1. Introduction

Consider a system S of any kind. Suppose that there is a way to make some number of copies of it, possibly with variations. Suppose that these systems are united into a new system S' which has the systems of the S type as its subsystems, and includes also an additional mechanism which somehow examines, controls, modifies and reproduces the S-subsystems. Then we call S' a metasystem with respect to S, and the creation of S' a metasystem transition. As a result of consecutive metasystem transitions a multilevel hierarchy of control arises, which exhibits complicated forms of behavior. In my book The Phenomenon of Science: a Cybernetic Approach to Evolution [39] I have interpreted the major steps in biological and cultural evolution, including the emergence of the thinking human being, as nothing else but metasystem transitions on a large scale. Even though my Ph.D. was in theoretical physics, I was so excited about my new cybernetic ideas that I shifted from physics to computer science and left the atomic center in Obninsk for the Institute for Applied Mathematics in Moscow. An extra-scientific factor came into play in the late 1960s: I became an active member of the human rights movement. My book was written about 1970, but it could not be published in the Soviet Union solely because of the author's name. It took seven years to smuggle it to the West and have it published in the English language. The first step towards MST in computers was to design an appropriate algorithmic language. I called the first version of such a language meta-algorithmic [36], since it was supposed to serve as a metalanguage for defining semantics of algorithmic languages. Then I simplified it into what was named REFAL (for REcursive Functions Algorithmic Language). Refal was conceived as the universal language of metasystem hierarchies, which is, on one hand, simple enough, so that the machine that executes programs in this language (the Refal machine) could become an object of theoretical analysis -- and, on the other hand, is rich enough to serve as a programming language for writing real-life algorithms, unlike such purely theoretical languages as the language of the Turing machine or Markov's normal algorithms (the latter, by the way, was one of the sources of Refal). Even to this day Refal remains, in my (biassed, no doubt) view, the most successful compromise between these two requirements. An efficient interpreter for Refal was first written in 1968 [26]. At that time Refal was very different from other languages, being a purely functional language with built-in pattern-matching. Now that functional languages are common place, I need only to summarize Refal's distinctive features to make it familiar to the reader. The most distinctive feature of Refal is its data domain. Refal uses expressions for symbol manipulation, not lists (skewed binary trees). Refal expressions can be seen as trees with arbitrary (unfixed) arity of nodes, or sequences of such trees. They are formally defined as:

term ::= symbol | variable | (expression) | <function expression>
expression ::= empty | term expression

Variables in Refal have their intrinsic types shown by a prefix e.g., s.1, t.2, e.x. Thus an s-variable s.i, where i is the index (name), stands for an arbitrary symbol, a t-variable t.i - for an arbitrary term, and an e-variable e.i - for any expression. In the case of an e-variable, the prefix may be dropped: x is the same as e.x. We use angular brackets to form function calls: <f x>. A program in Refal is a sequence of mutually recursive sentences (rewrite rules) which are tried in the order they are written. Here is an example of a function f which traverses its argument from left to right and replaces every 'a' by 'b':

<f 'a'> = 'b'<f x>
<f s.1 x> = s.1 <f x>
<f > =

  By the year 1970 I was lucky to have gathered a circle of young people, the Refal group, which met regularly, survived my emigration and is still alive and well. As in any informal group, some people went, new people came. I am deeply thankful to all of them, and especially to those most active and persistent: Sergei Romanenko (the first to have come), Nikolai Kondratiev, Elena Travkina, Andrei Klimov, Arkadi Klimov, Viktor Kistlerov, Igor Shchenkov, Sergei Abramov, Alexei Vedenov, Ruten Gurin, Leonid Provorov, Andrei Nemytykh, Vika Pinchuk. I will always remember Alexander Romanenko who died before his time in 1993.

Together with a smaller group at the Moscow Engineering and Physics Institute (Stanislav Florentsev, Alexander Krasovsky, Vladimir Khoroshevsky) we made Refal compilers for the most popular machines in the Soviet Union, and Refal became pretty well known in that part of the world (a bibliography on the development and use of Refal compiled a few years ago includes about 200 items). It is not my intention here to focus on Refal, but I want to mention two further outgrowths of Refal: The language FLAC for algebraic manipulation developed by V.Kistlerov [22,4], and Refal Plus by Sergei Romanenko and Ruten Gurin [17], which is a kind of logical closure of the ideas on which Refal is based. In my later work I used the extended version of Refal named Refal-5 [44], which is operational under DOS and UNIX.

