On repetitive right application of B-terms

TL;DR

This paper explores the cyclical behavior of B-terms under repetitive right application, providing conditions, examples, and an efficient algorithm to detect such properties, advancing understanding of their structural characteristics.

Contribution

It offers a sound and complete axiomatization for analyzing B-term properties and introduces a canonical form and algorithm for cycle detection.

Findings

Some B-terms have the property of circulation under repetitive right application.

There are infinitely many B-terms that do not have this property.

A canonical representation and efficient algorithm for cycle detection are proposed.

Abstract

B-terms are built from the B combinator alone defined by B f g x = f (g x), which is well-known as a function composition operator. This paper investigates an interesting property of B-terms, that is, whether repetitive right applications of a B-term circulates or not. We discuss conditions for B-terms to and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of B-terms which have the property and show that there are infinitely many B-terms which does not have the property. Also, we introduce a canonical representation of B-terms that is useful to detect cycles, or equivalently, to prove the property, with an efficient algorithm.

Equations

$$\rho (D)$$

$$\rho (F)$$

$$\rho (R)$$

$$\rho (T)$$

$$\rho (V)$$

$$S_{\left(2n+1\right)}$$

$$O_{\left(n+1\right)}$$

$$\rho (B^{0}B)$$

$$\rho (B^{1}B)$$

$$\rho (B^{2}B)$$

$$\rho (B^{3}B)$$

$$Bxyz$$

$$B(Bxy)$$

$$BB(Bx)$$

$$B(B{e}_1{e}_2)$$

$$=\lambda x.\lambda y.e_1(e_2(xy))$$

$$=\lambda x.\lambda y.e_1(B{e}_{2}xy)$$

$$=\lambda x.B{e}_1(B{e}_{2}x)$$

$$=B(B{e}_1)(B{e}_2)$$

$$=\lambda x.B(B{e}_1)(Bx)$$

$$=B(B(B{e}_1))B$$

$$B(e_1\circ e_2)=(B{e}_1)\circ (B{e}_2)$$

$$B\circ (B{e}_1)=(B(B{e}_1))\circ B\text{.}$$

$$B{e}_1(B{e}_2{e}_3)=B(B{e}_1{e}_2)e_3$$

$$B{e}_1(B{e}_2{e}_3)=B(B{e}_1)(B{e}_2)e_3=B(B{e}_1{e}_2)e_3\text{.}$$

$$e$$

$$e_B$$

$$(B^{n_1}B)\circ (B^{n_2}B)\circ \cdots \circ (B^{n_k}B)$$

$$(B^{m}B)\circ (B^{n}B)$$

$$=B^m(B\circ (B(B^{n-m-1}B)))$$

$$=B^m((B(B(B^{n-m-1}B)))\circ B)$$

$$=(B^{n+1}B)\circ (B^{m}B)$$

$$(B^{n_1}B)\circ \cdots \circ (B^{n_{k-2}}B)\circ (B^{n_{k-1}}B)\circ (B^{n}B)$$

$$=(B^{n_1}B)\circ \cdots \circ (B^{n_{k-2}}B)\circ (B^{n+1}B)\circ (B^{n_{k-1}}B)$$

$$e$$

$$B(B(B\circ (BB)))$$

$$=(B(BB))\circ (B(B(BB)))$$

$$=(B^{2}B)\circ (B^{3}B)\text{.}$$

$${\mathcal{L}}_i(\star )$$

$$\lambda x_1.\lambda x_2.\lambda x_3.\dots .\lambda x_{n+1}.\lambda x_{n+2}.\lambda x_{n+3}.x_1{x}_2{x}_3\dots x_{n+1}(x_{n+2}{x}_{n+3})$$

Full text

1 Introduction

The ‘bluebird’ combinator $B=\lambda f.\lambda g.\lambda x.f~{}(g~{}x)$ is well-known [8] as a bracketing combinator or composition operator, which has a principal type $(\alpha\to\beta)\to(\gamma\to\alpha)\to\gamma\to\beta$. A function $B~{}f~{}g$ (also written as $f\circ g$) synthesized from two functions $f$ and $g$ takes a single argument to apply $g$ and returns the result of an application $f$ to the output of $g$.

In the general case where $g$ takes $n$ arguments to pass the output to $f$, the synthesized function is defined by $\lambda x_{1}.\cdots\lambda x_{n}.f~{}(g~{}x_{1}~{}\dots~{}x_{n})$. Interestingly, the function can be expressed by $B^{n}~{}f~{}g$ where $e^{n}$ is an $n$-fold composition of function $e$ such that $e^{0}=\lambda x.x$ and $e^{n+1}=B~{}e^{n}~{}e$ for $n\geq 0$. We call $n$-argument composition for the generalized composition represented by $B^{n}$.

Now we consider the $2$-argument composition expressed by $B^{2}=\lambda f.\lambda g.\lambda x.\lambda y.~{}f~{}(g~{}x~{}y)$. From the definition, we have $B^{2}=B\circ B=B~{}B~{}B$ where function application is considered left-associative, that is, $f~{}a~{}b=(f~{}a)~{}b$. Thus $B^{2}$ is defined by an expression in which all applications nest to the left, never to the right. We call such an expression flat [7]. In particular we write $X_{\left(k\right)}$ for a flat expression involving only a combinator $X$, which is defined by $X_{\left(1\right)}=X$ and $X_{\left(k+1\right)}=X_{\left(k\right)}~{}X(k\geq 1)$. Okasaki [7] shows that there is no universal combinator $X$ that can represent any combinator by $X_{\left(k\right)}$ with some $k$. Using this notation, we can write $B^{2}=B_{\left(3\right)}$.

