Minimal Forms in lambda-Calculus Computations

The notion of a minimal form is defined as an extension of the notion of a normal form in A-:-calculus and its meaning is discussed in a computational environment. The features of the Knuth-Gross reduction strategy are used to prove that to possess a minimal form, for a generic term, is a semidecidable predicate. ?0. Introduction. Historically 2-calculus may be viewed as a computability model like a Turing machine (Church [3] and Turing [10]). The purpose of this work is to consider 2-calculus as a computation model. The stress is thus more on the properties of the reduction relation (>) than on the convertibility relation (=) between terms. The meaning of A>B is: there exists a (nondeterministic) computation starting from term (state) A and leading to B. The nonsymmetry of > reflects the general nonreversibility of computation: for instance 48 is univocally determined by 12-4 but not vice versa. The physical interpretation thereof is that a term during a computation suffers a loss of the quantity of information. What is unquestionable is that, in pure 2-calculus a normal from is minimal with respect to the quasi-ordering represented by the D, relation. Since there are minimal forms which are not in (and do not possess a) normal form, it follows that the concept of a minimal form is a proper extension of the concept of a normal form. The authors believe that a reasonable meaning for minimal forms (and hence for normal forms) could thus be that of a form with a minimal information content. Following this approach the main result of this paper may be stated by saying that, in a computation towards a minimal form, the information content of a term decreases unless a minimal form is constructively reached. If this form is not normal the information process begins to become a reversible one. In ?1 we define minimal forms of terms and consider their relationship with other wellknown concepts. ?2 is devoted to the construction of a semialgorithm detecting a minimal form, if any. ?1. The concept of a minimal form. Later on changes of bound variables will be omitted, thus F _ G will mean: F and G are the same up to a-conversion. We will write F 0 G if F reduces to G by the contraction of a single redex. We will write > for the transitive closure of P and D for the reflexive closure of >. We observe that N is the normal form (nf) of the term F iff (1) VG(F > G G D, N) and (2) there is no H such that N > H. Received June 27, 1978. 165 ? 1980 Association for Symbolic Logic 0022-4812/80/4501-001 5/$02.75 This content downloaded from 157.55.39.198 on Fri, 17 Jun 2016 05:17:54 UTC All use subject to http://about.jstor.org/terms 166 CORRADO BOHM AND SILVIO MICALI We want to generalize the definition of nf in the following way. DEFINITION 1. M is a minimal form (mf) of F iff VG(F > G :G >, M), i.e. G may be reduced to M whenever F reduces to G. The reason for the adjective "minimal" is clear if > is interpreted as quasiordering in the set of the reducts of F. We know that it is semidecidable for a generic term to have an nf. We want to show that the analogue is true for the mf's. As of now let us accept a weaker result. THEOREM 1. Let X4 be the set of terms having mf's. X is not a recursive set. PROOF. We shall make use of Scott's theorem [8]1. X is closed under convertibility: let X = Y and Y have an mf Q. We want to show that Q is an mf for X too. Let X > T; by a corollary of the Church-Rosser theorem, as T = Q there will be an S such that T > S and Q > S. But Q is an mf of Y, thus S > Q and T > S > Q. X is not trivial: later on we will give several examples of terms with mf's, so let us prove that not all terms have an mf: consider Tx = y(x(yy)) *Ay(x(yy)). T and all its reducts have only one redex. So it is meaningful to talk about the nth reduct which is