History and Future of Implicit and Inductionless Induction: Beware the Old Jade and the Zombie!

In this survey on implicit induction I recollect some memories on the history of implicit induction as it is relevant for future research on computer-assisted theorem proving, esp. memories that significantly differ from the presentation in a recent handbook article on “inductionless induction”. Moreover, the important references excluded there are provided here. In order to clear the fog a little, there is a short introduction to inductive theorem proving and a discussion of connotations of implicit induction like “descente infinie”, “inductionless induction”, “proof by consistency”, implicit induction orderings (term orderings), and refutational completeness. 1 What Is Inductive Theorem Proving (ITP)? Inductive reasoning can be seen as extending deductive reasoning in that infinite deductive proofs may be represented in a finite cyclic form, as suggested in the following example, where Γ (x0, y) is a proposition over the natural numbers, where ‘s’ denotes the successor function (x → x+1), and where the formulas below each line (sub-goals) imply the formula (goal) above: An infinite deductive proof of Γ (x0, y) Γ (x0, y) Γ (0, y) Γ (s(x1), y) .. Γ (s(0), y) Γ (s(s(x2)), y) .. Γ (s(s(0)), y) . . . .. . . . . . . (using xi =0 ∨ ∃xi+1. xi = s(xi+1) ) should be captured in something like D. Hutter, W. Stephan (Eds.): Mechanizing Mathematical Reasoning, LNAI 2605, pp. 192–203, 2005. c © Springer-Verlag Berlin Heidelberg 2005 History and Future of Implicit and Inductionless Induction 193 Γ (x0, y) Γ (0, y) Γ (s(x1), y) .. ∆ ⇒ Γ (s(x1), y) Π ⇒ Γ (s(x1), y) .. .. ∆′ ∨ Γ (x1, s) Π ′ ∨ Γ (x1, t) Π ′′ (back to top) (back to top) .. using ∆ ∨ Π. This kind of cyclic argument – namely inferring Γ (x1, s) and Γ (x1, t) from Γ (x0, y) – is sound if for each (ground) instantiation of the theorem (here: Γ (s(0), s(0))) the deductive proof terminates. This can be guaranteed by requiring that each cycle in the proof (graph) terminate, i.e. its preconditions (usually called induction hypotheses – here: Γ (x1, s) and Γ (x1, t)) be smaller than the “induction” conclusion (here: Γ (x0, y) or Γ (s(x1), y)) w.r.t. some wellfounded ordering, called induction ordering (here e.g. the usual ordering on the natural numbers applied to the first argument of Γ ). Thus, while the property of being an inductive theorem depends only on the specification (i.e. a language and a set of axioms) and the choice of a specific notion of inductive validity, cf. [74], an inductive proof of an inductive theorem also depends on an additional parameter, namely some induction ordering which must be chosen appropriately during the proof. 2 Explicit Versus Implicit Induction Although there is no generally accepted characterization of the two paradigms of explicit and implicit induction in the research community, in [67], which is a comprehensive survey on explicit induction, the following is said: Research on automated induction these days is based on two competing paradigms: Implicit induction (also termed inductive completion, inductionless induction, or, less confusingly, proof by consistency) evolved from the Knuth–Bendix Completion Procedure . . . . . . . The other research paradigm . . . is called explicit induction and resembles the more familiar idea of induction theorem proving using induction axioms. In accordance with this view, one reason to call the latter paradigm “explicit” is that in the underlying inference systems every cyclic argument must be made explicit in a single inference step applying a so-called induction rule. Besides generating induction base formulas, this step joins induction hypotheses and conclusions in induction step formulas and explicitly guarantees the termination of their cycles by a sub-proof or -mechanism for the wellfoundedness of the induction ordering resulting from the step formulas. In the explicit induction proof corresponding to the example induction proof of Section 1 the induction base formula is Γ (0, y) and the induction step formulas are ∆ ⇒ ( Γ (x1, s) ⇒ Γ (s(x1), y) ) and Π ⇒ ( Γ (x1, t) ⇒ Γ (s(x1), y) ). The explicit induction proof then has the following form: 194 Claus-Peter Wirth Γ (x0, y) Γ (0, y) ∆ ⇒ ( Γ (x1, s) ⇒ Γ (s(x1), y) ) Π ⇒ ( Γ (x1, t) ⇒ Γ (s(x1), y) ) .. .. .. Note that the first inference in this proof is an application of an induction axiom in the sense of [67]. As the example induction proof of Section 1 illustrates, the cyclic arguments and their termination in implicit induction proofs need not be confined to single inference steps, as is in explicit induction. Therefore, the induction axioms corresponding to the cyclic arguments in a finite implicit induction proof can only be determined by analyzing the whole proof, whereas in the case of explicit induction each applied induction axiom is given by a single application of the induction rule. The phrase “more familiar idea” attributed to explicit induction in the above quotation requires some remarks. Implicit induction in the style of descente infinie was already known to the ancient Greeks and – as will be explained below – is the standard method of mathematical induction since Pierre Fermat (1607?– 1665) rediscovered the method and named it descente infinie (ou indéfinie). Nevertheless more familiar is the idea of explicit induction to computer scientists who – inspired by J. Alan Robinson’s resolution method of the year 1963 – wanted to solve all problems of logical inference via reduction to machineoriented inference systems. Instead of implementing mathematical ITP, they decided to reduce it via the induction rule (cf. above) in the following fashion: Apply the induction rule backwards; then do purely deductive reasoning; if this fails, repeat the process! The so called “waterfall” of [15] refines this process into a fascinating heuristic and the ITP system Nqthm [15, 16] shows that in this case the reduction approach was quite successful – so successful actually that it is hard to understand why implicit induction was able to gain ground again, as we will see in the next section. 