Consider the $n$-argument composition expressed by $B^{n}$. Surprisingly, we have $B^{3}=\allowbreak B~{}B~{}B~{}B~{}B~{}B~{}B~{}B=B_{\left(8\right)}$. It is easy to check it by repeating $\beta$-reduction for $B_{\left(8\right)}~{}f~{}g~{}x~{}y~{}z=f~{}(g~{}x~{}y~{}z)$. For $n>3$, however, the $n$-argument composition cannot be expressed by flat $B$-terms. There is no integer $k$ such that $B^{n}=B_{\left(k\right)}$ with respect to $\beta\eta$-equality. It can be proved by $\rho$-property of combinator $B$, that is introduced in this paper. We say that a combinator $X$ has $\rho$-property if there exists two distinct integer $i$ and $j$ such that $X_{\left(i\right)}=X_{\left(j\right)}$. If such a pair $i,j$ is found, we have $X_{\left(i+k\right)}=X_{\left(j+k\right)}$ for any $k\geq 0$  (à la finite monogenic semigroup [5]). In the case of $B$, we can check $B_{\left(6\right)}=B_{\left(10\right)}=\lambda x.\lambda y.\lambda z.\lambda w.\lambda v.~{}x~{}(y~{}z)~{}(w~{}v)$ hence $B_{\left(i\right)}=B_{\left(i+4\right)}$ for $i\geq 6$. Fig. 1 shows a computation graph of $B_{\left(k\right)}$. The $\rho$-property is named after the shape of the graph.

The $\rho$-property implies that the set $\{~{}B_{\left(k\right)}\mid k\geq 1~{}\}$ is finite. Since none of the terms in the set is equal to $B^{n}$ with $n>3$ up to the $\beta\eta$-equivalence of the corresponding $\lambda$-terms, we conclude that there is no integer $k$ such that $B^{n}=B_{\left(k\right)}$.

This paper discusses the $\rho$-property of combinatory terms, particularly in $\mathbf{CL}(B)$, called $B$-terms, that are built from $B$ alone. Interestingly, $B~{}B$ enjoys the $\rho$ property with $(B~{}B)_{\left(52\right)}=(B~{}B)_{\left(32\right)}$ and so does $B~{}(B~{}B)$ with $(B~{}(B~{}B))_{\left(294\right)}=(B~{}(B~{}B))_{\left(258\right)}$ as reported [6]. Several combinators other than $B$-terms can be found enjoy the $\rho$-property, for example, $K=\lambda x.\lambda y.x$ and $C=\lambda x.\lambda y.\lambda z.~{}x~{}z~{}y$ because of $K_{\left(3\right)}=K_{\left(1\right)}$ and $C_{\left(4\right)}=C_{\left(3\right)}$. They are not so interesting in the sense that the cycle starts immediately and its size is very small, comparing with $B$-terms like $B~{}B$ and $B~{}(B~{}B)$. As we will see later, $B~{}(B~{}(B~{}(B~{}(B~{}(B~{}B)))))(\equiv B^{6}~{}B)$ has the $\rho$-property with the cycle of the size more than $3\times 10^{11}$ which starts after more than $2\times 10^{12}$ repetitive right application. This is why the $\rho$-property of $B$-terms is intensively discussed in the present paper.

The contributions of the paper are two-fold. One is to give a characterization of $\mathbf{CL}(B)$ and another is to provide a sufficient condition for the $\rho$-property and anti-$\rho$-property of $B$-terms. In the former, a canonical representation of $B$-terms is introduced and sound and complete equational axiomatization for $\mathbf{CL}(B)$ is established. In the latter, the $\rho$-property of $B^{n}B$ with $n\leq 6$ is shown with an efficient algorithm and the anti-$\rho$-property for $B$-terms in particular forms is proved.

2 $\rho$-property of terms

The $\rho$-property of combinator $X$ is that $X_{\left(i\right)}=X_{\left(j\right)}$ holds for some $i>j\geq 1$. We adopt $\beta\eta$-equivalence of corresponding $\lambda$-terms for the equivalence of combinatory terms in this paper. We could use other equivalence, for example, induced by the axioms of the combinatory logic. The choice of equivalence does not have an essential influence, e.g., $B_{\left(9\right)}$ and $B_{\left(13\right)}$ are equal even up to the combinatory axiom of $B$, while $B_{\left(6\right)}=B_{\left(10\right)}$ holds for $\beta\eta$-equivalence. Furthermore, for simplicity, we only deal with the case where $X_{\left(n\right)}$ is normalizable for all $n$. If $X_{\left(n\right)}$ is not normalizable, it is much difficult to check equivalence with the other terms.

Let us write $\rho(X)=(i,j)$ when a combinator $X$ has the $\rho$-property due to $X_{\left(i\right)}=X_{\left(i+j\right)}$ with minimum positive integers $i$ and $j$. For example, we can write $\rho(B)=(6,4)$, $\rho(C)=(3,1)$, $\rho(K)=(1,2)$ and $\rho(I)=(1,1)$. Besides them, several combinators introduced in the Smullyan’s book [8] have the $\rho$-property:

[TABLE]

Except the $B$ and $D~{}(=B~{}B)$ combinators, the property is ‘trivial’ in the sense that loop starts early and the size of cycle is very small.

On the other hand, the combinators $S=\lambda x.\lambda y.\lambda z.x~{}z~{}(y~{}z)$ and $O=\lambda x.\lambda y.y~{}(x~{}y)$ in the book do not have the $\rho$-property for reason (A), which is illustrated by

[TABLE]

The definition of the $\rho$-property is naturally extended from single combinators to terms obtained by combining several combinators. We found by computation that several $B$-terms, built from the $B$ combinator alone, have a nontrivial $\rho$-property as shown in Fig. 2. The detail will be shown in Section 4.

