abadi and Leino[2]. The underlying language of AL has features such as objects, mutually recursive procedures (methods) and aliasing. As the underlying language has become more elaborate, so has the logic itself. One soon finds using such a logic slow and laborious to use in the absence of machine assistance. Embedding logics in theorem provers demonstrates to be a useful tool for checking examples. There are broadly two differing approaches to embedding program logics in higher-order logic. On one hand, we have the deep-style embedding, where the first-order syntax of the language (including variables, substitution and binding) are first encoded, followed by the operational semantics. Then the axiomatic rules can be derived directly from the operational semantics, using the proof tactics provided by the theorem prover. Thus correctness of the resulting axiomatic rules is guaranteed by the correctness of the theorem prover itself. However, it is necessary to define the notions of variable, substitution and binding: notions that are already present in the theorem prover. On the other hand, we have the traditional higher-order abstract syntax (HOAS) style of embedding. In this case, we take advantage of the existing notions of subsitution and binding of the theorem prover. Traditionally, the assertion logic is given a “deep” embedding: we embed the logic syntax and rules. The effort saved in reusing the theorem prover for the syntax, is offset by the need to define a whole logic again. However, this deep embedding of the assertion logic allows us to translate proofs in the embedding, back directly into proofs of the logic. We now propose a different style of encoding where we use both HOAS and a shallow embedding of the logic. The advantage is of course we inherit variable management and the proof tactics from the underlying theorem prover. However, soundness of the embedding becomes more involved.