       The next conceptual station after fixing the language was driving [37,38]. Suppose we have the call <f 'a'x> of the function f above. Obviously, it can be replaced by 'b'<f x>, because this is what the Refal machine will do in one step of function evaluation. Partial evaluation took place, which is a simple case of driving. Now take the call <f x> where a partial evaluator has nothing to do. Driving, however, is still possible, because it is simulation of one step of computation in any circumstances. When we see equivalent transformation of programs as the use of some equations, our partial evaluation is the use of the equation <f 'a'x> ='b'<f x>, and this makes perfect sense. But, given the call <f x>, there is no equation in sight that would improve the program. Driving is a product of cybernetic thinking. We create a metamachine, and the first thing it must be able to do with the Refal machine is to simulate its behavior. Thus, the metamachine drives the Refal machine, forcing it to do something for which it was not originally prepared: make computation over expressions with free variables. Such computation should better be called metacomputation; its result is not the value of the function, but a graph of states and transitions of the Refal machine which describes one step of computation towards that value. In our case driving will produce the graph:

[1]<f x> x:'a'x; [2]'b'<f x>
+ x:s.1 x; [3] s.1 <f x>
+ x:[]; []

where the original call is the root node labeled by [1], and there are three edges, separated by +, which lead to three resulting expressions, according to the three cases (sentences) in the definition of f. By e:p we denote the operation of matching expression e to pattern p; a pattern is a special kind of an expression (referred to as rigid), which guarantees the uniqueness of the operation. An expression E is rigid if (1) only s-variables may enter E more than once, and (2) no subexpression of E (inlcuding E itself) includes two e-variables separated by an expression. [] stands for an empty expression for readability.

     From driving I came to supercompilation (SCP for short). Let me briefly describe this technique of program transformation, leaving aside many details and variations. A supercompiler is an upgrade of a driver. While driving describes one step of the Refal machine, supercompilation may include any number of steps. When a new active (i.e., representing a function call) node C appears in driving, the supercompiler examines its ancestors and with regard to each ancestor C' takes one of the three decisions:

1.     Reduce C to C' with a substitution; this can be done only if  img7.gif (308 bytes)
Generalize C and C', i.e., find such Cg that $C\subseteq C^g$ and $C'\subseteq C^g$; then erase the driving subtree which starts at C', reduce C' to Cg and go on driving Cg.
Do nothing on C' (the nodes C and C' are "far from each other"), and examine the next ancestor. If there are no more ancestors to loop back to, go on driving C.

      Supercompilation ends when the graph becomes self-sufficient i.e., it shows, for every active node, how to make a transition to the next node which corresponds to at least one step of the Refal machine. This graph is a program for computing the function call represented by the initial node.

     Note the difference between the principle of supercompilation and the usual idea of program transformation, where the program is changed step by step by applying some equivalences. In supercompilation we never change the original program. We regard it as a sort of "laws of nature" and construct a model of a computation process governed by these laws. When the model becomes self-sufficient, we simply throw away the unchanged original program. I see supercompilation as a computer implementation of the general principle of human knowledge, which, in my view, is the search for such generalized states in terms of which we can construct a self-sufficient model of some part of the world.

     When we discussed supercompilation in the seminars of the Refal group, I always stressed my belief that metasystem transition is important in itself. If indeed it has been the key at all stages of biological and technical evolution, how can we hope to create really intelligent machines without the use of this principle? We should develop techniques of dealing with repeated metasystem transitions; applications are bound to follow.

     The first confirmation of this belief came when I figured out that supercompilation of an interpreter is compilation, and supercompilation of the supercompiler that does compilation (second MST) yields a compiled (efficient) compiler. Moreover, making the third MST we can obtain a compiled compiler generator. I was very excited about my discovery, and told Andrei Ershov about it. Ershov at that time worked on partial evaluation - the first MST - but he did not know about the second MST, and he also got excited (Ershov describes our meeting in detail in his 1987 keynote speech [8]). Neither he, nor I knew at that time that Yoshihiko Futamura [9] made this discovery a few years before me. Ershov saw a reference to Futamura's paper, but could not get the journal. In his big paper [7] he referred to my result as "Turchin's theorem of double driving".

     That was in 1976. Since 1974 I had been jobless, home-searched and periodically interrogated by the KGB. Meanwile a book on Refal and its implementation was written by several members of the Refal group, including myself [3]. My friends managed to get permission for publishing it as a technical material of the institute where I worked before being fired - on the condition that I do not figure in the list of authors. To meet this requirement they decided not to mention any authors at all. The book was thus published anonymously. But I smuggled into it a few pages about my results on automatic production of compilers.

     I emigrated in 1977, and it took some time to take roots in the new environment: a process which can never be fully successful. For a year and a half I stayed at the Courant Institute of NYU and used this time to write a 250 pages report [40], where I summarized the ideas and results of the past years and also sketched some new ideas to turn to them later. Then I got a position at the City College of the City University of New York.

    I wrote the first supercompiler (SCP-1) in 1981-82 [41] with the help of Bob Nirenberg and my son Dimitri in carrying over the implementation of Refal from Russia to the US and upgrading it.   From the examples in [41] one could see that supercompilation includes, but is much stronger as a transformation technique than partial evaluation (which takes place automatically in driving). The examples in [41] included program specialization, but when I tried self-application (2nd MST), I met a dozen of technical difficulties.

    Partial evaluation is simpler then supercompilation, so automatic generation of a compiler from an interpreter was first achieved by self-application of a partial evaluator (2nd and 3rd MST). This was done by Neil Jones and co-workers at DIKU, Copenhagen [18,27,19,20], and was an important step forward and a great success. Self-applicable partial evaluation became an established field of research. Of the members of the Moscow Refal group, Sergei Romanenko and Andrei Klimov contributed to this field [23,30,31], but I decided to concentrate on further development of techniques of supercompilation, and wrote a stronger and better supercompiler, SCP-2.