3 Characterization of $B$-terms

The set of all $B$-terms, $\mathbf{CL}(B)$, is closed under application, that is, the repetitive right application of a $B$-term always generates a sequence of $B$-terms. Hence, the $\rho$-property can be decided by checking ‘equivalence’ among generated $B$-terms, where the equivalence should be checked through $\beta\eta$-equivalence of their corresponding $\lambda$-terms in accordance with the definition of the $\rho$-property. It would be useful if we have a simple decision procedure for deciding equivalence over $B$-terms.

In this section, we give a characterization of the $B$-terms to make it easy to decide their equivalence. We introduce a method for deciding equivalence of $B$-terms without calculating the corresponding $\lambda$-terms. To this end, we first investigate equivalence over $B$-terms with examples and then present an equation system as characterization of $B$-terms so as to decide equivalence between two $B$-terms. Based on the equation system, we introduce a canonical representation of $B$-terms. The representation makes it easy to observe the growth caused by repetitive right application of $B$-terms, which will be later shown for proving the anti-$\rho$-property of $B^{2}$. We believe that this representation will be helpful to prove the $\rho$-property or the anti-$\rho$-property for the other $B$-terms.

3.1 Equivalence over $B$-terms

Two $B$-terms are said equivalent if their corresponding $\lambda$-terms are $\beta\eta$-equivalent. For instance, $B~{}B~{}(B~{}B)$ and $B~{}(B~{}B)~{}B~{}B$ are equivalent. This can be easily shown by the definition $B~{}x~{}y~{}z=x~{}(y~{}z)$. For another (non-trivial) instance, $B~{}B~{}(B~{}B)$ and $B~{}(B~{}(B~{}B))~{}B$ are equivalent. This is illustrated by the fact that they are equivalent to $\lambda x.\lambda y.\lambda z.\lambda w.\lambda v.x~{}(y~{}z)~{}(w~{}v)$. where $B$ is replaced with $\lambda x.\lambda y.\lambda z.~{}x~{}(y~{}z)$ or the other way around at the $=_{\beta}$ equation. Similarly, we cannot show equivalence between two $B$-terms, $B~{}(B~{}B)~{}(B~{}B)$ and $B~{}(B~{}B~{}B)$, without long calculation. This kind of equality makes it hard to investigate the $\rho$-property of $B$-terms. To solve the annoying issue, we will later introduce a canonical representation of $B$-terms.

3.2 Equational axiomatization for $B$-terms

Equality between two $B$-terms can be effectively decided by an equation system.

Figure 3 shows a sound and complete equation system as described in the following theorem.

Theorem 3.1

Two $B$-terms are $\beta\eta$-equivalent if and only if their equality is derived by equations (B1), (B2), and (B3).

The proof of the if-part is given here, which corresponds to the soundness of the equation system. We will later prove the only-if-part with the uniqueness of the canonical representation of $B$-terms.

Proof of if-part of Theorem 3.1.  Equation (B1) is immediate from the definition of $B$. Equation (B2) and (B3) are shown by

[TABLE]

where the $\alpha$-renaming is implicitly used. $\Box$

Equation (B2) has been employed by Statman [9] to show that no $B\omega$-term can be a fixed-point combinator where $\omega=\lambda x.x~{}x$. This equation exposes an interesting feature of the $B$ combinator. Write equation (B2) as

[TABLE]

by replacing every $B$ combinator with $\circ$ infix operator if it has exactly two arguments. The equation is an distributive law of $B$ over $\circ$, which will be used to obtain the canonical representation of $B$-terms. Equation (B3) is also used for the same purpose as the form of

[TABLE]

One may expect a natural equation

[TABLE]

which represents associativity of function composition, i.e., $e_{1}\circ(e_{2}\circ e_{3})=(e_{1}\circ e_{2})\circ e_{3}$ This is shown with equations (B1) and (B2) by

[TABLE]

3.3 Canonical representation of $B$-terms

To decide equality between two $B$-terms, it does not suffices to compute their normal forms under the definition of $B$, $B~{}x~{}y~{}z\to x~{}(y~{}z)$. This is because two distinct normal forms may be equal up to $\beta\eta$-equivalence, e.g., $B~{}B~{}(B~{}B)$ and $B~{}(B~{}(B~{}B))~{}B$. We introduce a canonical representation of $B$-terms, which makes it easy to check equivalence of $B$-terms. We will finally find that for any $B$-term $e$ there exists a unique finite non-empty weakly-decreasing sequence of non-negative integers $n_{1}\geq n_{2}\geq\dots\geq n_{k}$ such that $e$ is equivalent to $(B^{n_{1}}B)\circ(B^{n_{2}}B)\circ\dots\circ(B^{n_{k}}B)$. Ignoring the inequality condition gives polynomials introduced by Statman [9]. We will use decreasing polynomials for our canonical representation as presented later.

First, we explain how its canonical form is obtained from every $B$-term. We only need to consider $B$-terms in which every $B$ has at most two arguments. One can easily reduce the arguments of $B$ to less than three by repeatedly rewriting occurrences of $B~{}e_{1}~{}e_{2}~{}e_{3}~{}e_{4}~{}\dots~{}e_{n}$ into $e_{1}~{}(e_{2}~{}e_{3})~{}e_{4}~{}\dots~{}e_{n}$. The rewriting procedure always terminates because it reduces the number of $B$. Thus, every $B$-term in $\mathbf{CL}(B)$ is equivalent to a $B$-term built by the syntax

[TABLE]

where $e_{1}\circ e_{2}$ denotes $B~{}e_{1}~{}e_{2}$. We prefer to use the infix operator $\circ$ instead of $B$ that has two arguments because associativity of $B$, that is, $B~{}e_{1}~{}(B~{}e_{2}~{}e_{3})=B~{}(B~{}e_{1}~{}e_{2})~{}e_{3}$ can be implicitly assumed. This simplifies the further discussion on $B$-terms. We will deal with only $B$-terms in syntax (2) from now on. The $\circ$ operator has a lower precedence than application in this paper. Terms $B~{}B\circ B$ and $B\circ B~{}B$ represent $(B~{}B)\circ B$ and $B\circ(B~{}B)$, respectively.

