analysis, of well known results concerning CL and weak reduction (e.g. the Church-Rosser theorem and the leftmost reduction theorem) and to generalize these results to combinatory systems as well. Also, some new (as far as we know) results concerning the decidability of certain fragments of CL+extensionality can be obtained in the same style.
In the present paper, we shall give a survey of the results so far discussed (unless otherwise specified, detailed proofs are to be found in [10]), and we shall hint as well at some new (still partial) results concerning the proof-theory of λ-calculus.
2
Combinatory Systems and Analytic Combinatory
Calculi
Given a non-empty, possibly countably infinite setX={F,G,H, . . .}of individual constants (combinators), we denote by TX the set of all terms that are induc- tively generated from variables inV ={v1, v2, v3, . . .}and combinators inXby
means of the binary function symbol ap forapplication. We letx, y, z, . . ., resp. t, s, r, . . .(occasionallyΦ, Ψ, . . .) vary overV, resp.TX. The standard conventions for writing application terms are adopted throughout: ts is an abbreviation of ap(t, s), and missing parentheses are to be restored by associating to the left. Further notational conventions include:
– V(t) := the set of all variables occurring int; – Tn
X := the set of all terms t such that V(t) ⊆ {v1, . . . , vn} (the set of all closed terms, ifn= 0);
– t[x1/s1, . . . xn/sn] := the term resulting fromtby simultaneous substitution
ofsi forxi(1≤i≤n);
– forΦ∈ Tn
X,Φ[s1, . . . , sn] :=Φ[v1/s1, . . . vn/sn].
The symbol ≡ denotes syntactic equality between terms, and ktk denotes the length oft, i.e. the number of occurrences of individual variables and individual constants int.
A combinatory system X over X is defined as a CL-like system built over the set X of primitive combinators. Formally, X is a map associating to each combinator F∈Xa pairhkF, ΦFi, wherekF ≥0 (theindex ofF underX) and ΦF∈ TXkF (thedefinition ofFunder X). Intuitively, the pairX(F) describes the intendedreduction for the combinatorFin the systemX:
Ft1. . . tkF→ΦF[t1, . . . , tkF] .
Accordingly, CLmay be represented as the combinatory system C over L{K,S} such that:
Proof-Theoretical Methods in Combinatory Logic andλ-Calculus 151 A combinatory systemX overXis said to belinear, resp.pure, iff in the defi- nitionΦFof an arbitraryF∈Xeach variable hasat most oneoccurrence, resp. no combinator does occur. It is said to berecursive iff, beingX={F0,F1, . . .}, the
mapn7→ hkF, ΦFiis recursive. Note that in combinatory systems which areboth pure and linear each combinator F satisfies, for everyt1, . . . , tkF: kFt1. . . tkFk>
kΦF[t1, . . . , tkF]k.
We are going to associate to each combinatory systemXa standardsynthetic calculusC[X] and ananalytic calculusG[X]. To this aim let, forF∈X, k=kF andm≥k:
– [Ax−F]X be the equation schema:
Ft1. . . tk =ΦF[t1, . . . , tk] ;
– [Fr]X and [Fl]X be the two inference rules (right andleft F-introducton): s=ΦF[t1, . . . , tk]tk+1. . . tm s=Ft1. . . tm [Fr]X ΦF[t1. . . tk]tk+1. . . tm=s Ft1. . . tm=s [Fl]X .
The terms tk+1, . . . , tm are said to form the context of the introduction rule
(which is empty in casek=m).
Next, let [%],[σ],[τ] and [App] be the inference rules ofreflexivity,symmetry, transitivity andap-congruence of equality (reflexivity being conveniently treated as a 0-premises rule): t=t[%] t=s s=t[σ] t=s s=r t=r [τ] t1=s1 t2=s2 t1t2=s1s2 [App] . Now, the synthetic proof-systemC[X] is defined as the equational calculus deter- mined by the axiom schemas [Ax−F]X (for eachF∈X) and the inference rules [%],[σ],[τ] and [App]; and the analytic proof-systemG[X] as the equational cal- culus determined by the “combinatory” introduction rules [Fr]X and [Fl]X (for eachF∈X), plus the“structural”inference rules [%],restricted to atomic terms, [τ] and [App].
Furthermore, byCext[X] andGext[X] we shall denote the extension ofC[X], resp.G[X], by means of the extensionality rule:
tx=sx
t=s [Ext] (x /∈V(ts)) .
Note that the symmetry ruleis not a primitive rule of analytic calculi, yet the latter are readily seen to be closed under [σ]. As a consequence, it can be easily verified that theC- andG-calculi are equivalent.
Proposition 1. For every combinatory systemX and every equationE, `C(ext)[X] E iff `G(ext)[X] E .
152 P. Minari
As a concluding remark, we call the reader’s attention to the fact that, as far as equational deductive power is concerned, “almost every” calculus C[X] can be embedded in CL. More precisely, given an arbitraryrecursive combinatory system X over X = {F0,F1, . . .} we can prove (by applying the multiple fixed
point theoremforCLin the simplest case in whichXis finite, and Klop’sinfinite fixed point theorem, see e.g. [1], p. 184, in the general case) that there exists a setX0={F0
0,F01. . .}ofclosed T{K,S}-terms such that, for everyn≥0 and every listt1, . . . , tkFn ofT{K,S}-terms:
`CLF0nt1. . . tkFn =Φ
0
n[t1, . . . , tkFn] ,
whereΦ0n is obtained fromΦn by replacing everyFi with the correspondingF0i.
By contrast, it is evident that X-reduction (to be defined in the next section) cannot, in general, be faithfully simulated by means of standard CL w(eak)- reduction.