    The fall semester of 1985 I spent at DIKU in Copenhagen invited by Neil Jones. This was a very important visit for me. I got a chance to discuss in detail my ideas on MST and SCP with Neil and other people at DIKU. Among other things, these discussions helped me to finalize and have published my paper on supercompilation [42]. Since then I visited DIKU on several occasions and always had useful discussions and enjoyed the friendly atmosphere there.

     In 1988-89 Robert Glück, then at the Vienna Technical University, joined the MST-SCP Project. He spent a year and a half at the City College working with me on its various aspects. We managed to achieve self-application of SCP-2 in some simple cases, but it became clear that a thorough overhaul of the supercompiler is needed. After returning to Europe, Glück carried over the techniques of metacoding and doing MSTs, which was originally developed for the Refal supercompiler, to partial evaluation and list-based languages. In the early work on partial evaluation by N.Jones with co-workers [20] it was stated that in order to achieve good results in self-application, a preliminary binding time analysis (``off-line'') was necessary. However, Glück showed ([12]) that with a correct use of metacoding, partial evaluation is self-applicable without any binding time analysis: ``on line''.

    Little by little, more people got involved in the work on MST+SCP. The Moscow Refal group has never ceased discussing and working on these ideas, but now their work began being published in English. Andrei Klimov found a mate in Robert Glück [14,15,16]; their paper [14] helped arouse interest towards supercompilation in the partial evaluation community by bridging the two methods. Alexander Romanenko started work on function inversion [28,29]. Sergei Abramov did an excellent and potentially very important work [1] on program testing on the basis of driving with a neighborhood (see Sec.3.3), and wrote a monograph on metacomputation other than supercompilation, [2] (it is still in Russian, though).

    Neil Jones [21] gave a thorough theoretical analysis of driving as compared to partial evaluation. More papers on supercompilation have appeared recently from DIKU [13,32,33,34,35]. The role of Glück in that, as one can see from the references, has been of primary importance.

    Morten Sørensen wrote a Master thesis on supercompilation [32]. Its title, Turchin's Supercompiler Revisited, symbolized the emerging interest to my work. Later Sørensen made a suggestion that the Higman-Kruskal theorem on homeomorphic embedding be used as the criterion for generalization in supercompilation [33], which probably will be judged as one of the most important contributions to the SCP techniques during the last few years; I discuss it in more detail in Sec.2.2.

     In 1993 I decided to restrict Refal as the object language of SCP to its flat subset where the right side of a sentence cannot include nested function calls: it is either a passive expression, or a single function call. The purpose, of course, was to simplify the supercompiler for self-application to become possible. I started writing such a supercompiler, SCP-3. In September 1993 Andrei Nemytykh from the Programming Systems Institute (Pereslavl, Russia), came to CCNY for an academic year under a grant from the National Research Council. Working together, we have, at long last, made SCP-3 self-applicable [46,47]. That was in the spring of 1994. Returning to Russia, Nemytykh continued the work on SCP-3 with Vika Pinchuk. I do not speak more on this because SCP-3 with some examples of its performance is presented in a separate paper at this symposium [25].

    In the summer of 1993 Futamura invited me to spend a month in Tokyo to discuss supercompilation in relation to his concept of generalized partial computation [10,11]. There are common aspects, indeed. In both approaches the information about the states of a computing machine goes beyond listing the values of some variables. The main difference is that Futamura relies on some unspecified theorem proving, while my point is to do everything by supercompilation, including theorem proving.

     In July 1995, invited by José Meseguer, I spent a pleasant week at Stanford Research Institute in California explaining and discussing the details of SCP and MST. Meseguer and his graduate student Manuel Clavel are working on reflective logics and languages in the frame of Meseguer's theory of general logics [24,5]. They set a goal of extending the techniques of supercompilation to their systems in order to improve their efficiency, and I know that they have already made some progress along this path.

     My review of events and ideas concerning MST and SCP is, no doubt, incomplete, and so is my bibliography. I ask for forgiveness in advance.

     In Sec.2  I discuss a few important aspects of supercompilation. In Sec.3 I give an outline of a few ideas which have not yet been properly translated into computer programs. My exposition is very informal and sketchy. I try to do it through a few simple examples. The field of MST+SCP has its own formalism, which I, obviously, cannot systematically present here. Yet I hope that the reader unfamiliar with it will still be able to figure out what it is all about, without going into formal details.

    A word on terminology. In 1987 I suggested metacomputation as an umbrella term covering all computations which include at least one metasystem transition. By this definition, partial evaluation is also a variety of metacomputation, and the term could be used for identifying such meetings as this seminar. However, partial evaluation people have been in no hurry to use it. In contrast, it is readily used by people in the field I denote in this paper as MST plus SCP (sounds barbarian, of course). Thus, for the time being, at least,

metacomputation  = MST + SCP

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