The syntactic restriction by (2) does not suffice to proffer a canonical representation of $B$-terms. There are many pairs of $B$-terms which are equivalent even in the form of (2). For example, $B\circ B~{}B=B~{}(B~{}B)\circ B$ holds according to equation (B3’).

A polynomial form of $B$-terms is obtained by putting a restriction to the syntax so that no $B$ combinator occurs outside of the $\circ$ operator while syntax (2) allows the $B$ combinators and the $\circ$ operators to occur in arbitrary position. The restricted syntax is given as

[TABLE]

where terms in $e_{B}$ have a form of $\underbrace{B(\dots(B}_{n}B)\dots)$, that is $B^{n}B$, called monomial. The syntax can be simply rewritten into $e::=B^{n}B~{}\mid~{}e\circ e$, which is called polynomial.

Definition 3.2

A $B$-term $B^{n}B$ is called monomial. A polynomial is given as the form of

[TABLE]

where $k>0$ and $n_{1},\dots,n_{k}\geq 0$ are integers. In particular, a polynomial is called decreasing when $n_{1}\geq n_{2}\geq\dots\geq n_{k}$. The length of a polynomial $P$ is defined by adding 1 to the number of $\circ$ in $P$. The numbers $n_{1},n_{2},\dots,n_{k}$ are called degree.

In the rest of this subsection, we prove that for any $B$-term $e$ there exists a unique decreasing polynomial equivalent to $e$. First, we show that $e$ has an equivalent polynomial.

Lemma 3.3** ([9])**

For any $B$-term $e$, there exists a polynomial to $e$.

Proof.  We prove the statement by induction on structure of $e$. In the case of $e\equiv B$, the term itself is polynomial. In the case of $e\equiv B~{}e_{1}$, assume that $e_{1}$ has equivalent polynomial $(B^{n_{1}}B)\circ(B^{n_{2}}B)\circ\dots\circ(B^{n_{k}}B)$. Repeatedly applying equation (B2’) to $B~{}e_{1}$, we obtain a polynomial equivalent to $B~{}e_{1}$ as $(B^{n_{1}+1}B)\circ(B^{n_{2}+1}B)\circ\dots\circ(B^{n_{k}+1}B)$. In the case of $e\equiv e_{1}\circ e_{2}$, assume that $e_{1}$ and $e_{2}$ have equivalent polynomials $P_{1}$ and $P_{2}$, respectively. A polynomial equivalent to $e$ is given by $P_{1}\circ P_{2}$. $\Box$

Next we show that for any polynomial $P$ there exists a decreasing list equivalent to $P$. A key equation of the proof is

[TABLE]

which is shown by

[TABLE]

using equations (B2’) and (B3’).

Lemma 3.4

Any polynomial $P$ has an equivalent decreasing polynomial $P^{\prime}$ such that

• •

the length of $P$ and $P^{\prime}$ are equal, and

• •

the lowest degrees of $P$ and $P^{\prime}$ are equal.

Proof.  We prove the statement by induction on the length of $P$. When the length is 1, that is, $P$ includes no $\circ$ operator, $P$ itself is decreasing and the statement holds. When the length of $P$ is $k>1$, take $P_{1}$ such that $P\equiv P_{1}\circ(B^{n}B)$. From the induction hypothesis, there exists a decreasing polynomial $P_{1}^{\prime}\equiv(B^{n_{1}}B)\circ(B^{n_{2}}B)\circ\dots\circ(B^{n_{k-1}}B)$ equivalent to $P_{1}$, and the lowest degree of $P_{1}$ is $n_{k-1}$. If $n_{k-1}\geq n$, then $P^{\prime}\equiv P_{1}^{\prime}\circ(B^{n}~{}B)$ is decreasing and equivalent to $P$. Since the lowest degrees of $P$ and $P^{\prime}$ are $n$, the statement holds. If $n_{k-1}<n$, $P$ is equivalent to

[TABLE]

due to equation (3). Putting the last term as $P_{2}\circ(B^{n_{k-1}}B)$, the length of $P_{2}$ is $k-1$ and the lowest degree of $P_{2}$ is greater than or equal to $n_{k-1}$. From the induction hypothesis, $P_{2}$ has an equivalent decreasing polynomial $P_{2}^{\prime}$ of length $k-1$ and the lowest degree of $P_{2}^{\prime}$ greater than or equal to $n_{k-1}$. Thereby we obtain a decreasing polynomial $P_{2}^{\prime}\circ(B^{n_{k-1}}B)$ equivalent to $P$ and the statement holds. $\Box$

Example 3.5

Consider a $B$-term $e=B~{}(B~{}B~{}B)~{}(B~{}B)~{}B$. First, applying equation (B1),

[TABLE]

so that every $B$ has at most two arguments. Then replace each $B$ to the infix $\circ$ operator if it has two arguments and obtain $B~{}(B~{}(B\circ(B~{}B)))$ Applying equation (B2’), we have

[TABLE]

Applying equation (3), we obtain the decreasing polynomial $(B^{4}B)\circ(B^{2}B)$ equivalent to $e$.

Every $B$-term has at least one equivalent decreasing polynomial as shown so far. To conclude this subsection, we show the uniqueness of decreasing polynomial equivalent to any $B$-term, that is, every $B$-term $e$ has no two distinct decreasing polynomials equivalent to $e$.

The proof is based on the idea that $B$-terms correspond to unlabeled binary trees. In every $\lambda$-term in $\mathbf{CL}(B)$, all variables are referred exactly once (linear) in the order they it was introduced (ordered). More precisely, this fact is formalized as follows.