3 1988 – Start of Practical Interest in Implicit Induction Cf. e.g. [62, 19] for historical surveys on implicit induction, which, however, have to be taken with a grain of salt. After several papers on implicit induction in purely equational theories already in the year 1980, [32, 36, 47, 51], there was a sequence of papers on technical improvements, [37, 23, 30, 44], which was topped by Leo Bachmair’s paper [6] in the year 1988. Up to 1987 the papers on implicit induction had a theoretical nature, but some of them were very inspiring, [38] being my favorite. Bachmair’s paper [6] is the first one that gave good reason for a hope to develop the method into practical usefulness. And – in 1990 – there was real hope, cf. [34, p. 1]: This approach has some very attractive and promising characteristics compared to classical approaches for explicit inductive theorem proving based on induction schemas, which will be pointed out later on. History and Future of Implicit and Inductionless Induction 195 As a consequence, in the end of the 1980s and the first half of the 1990s several researchers tried to clearly understand what “implicit induction” means from a practical point of view and whether it would be useful for practical ITP. 4 1996 – End of Most Interest in Implicit Induction During the Induction Workshop on the 13th Int. Conf. on Automated Deduction, New Brunswick (NJ), 1996, there was an agreement on better not to use the term “implicit induction” in the future, for the following two reasons, which will be discussed in the next two subsections. 1. Different researchers understand this term differently. 2. The notion has lost its potential relevance for the practice of ITP. 4.1 What Is Implicit Induction? While it is generally accepted that [6] is implicit and [15] is explicit, we report the following different views on what it is that makes induction implicit: Descente Infinie. In explicit induction there is something like an “induction rule” (cf. Section 2) whose addition turns a deductive inference system into an inductive inference system without further changes on the deductive part. Concerning the concept of induction hypothesis, the explicit induction “hides” several (applications of) induction hypotheses in a single inference step. To the contrary, in descente infinie, the inference system “knows” what an induction hypothesis is, i.e. it includes inference rules that provide or apply induction hypotheses, given that certain ordering conditions resulting from these applications can be met by an induction ordering. Note that descente infinie is important for human-oriented ITP because this is the style in which working mathematicians do induction since Pierre Fermat (1607?–1665) rediscovered and named the method which was already known to the ancient Greeks. The working mathematician applies it in the following fashion. He starts with the conjecture and simplifies it by case analysis. When he realizes that the current goal becomes similar to an instance of the conjecture, he applies the instantiated conjecture just like a lemma, but keeps in mind that he has actually applied an induction hypothesis. Finally, he searches for some wellfounded ordering in which all the instances of the conjecture he has applied as an induction hypothesis are smaller than the original conjecture itself. This view on the notion of “implicit induction” (i.e. descente infinie) was the one of the majority on the Induction Workshop in 1996. The name descente infinie for this aspect of implicit induction was coined later in [71] after suggestions and complaints on the occurrence of “implicit induction” in the title of [70]. Researchers introduced to descente infinie by [57] (entitled “Lazy Generation of Induction Hypotheses”) sometimes speak of “lazy induction” instead of descente infinie. 196 Claus-Peter Wirth Inductionless Induction/Proof byConsistency. “Inductionless Induction” means that no induction can be observed explicitly. E.g. some Knuth– Bendix [35] or superposition calculus [25] completion procedure produces a huge number of irrelevant inferences under which the ones relevant for induction can hardly be made explicit in an automatic way. “Inductionless induction” has shown to be practically useless, mainly due to too many superfluous inferences, typically infinite runs, and too restrictive admissibility conditions. The approach in [25] is an interesting theoretical possibility but definitely useless for practical purposes. Roughly speaking, the conceptual flaw in “inductionless induction” seems to be that, instead of finding a sufficient set of reasonable inferences, the research follows the paradigm of ruling out as many irrelevant inferences as possible. A proof attempt is successful when the prover has drawn all necessary inferences and stops without having detected an inconsistency (empty clause). Christoph Walther (cf. [67]) prefers to use the name “proof by consistency” instead of the contradictio in eo ipso “inductionless induction”, which, however, is used again in the year 2001 in the title of [19]. Indeed, “proof by consistency” seems to be a better name for this aspect of implicit induction than “inductionless induction” because the former highlights on the fact that – roughly speaking – the derivation of the empty clause (or inconsistency) means disproof, not proof. E.g., in the case of the toy example proofs in [20] the number of irrelevant inferences is cut down to zero (unless they are hidden in the omitted parts of the proofs) and it is quite obvious