Transformational synthesis

Since deductive inference is generally a familiar notion, we do not survey it here. We rather directly propose a taxonomy of the different ways of applying deductive inference to program synthesis from axiomatic specifications. Thus, Sections 2.2.1 to 2.2.3 respectively contain general introductions to transformational synthesis, proofs-as-programs synthesis, and schema-guided synthesis. 

In transformational synthesis, a sequence of meaning-preserving transformation rules is applied directly to a specification, until a program is obtained. This kind of stepwise forward reasoning is akin to constructing a derivation tree, and is feasible with axiomatic specifications that link their parameters in a relational way  

Applicability conditions are often attached to the transformation rules. Transformational synthesis is obviously an outgrowth of program transformation (optimization) research. And, as hinted before, there is only a fine line—if any— between program synthesis and program transformation. 

There are atomic transformation rules, such as unfolding (which mimics the execution mechanism of the target ImguagQ), folding (which performs the reverse transformation of unfolding), universal instantiation, abstraction, predicate definition, and various (possibly conditional) rewrite rules for the target language and the lemmas of the application domain. 

The objective of applying transformations is to filter out a new version of the specification where recursion (or a loop) may be introduced by a folding step. This usually involves a sequence of unfolding steps, then some rewriting, and finally a folding step. 

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To execute this proposal (route A) by chemical transformation, synthesis of a hypothetical intermediate 159 and its oxidative rearrangement to the oxindole derivative have been examined. The C-21 aldehyde carbon in ajmaline (14) was removed by C-20,21 bond cleavage in the glycol derivative 162, which was prepared from 14 by a six-step operation (Scheme 24). 

The formation of mixed anhydrides 2 is critical for overall yield of the conversion of H-phosphonate monoesters 1 into H-phosphonothioate 4. The transformation of an H-phos-phonate into an H-phosphonothioate function in the ribo series proved to be a stereoselective process. In contradistinction to (his transformation, synthesis of the ribonucleoside 3 -H-phosphonothioate via phosphinate intermediates [202] hardly showed any stereoselectivity. 

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There is an obvious interest in proofs-by-induction, because these allow the synthesis of recursive programs. Note that there often is a similar proof content in transformational synthesis and proofs-as-programs synthesis, and it seems that the same proof construction techniques should be applicable to both. This may suggest that these approaches are probably two facets of the same process. For instance, the work of [Neugebauer 93] shows how the specification forms of these approaches may be reconciled. 

Compared to transformational synthesis, there is no problem here about when to stop synthesis halts when the proof is completed. Transformational synthesis seems more appropriate for synthesis from specifications that are almost programs (in which case synthesis is more like an optimizing transformation), whereas proofs-as-programs synthesis seems more appropriate for synthesis from highly descriptive specifications. Note that program transformation may actually be performed by the transformation of synthesis proofs. 

The 15-year SAFE project [Balzer 85] at the Information Sciences Institute of the University of Southern California aimed at transformational synthesis from specifications written in the GIST language. Some of the used transformation rules are explained by [London and Feather 82]. Ways of mechanizing the application of transformation rules are embodied by the GLITTER sub-system of [Fickas 85]. 

In order to give a rough feel for transformational synthesis of logic programs, we first present the pioneering mechanism in some detail. Then, we cite the achievements of some of the more recent mechanisms. 

Synthesis by symbolic execution obviously features the usual advantages and drawbacks of transformational synthesis. 

The theoretical foundations to transformational synthesis are being laid out by [Lau and Omaghi 93, 94ab] [Lau et al. 94]. (Note that their deductive synthesis corresponds to what we here call transformational synthesis.) This work introduces a realistic specification elaboration scenario, based on specification frameworks, where the synthesis process may provide feedback to the specification elaboration process. 

In Chapter 2, we survey the use of deductive inference in automatic programming. Axiomatic specifications are expected to be complete and non-ambiguous, but are usually also quite lengthy and artificial. Deductive synthesis from axiomatic specifications can be classified into transformational synthesis, proofs-as-programs synthesis (or constructive synthesis), and schema-guided synthesis. We survey the achievements of deductive synthesis of LISP functions and Prolog predicates. 

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