Lemma 3.6

Every $\lambda$-term in $\mathbf{CL}(B)$ is $\beta\eta$-equivalent to a $\lambda$-term of the form $\lambda x_{1}.\lambda x_{2}.\dots.\lambda x_{k}.~{}M$ with some $k>2$ where $M$ is built by putting parentheses to appropriate positions in the sequence $x_{1}~{}x_{2}~{}\dots~{}x_{k}$.

Proof.  This can be proved by induction on structure of terms in $\mathbf{CL}(B)$. $\Box$

This lemma implies that every $\lambda$-term in $\mathbf{CL}(B)$ is characterized by an unlabeled binary tree. A $\lambda$-term in $\mathbf{CL}(B)$ is constructed for any unlabeled binary tree by putting a variable to each leaf in the order of $x_{1},x_{2},\dots$ and enclosing it with $k$-fold lambda abstraction $\lambda x_{1}.\lambda x_{2}.\dots.\lambda x_{k}.~{}[~{}]$ where $k$ is a number of leaves of the binary tree. Let us use the notation $\star$ for a leaf and $\langle t_{1},t_{2}\rangle$ for a tree with left subtree $t_{1}$ and right subtree $t_{2}$. For example, $B$-terms $B=\lambda x.\lambda y.\lambda z.~{}x~{}(y~{}z)$ and $B~{}B=\lambda x.\lambda y.\lambda z.\lambda w.~{}x~{}y~{}(z~{}w)$ are represented by $\langle\star,\langle\star,\star\rangle\rangle$ and $\langle\langle\star,\star\rangle,\langle\star,\star\rangle\rangle$, respectively.

We will present an algorithm for constructing the corresponding decreasing polynomial from a given binary tree. First let us define a function $\mathcal{L}_{i}$ with integer $i$ which maps binary trees to lists of integers:

[TABLE]

where $\mathbin{{+}\mspace{-8.0mu}{+}}$ concatenates two lists and $\left|\!\left|t\right|\!\right|$ denotes a number of leaves. For example, $\mathcal{L}_{0}(\langle\langle\star,\star\rangle,\langle\star,\star\rangle\rangle)=[2,0,0]$ and $\mathcal{L}_{1}(\langle\langle\star,\langle\star,\star\rangle\rangle,\langle\star,\langle\star,\star\rangle\rangle\rangle)=[4,4,2,1,1]$. Informally, the $\mathcal{L}_{i}$ function returns a list of integers which is obtained by labeling both leaves and nodes in the following steps. First each leaf of a given tree is labeled by $i,i+1,i+2,\dots$ in left-to-right order. Then each binary node of the tree is labeled by the same label as its leftmost descendant leaf. The $\mathcal{L}_{i}$ functions returns a list of only node labels in decreasing order. The length of the list equals the number of nodes, that is, smaller by one than the number of variables in the $\lambda$-term.

We define a function $\mathcal{L}$ which takes a binary tree $t$ and returns a list of non-negative integers in $\mathcal{L}_{-1}(t)$, that is, the list obtained by excluding trailing all $-1$’s in $\mathcal{L}_{-1}(t)$. Note that by excluding the label $-1$’s it may happen to be $\mathcal{L}(t)=\mathcal{L}(t^{\prime})$ for two distinct binary trees $t$ and $t^{\prime}$ even though the $\mathcal{L}_{i}$ function is injective. However, those binary trees $t$ and $t^{\prime}$ must be ‘$\eta$-equivalent’ in terms of the corresponding $\lambda$-terms.

The following lemma claims that the $\mathcal{L}$ function computes a list of degrees of a decreasing polynomial corresponding to a given $\lambda$-term.

Lemma 3.7

A decreasing polynomial $(B^{n_{1}}B)\circ(B^{n_{2}}B)\circ\dots\circ(B^{n_{k}}B)$ is $\beta\eta$-equivalent to a $\lambda$-term $e\in\mathbf{CL}(B)$ corresponding a binary tree $t$ such that $\mathcal{L}(t)=[n_{1},n_{2},\dots,n_{k}]$.

Proof.  We prove the statement by induction on the length of the polynomial $P$.

When $P\equiv B^{n}B$ with $n\geq 0$, it is found to be equivalent to the $\lambda$-term

[TABLE]

by induction on $n$. This $\lambda$-term corresponds to a binary tree t=\langle\langle\dots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\dots,\star\rangle}_{\text{n leaves}},\langle\star,\star\rangle\rangle. Then we have $\mathcal{L}(t)=[n]$ holds from $\mathcal{L}_{-1}(t)=[n,\underbrace{-1,-1,\dots,-1}_{n+1}]$.

When $P\equiv P^{\prime}\circ(B^{n}B)$ with $P^{\prime}\equiv(B^{n_{1}}B)\circ\dots\circ(B^{n_{k}}B)$, $k\geq 1$ and $n_{1}\geq\dots\geq n_{k}\geq n\geq 0$, there exists a $\lambda$-term $\beta\eta$-equivalent to $P^{\prime}$ corresponding a binary tree $t^{\prime}$ such that $\mathcal{L}(t^{\prime})=[n_{1},\dots,n_{k}]$ from the induction hypothesis. The binary tree $t^{\prime}$ must have the form of \langle\langle\langle\dots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\dots,\star\rangle}_{\text{n_{k} leaves}},t_{1}\rangle,\dots,t_{m}\rangle with $m\geq 1$ and some trees $t_{1},\dots,t_{m}$, otherwise $\mathcal{L}(t^{\prime})$ would contain an integer smaller than $n_{k}$. From the definition of $\mathcal{L}$ and $\mathcal{L}_{i}$, we have

