In and Out of SSA: A Denotational Specification

We present non-standard denotational specifications of the SSAform and of its conversion processes from and to imperative pro-gramming languages. Thus, we provide a strong mathematicalfoundation for this intermediate code representation language usedin modern compilers such as GCC or Intel CC.More specifically, we provide (1) a new functional approachto SSA, the Static Single Assignment form, together with its de-notational semantics, (2) a collecting denotational semantics for asimple imperative language Imp, (3) a non-standard denotationalsemantics specifying the conversion of Imp to SSA and (4) a non-standard denotational semantics for the reverse SSA to Imp conver-sion process. These translations are proven correct, ensuring thatthe structure of the memory states manipulated by imperative con-structs is preserved in compilers’ middle ends that use the SSAform as control-flow data representation. Interestingly, as unex-pected by-products of our conversion procedures, we offer (1) anew proof of the reducibility of the RAM computing model to thedomain of Kleene’s partial recursive functions, to which SSA isstrongly related, and, on a more practical note, (2) a new algorithmto perform program slicing in imperative programming languages.All these specifications have been prototyped using GNU CommonLisp.These fundamental results prove that the widely used SSA tech-nology is sound. Our formal denotational framework further sug-gests that the SSA form could become a target of choice for otheroptimization analysis techniques such as abstract interpretationor partial evaluation. Indeed, since the SSA form is language-independent, the resulting optimizations would be automaticallyenabled for any source language supported by compilers such asGCC.