Interpretation Bondorf's definition can be simplified considerably. To see why, consider the second component of CMap(E) \Theta CEnv(E). This component is updated only in Closure Analysis in Constraint Form \Delta 9 b(E 1 @ i E 2 )¯ae and read only in b(x l )¯ae. The key observation is that both these operations can be done on the first component instead. Thus, we can omit the use of CEnv(E). By rewriting Bondorf's definition according to this observation, we arrive at the following definition. As with Bondorf's definition, we assume that all labels are distinct. Definition 2.3.1. We define m : (E : ) ! CMap(E) ! CMap(E) m(x l )¯ = ¯ m( l x:E)¯ = (m(E)¯) t h[[ l ]] 7! flgi m(E 1 @ i E 2 )¯ = (m(E 1 )¯) t (m(E 2 )¯) t F l2¯(var(E1 )) (h[[ l ]] 7! ¯(var(E 2 ))i t h[[@ i ]] 7! ¯(var(body(l)))i) . We can now do closure analysis of E by computing fix(m(E)). A key question is: is the simpler abstract interpretation equivalent to Bondorf's? We might attempt to prove this u...

Selective eta-expansion is a powerful "binding-time improvement", i.e., a source-program modification that makes a partial evaluator yield better results. But like most binding-time improvements, the exact problem it solves and the reason why have not been formalized and are only understood by few. In this paper, we describe the problem and the effect of eta-redexes in terms of monovariant binding-time propagation: eta-redexes preserve the static data ow of a source program by interfacing static higher-order values in dynamic contexts and dynamic higher-order values in static contexts. They contribute to two distinct binding-time improvements. We present two extensions of Gomard's monovariant binding-time analysis for the pure-calculus. Our extensions annotate and eta-expand-terms. The rst one eta-expands static higher-order values in dynamic contexts. The second also eta-expands dynamic higher-order values in static contexts. As a significant application, we show that our first binding-time analysis suffices to reformulate the traditional formulation of a CPS transformation into a modern onepass CPS transformer. This binding-time improvement is known, but it is still left unexplained in contemporary literature, e.g., about "cps-based" partial evaluation. We also outline the counterpart of eta-expansion for partially static data structures.

Our results formalize and confirm a folklore theorem about traditional bindingtime analysis, namely that CPS has a positive effect on binding times. What may be more surprising is that the benefit does not arise from a standard refinement of program analysis, as, for instance, duplicating continuations.

Partial-evaluation folklore has it that massaging one's source programs can make them specialize better. In Jones, Gomard, and Sestoft's recent textbook, a whole chapter is dedicated to listing such "binding-time improvements": nonstandard use of continuationpassing style, eta-expansion, and a popular transformation called "The Trick". We provide a unified view of these binding-time improvements, from a typing perspective. Just as a

We show that using deductive systems to specify an offline partial evaluator allows its correctness to be mechanically verified. For a -mix-style partial evaluator, we specify binding-time constraints using a natural-deduction logic, and the associated program specializer using natural (aka "deductive") semantics. These deductive systems can be directly encoded in the Elf programming language --- a logic programming language based on the LF logical framework. The specifications are then executable as logic programs. This provides a prototype implementation of the partial evaluator. Moreover, since deductive system proofs are accessible as objects in Elf, many aspects of the partial evaluation correctness proofs (e.g., the correctness of binding-time analysis) can be coded in Elf and mechanically verified. This work illustrates the utility of declarative programming and of using deductive systems for defining program specialization systems: by exploiting the logical character of definit...

Binding-time analysis is important in partial evaluators. Its task is to determine which parts of a program can be evaluated if some of the expected input is known. Two approaches to do this are abstract interpretation and type inference. We compare two specific such analyses to see which one determines most program parts to be eliminable. The first is a an abstract interpretation approach based on closure analysis and the second is the type inference approach of Gomard and Jones. Both apply to the pure -calculus. We prove that the abstract interpretation approach is more powerful than that of Gomard and Jones: the former determines the same and possibly more program parts to be eliminable as the latter.

A binding-time analysis is the first pass of an offline partial evaluator. It determines which parts of a program may be executed at specialization time. Region-based binding-time analysis applies to higher-order programming languages with firstclass references. The consideration of effects in the determination of binding time properties makes it possible to have a partial evaluator perform assignments at specialization time. We present such a region-based binding-time analysis and prove its correctness with respect to a continuation-style semantics for an annotated call-by-value lambda calculus with ML-style references. We provide a relative correctness proof that relies on the correctness of region inference and on the correctness of a binding-time analysis for an applied lambda calculus. The main tool in the proof is a translation from terms with explicit region annotations to an extended continuation-passing store-passing style. The analysis is monovariant/monomorphic, however, ess...

We demonstrate the usefulness of higher-order rewriting techniques for specializing programs, i.e., for partial evaluation. More precisely, we demonstrate how casting program specializers as combinatory reduction systems (CRSs) makes it possible to formalize the corresponding program transformations as meta-reductions, i.e., reductions in the internal "substitution calculus." For partial-evaluation problems, this means that instead of having to prove on a case-by-case basis that one's "two-level functions" operate properly, one can concisely formalize them as a combinatory reduction system and obtain as a corollary that static reduction does not go wrong and yields a well-formed residual program.

We formalize a partial evaluator for Eugenio Moggi's computational metalanguage. This formalization gives an evaluation-order independent view of binding-time analysis and program specialization, including a proper treatment of call unfolding, and enables us to express the essence of "control-based binding-time improvements" for let expressions. Specifically,

iii Acknowledgment My graduate study and life, of which this dissertation is one of the outcomes, benefitted greatly from the time, energy, and enthusiasm of many people.