[TABLE]

where $s_{j}=n_{k}+1+\sum_{i=1}^{j-1}\left|\!\left|t_{i}\right|\!\right|$. Additionally, the structure of $t^{\prime}$ implies $P^{\prime}=\lambda x_{1}.\dots.\lambda x_{l}.~{}\allowbreak x_{1}~{}x_{2}~{}\dots~{}x_{n_{k}+1}~{}e_{1}\dots e_{m}$ where $e_{i}$ corresponds to a binary tree $t_{i}$ for $i=1,\dots,m$. From $B^{n}~{}B=\lambda y_{1}.\dots.\lambda y_{n+3}.~{}\allowbreak y_{1}~{}y_{2}\dots y_{n+1}~{}(y_{n+2}~{}y_{n+3})$, we compute a $\lambda$-term $\beta\eta$-equivalent to $P\equiv P^{\prime}\circ(B^{n}B)$ by

[TABLE]

where $n_{k}\geq n$ is taken into account. We split into four cases: (i) $n_{k}=n$ and $m=1$, (ii) $n_{k}=n$ and $m>1$, (iii) $n_{k}=n+1$, and (iv) $n_{k}>n+1$. In the case (i) where $n_{k}=n$ and $m=1$, we have

[TABLE]

whose corresponding binary tree $t$ is \langle\langle\dots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\dots,\star\rangle}_{\text{n leaves}},\langle t_{1},\star\rangle\rangle. From equation (4), $\mathcal{L}(t)=\mathcal{L}_{n+1}(t_{1})\mathbin{{+}\mspace{-8.0mu}{+}}[n+1]=\mathcal{L}(t^{\prime})\mathbin{{+}\mspace{-8.0mu}{+}}[n+1]=[n_{1},\dots,n_{k},n+1]$, thus the statement holds. In the case (ii) where $n_{k}=n$ and $m>1$, we have

[TABLE]

whose corresponding binary tree $t$ is \langle\langle\langle\dots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\dots,\star\rangle}_{\text{n leaves}},\langle t_{1},t_{2}\rangle,t_{3}\rangle,\dots,t_{m}\rangle. Hence, $\mathcal{L}(t)=\mathcal{L}(t^{\prime})\mathbin{{+}\mspace{-8.0mu}{+}}[n+1]$ holds again from equation (4). In the case (iii) where $n_{k}=n+1$, we have

[TABLE]

whose corresponding binary tree $t$ is \langle\langle\langle\dots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\dots,\star\rangle}_{\text{n leaves}},\langle\star,t_{1}\rangle,t_{2}\rangle,\dots,t_{m}\rangle. Hence, $\mathcal{L}(t)=\mathcal{L}(t^{\prime})\mathbin{{+}\mspace{-8.0mu}{+}}[n+1]$ holds from equation (4). In the case (iv) where $n_{k}\geq n+2$, we have

[TABLE]

whose corresponding binary tree $t$ is \langle\langle\langle\dots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\dots,\star\rangle}_{\text{n leaves}},\langle\star,\star\rangle,\dots,t_{1}\rangle,\dots,t_{m}\rangle. Hence, $\mathcal{L}(t)=\mathcal{L}(t^{\prime})\mathbin{{+}\mspace{-8.0mu}{+}}[n+1]$ holds from equation (4). $\Box$

Example 3.8

A $\lambda$-term $\lambda x_{1}.\lambda x_{2}.\lambda x_{3}.\lambda x_{4}.\lambda x_{5}.\lambda x_{6}.\lambda x_{7}.\lambda x_{8}.~{}x_{1}~{}(x_{2}~{}x_{3})~{}(x_{4}~{}x_{5}~{}x_{6}~{}(x_{7}~{}x_{8}))$ is $\beta\eta$-equivalent to $(B^{5}~{}B)\circ(B^{2}~{}B)\circ(B^{2}~{}B)\circ(B^{2}~{}B)\circ(B^{0}~{}B)$ because its corresponding binary tree $t=\langle\langle\langle\star,\star\rangle\rangle,\langle\langle\langle\star,\star\rangle,\star\rangle,\langle\star,\star\rangle\rangle\rangle$ satisfies $\mathcal{L}(t)=[5,2,2,2,0]$.

The previous lemmas immediately conclude the uniqueness of of decreasing polynomials for a $B$-term shown in the following theorem.

Theorem 3.9

Every $B$-term $e$ has a unique decreasing polynomial.

Proof.  For any given $B$-term $e$, we can find a decreasing polynomial for $e$ from Lemma 3.3 and Lemma 3.4. Since no other decreasing polynomial an be equivalent to $e$ from Lemma 3.7, the present statement holds. $\Box$

This theorem implies that the decreasing polynomial of $B$-terms can be used as their canonical representation, which is effectively derived as shown in Lemma 3.3 and Lemma 3.4.

As a corollary of the theorem, we can show the only-if statement of Theorem 3.1, which corresponds to the completeness of the equation system.

Proof of only-if-part of Theorem 3.1.  Let $e_{1}$ and $e_{2}$ be equivalent $B$-terms, that is, their $\lambda$-terms are $\beta\eta$-equivalent. From Theorem 3.9, their decreasing polynomials are the same. Since the decreasing polynomial is derived from $e_{1}$ and $e_{2}$ by equations (B1), (B2), and (B3) according to the proofs of Lemma 3.3 and Lemma 3.4, equivalence between $e_{1}$ and $e_{2}$ is also derived from these equations. $\Box$

4 Results on the $\rho$-property of $B$-terms

We investigate the $\rho$-property of concrete $B$-terms, some of which have the property and others do not. For $B$-terms having the $\rho$-property, we introduce an efficient implementation to compute the entry point and the size of the cycle. For $B$-terms not having the $\rho$-property, we give a proof why they do not have.