where the induction takes place, so that “proof by consistency” is still appropriate, but “inductionless induction” is not. While the opinion that the “inductionless induction” aspect (i.e. no induction explicitly observable) is the crucial one for “implicit induction” is now shared by several researchers who had worked on implicit induction in the past, at the Induction Workshop in 1996, only a minority held this opinion. Note that this view on the notion of implicit induction is held by Martin Protzen and places his important work [57] (which is implicit induction in the sense of descente infinie) on the side of explicit induction, where the practically useful ITP systems use to dwell, such as the powerful up-to-date systems Inka [3] and Acl2 [41]. The name “proof by consistency” was coined in [39], which is the forerunner of the earlier published improved version [38]. The name “inductionless induction” was coined already before [48] (to which it is erroneously attributed in [19, 62]), namely in [47] in the year 1980, when also the title of [32] included a similar phrase. Implicit Induction Ordering. This means that there is no explicit induction ordering in the signature and the model. Instead of such semantical orderings, the induction is performed on syntactical term orderings that are not part of the logic language. The semantical orderings (cf. Definition 13.7 of [69]) cannot depend on the syntactical term structure of a weight w but only on the value of w under the evaluation function. In [69] we have rigorously investigated the price one has to pay for the possibility to have History and Future of Implicit and Inductionless Induction 197 induction orderings also depending on the syntax of weights. For powerful concrete inference systems this price turned out to be surprisingly high. Besides this, after improving the ordering information in descente infinie by our introduction of explicit weights (cf. [72]) the former necessity of sophisticated induction orderings that exploit the term structure (cf. e.g. [6, 64]) does not seem to exist anymore. This view on the notion of “implicit induction” (i.e. no explicit induction ordering in the language) is held by a minority, e.g. by Peter Padawitz. Note that his view places his ITP system Expander, [54, 55], on the explicit side, where the practically useful ITP systems use to dwell. Note that Expander is also explicit in the sense that it does not perform “inductionless induction”. Expander is, however, implicit in the sense that it realizes descente infinie, and – to my knowledge – it is the only integration of descente infinie into a framework of refutational resolution (or – more precisely – inverse method, [49]) where – roughly speaking – the empty clause (or inconsistency) means proof, not disproof. Moreover, this view on the notion of “implicit induction” (i.e. implicit induction ordering) is also the view found in [19], which, however, concentrates on “inductionless induction”. It does not seem to be completely superfluous to note that “refutational completeness” in itself cannot be crucial for implicitness of induction, simply because it is not a very important property: – For practical purposes, it is not important that the invalidity of a theorem would be detected sometime (= refutational completeness), but that it is detected efficiently, cf. [56]. – In theoretical terms, refutational completeness used to be trivial for the very restricted logics in the scope of implicit induction. Nevertheless, since Leo Bachmair’s important paper [6] various authors in the field of implicit induction emphasized refutational completeness as if it were a major asset. So let us have a final look at it: In general, for all significant notions of inductive validity, the set of inductively valid theorems is not enumerable, cf. [31, 52, 50]. Therefore, refutational completeness is an optimal theoretical quality of inference systems for ITP. In practice, however, refutational completeness by itself does not help in refuting invalid conjectures or in finding finite proofs for inductively valid formulas. Only theoreticians completely detached from reality can consider nonterminating proof attempts in refutationally complete inference systems to be successful proofs. 4.2 End of the Schism In 1996, the theoretical aspects of “inductionless induction” were clearly understood by the experts. Moreover, the severe limitations of “inductionless induction” were admitted by all researchers in the field, at least by those who attended 198 Claus-Peter Wirth the Induction Workshop. There was the general opinion that “inductionless induction” was dead . The practitioners in automated theorem proving were also not too interested in the remaining aspects of implicit induction, i.e. “descente infinie” and “implicit induction orderings”. My main interest, however, was and still is to convince the community of the practical importance of “descente infinie”, cf. [71]. All participants agreed that the general challenge would be the development of practically more useful ITP systems where the separation of implicit vs. explicit would not play a splitting role because (contrary to “inductionless induction”) “descente infinie” goes together well with the standard ideas of explicit induction. The general opinion was – roughly speaking – the following. To succeed in proving an inductive theorem in finite time, implicit inductive theorem provers have to solve the same problems as explicit inductive theorem provers, namely to find a finite cyclic representation for an infinite deductive proof as well as an induction ordering guaranteeing the termination of its cycles. Thus, the hatchet between the tribe of explicit ITP and the (rather small) tribe of implicit ITP was buried at the Induction Workshop in 1996, at least between the attendees.