4.1 $B$-terms having the $\rho$-property

As shown in Section 2, we can check that $B$-terms equivalent to $B^{n}B$ with $n\leq 6$ have the $\rho$-property by computing $(B^{n}B)_{\left(i\right)}$ for each $i$. However, it is not easy to check it by computer without an efficient implementation because we should compute all $(B^{6}B)_{\left(i\right)}$ with $i\leq 2980054085040~{}(=2641033883877+339020201163)$ to know $\rho(B^{6}B)=(2641033883877,339020201163)$. A naive implementation which computes terms of $(B^{6}B)_{\left(i\right)}$ for all $i$ and stores all of them has no hope to detect the $\rho$-property.

We introduce an efficient procedure to find the $\rho$-property of $B$-terms which can successfully compute $\rho(B^{6}B)$. The procedure is based on two orthogonal ideas, Floyd’s cycle-finding algorithm [4] and an efficient right application algorithm over decreasing polynomials presented in Section 3.3.

The first idea, Floyd’s cycle-finding algorithm (also called the tortoise and the hare algorithm), enables to detect the cycle with a constant memory usage, that is, the history of all terms $X_{\left(i\right)}$ does not need to be stored to check the $\rho$-property of the $X$ combinator. The key of this algorithm is the fact that there are two distinct integers $i$ and $j$ with $X_{\left(i\right)}=X_{\left(j\right)}$ if and only if there are an integer $m$ with $X_{\left(m\right)}=X_{\left(2m\right)}$, where the latter requires to compare $X_{\left(i\right)}$ and $X_{\left(2i\right)}$ from smaller $i$ and store only these two terms for the next comparison between $X_{\left(i+1\right)}=X_{\left(i\right)}X$ and $X_{\left(2i+2\right)}=X_{\left(2i\right)}XX$ when $X_{\left(i\right)}\neq X_{\left(2i\right)}$. The following procedure computes the entry point and the size of the cycle if $X$ has the $\rho$-property.

Find the smallest $m$ such that $X_{\left(m\right)}=X_{\left(2m\right)}$. 2. 2.

Find the smallest $k$ such that $X_{\left(k\right)}=X_{\left(m+k\right)}$. 3. 3.

Find the smallest $0<c\leq k$ such that $X_{\left(m\right)}=X_{\left(m+c\right)}$. If not found, put $c=m$.

After this procedure, we find $\rho(X)=(k,c)$. The third step can be simultaneously run during the second one. See [4, exercise 3.1.6] for the detail. One could use slightly more (possibly) efficient algorithms by Brent [2] and Gosper [1, item 132] for cycle detection.

Efficient cycle-finding algorithms do not suffice to compute $\rho(B^{6}B)$. Only with the idea above running on a laptop (1.7 GHz Intel Core i7 / 8GB of memory), it takes about 2 hours even for $\rho(B^{5}B)$ fails to compute $\rho(B^{6}B)$ due to out of memory.

The second idea enables to efficiently compute $X_{\left(i+1\right)}$ from $X_{\left(i\right)}$ for $B$-terms $X$. The key of this algorithm is to use the canonical representation of $X_{\left(i\right)}$, that is a decreasing polynomial, and directly compute the canonical representation of $X_{\left(i+1\right)}$ from that of $X_{\left(i\right)}$. Our implementation based on both ideas can compute $\rho(B^{5}B)$ and $\rho(B^{6}B)$ in 10 minutes and 59 days (!), respectively.

For two given decreasing polynomials $P_{1}$ and $P_{2}$, we show how a decreasing polynomial $P$ equivalent to $(P_{1}~{}P_{2})$ can be obtained. The method is based on the following lemma about application of one $B$-term to another $B$-term.

Lemma 4.1

*For $B$-terms $e_{1}$ and $e_{2}$, there exists $k\geq 0$ such that $e_{1}\circ(B~{}e_{2})=B~{}(e_{1}~{}e_{2})\circ B^{k}$. *

Proof.  Let $P_{1}$ be a decreasing polynomial equivalent to $e_{1}$. We prove the statement by case analysis on the maximum degree in $P_{1}$. When the maximum degree is 0, we can take $k^{\prime}\geq 1$ such that $P_{1}\equiv\underbrace{B\circ\dots\circ B}_{k^{\prime}}=B^{k^{\prime}}$. Then,

[TABLE]

where equation (B3’) is used $k^{\prime}$ times in the second equation. Therefore the statement holds by taking $k=k^{\prime}$. When the maximum degree is greater than 0, we can take a decreasing polynomial $P^{\prime}$ for a $B$-term and $k^{\prime}\geq 0$ such that $P_{1}=(B~{}P^{\prime})\circ\underbrace{B\circ\dots\circ B}_{k^{\prime}}=(B~{}P^{\prime})\circ B^{k^{\prime}}$ due to equation (B2’). Then,

[TABLE]

Therefore, the statement holds by taking $k=k^{\prime}$. $\Box$

This lemma indicates that, from two decreasing polynomials $P_{1}$ and $P_{2}$, a decreasing polynomial $P$ equivalent to $(P_{1}~{}P_{2})$ can be obtained by the following steps:

Build $P_{2}^{\prime}$ by raising each degree of $P_{2}$ by 1, i.e., when $P_{2}\equiv(B^{n_{1}}B)\circ\dots\circ(B^{n_{l}}B)$, $P_{2}^{\prime}\equiv(B^{n_{1}+1}B)\circ\dots\circ(B^{n_{l}+1}B)$. 2. 2.

Find a decreasing polynomial $P_{12}$ corresponding to $P_{1}\circ P_{2}^{\prime}$ by equation (B3’); 3. 3.

Obtain $P$ by lowering each degree of $P_{12}$ after eliminating the trailing 0-degree units, i.e., when $P_{12}\equiv(B^{n_{1}}B)\circ\dots\circ(B^{n_{l}}B)\circ(B^{0}B)\circ\dots\circ(B^{0}B)$ with $n_{1}\geq\dots\geq n_{l}>0$, $P\equiv(B^{n_{1}-1}B)\circ\dots\circ(B^{n_{l}-1}B)$.

In the first step, a decreasing polynomial $P_{2}^{\prime}$ equivalent to $B~{}P_{2}$ is obtained. The second step yields a decreasing polynomial $P_{12}$ for $P_{1}\circ P_{2}^{\prime}=P_{1}\circ(B~{}P_{2})$. Since $P_{1}$ and $P_{2}$ are decreasing, it is easy to find $P_{12}$ by repetitive application of equation (B3’) for each unit of $P_{2}^{\prime}$, à la insertion operation in insertion sort. In the final step, a polynomial $P$ that satisfies $(B~{}P)\circ B^{k}=P_{12}$ with some $k$ is obtained. From Lemma 4.1 and the uniqueness of decreasing polynomials, $P$ is equivalent to $(P_{1}~{}P_{2})$.

Example 4.2

Let $P_{1}$ and $P_{2}$ be decreasing polynomials given as $P_{1}=(B^{4}B)\circ(B^{1}B)\circ(B^{0}B)$ and $P_{2}=(B^{2}B)\circ(B^{0}B)$. Then a decreasing polynomial $P$ equivalent to $(P_{1}~{}P_{2})$ is obtained by three steps:

Raise each degree of $P_{2}$ to get $P_{2}^{\prime}=(B^{3}B)\circ(B^{1}B)$. 2. 2.

Find a decreasing polynomial by

[TABLE]

where equation (B3’) is applied in each. 3. 3.

By lowering each degree after removing trailing $(B^{0}B)$’s,

[TABLE]

is obtained.

The implementation based on the right application over decreasing polynomials e is available at https://github.com/ksk/Rho. Note that the program does not terminate for the combinator which does not have the $\rho$-property. It will not help to decide if a combinator has the $\rho$-property. One might observe how the terms grow by repetitive right applications thorough running the program, though.

4.2 $B$-terms not having the $\rho$-property

We prove that the $B$-terms $(B^{k}B)^{(k+2)n}$ ($k\geq 0,\ n>0$) do not have the $\rho$-property. In particular, we show that the number of variables in the $\beta\eta$-normal form of $((B^{k}B)^{(k+2)n})_{(i)}$ is monotonically non-decreasing and that it implies the anti-$\rho$-property. Additionally, after proving that, we consider a sufficient condition not to have the $\rho$-property through the monotonicity.

First we introduce some notation. Suppose that the $\beta\eta$-normal form of a $B$-term $X$ is given by $\lambda x_{1}.\dots\lambda x_{n}.~{}\allowbreak x_{1}~{}e_{1}~{}\cdots~{}e_{k}$ for some terms $e_{1},\dots,e_{k}$. Then we define $l(X)=n$ (the number of variables), $a(X)=k$ (the number of arguments of $x_{1}$), and $N_{i}(X)=e_{i}$ for $i=1,\dots,k$. Let $X^{\prime}$ be another $B$-term and suppose its $\beta\eta$-normal form is given by $\lambda x^{\prime}_{1}.\dots\lambda x^{\prime}_{n^{\prime}}.~{}e^{\prime}$, We can see $X~{}X^{\prime}=(\lambda x_{1}.\ldots\lambda x_{n}.~{}x_{1}~{}e_{1}~{}\cdots~{}e_{k})~{}X^{\prime}=\lambda x_{2}.\dots\lambda x_{n}.~{}X^{\prime}~{}e_{1}~{}\cdots~{}e_{k}$ and from Lemma 3.6, its $\beta\eta$-normal form is

[TABLE]

Here $[e_{1},\dots,e_{k}]e^{\prime}$ is the term which is obtained by substituting $e_{1},\dots,e_{k}$ to the variables $x^{\prime}_{1},\dots,x^{\prime}_{k}$ in $e^{\prime}$ in order.

By simple computation with this fact, we get the following lemma:

Lemma 4.3

Let $X$ and $X^{\prime}$ be $B$-terms. Then

[TABLE]

where $m=\min\{l(N_{1}(X^{\prime})),\ a(X)\}$.

The $\beta\eta$-normal form of $(B^{k}B)^{(k+2)n}$ is given by

[TABLE]

This is deduced from Lemma 3.7 since the binary tree corresponding to the above $\lambda$-term is $t=\allowbreak\langle\langle\ldots\langle\langle\underbrace{\star,\star\rangle,\star\rangle,\ldots,\star}_{k+1}\rangle,\allowbreak\langle\ldots\langle\langle\star,\underbrace{\star\rangle,\star\rangle,\ldots,\star}_{(k+2)n}\rangle\rangle$ and $\mathcal{L}(t)=[\underbrace{k,\ldots,k}_{(k+2)n}]$. Especially, we get $l((B^{k}B)^{(k+2)n})=k+(k+2)n+2$. In this section, we write $\langle\star,\star,\star,\ldots,\star\rangle$ for $\langle\ldots\langle\langle\star,\star\rangle,\star\rangle,\ldots,\star\rangle$ and identify $B$-terms with their corresponding binary trees.

To describe properties of $(B^{k}B)^{(k+2)n}$, we introduce a set $T_{k,n}$ which is closed under right application of $(B^{k}B)^{(k+2)n}$, that is, $T_{k,n}$ satisfies that “if $X\in T_{k,n}$ then $X~{}(B^{k}B)^{(k+2)n}\in T_{k,n}$ holds”. First we inductively define a set of terms $T^{\prime}_{k,n}$ as follows:

$\star\in T^{\prime}_{k,n}$ 2. 2.

$\langle\star,~{}s_{1},~{}\ldots,~{}s_{(k+2)n}\rangle\in T^{\prime}_{k,n}$ if $s_{i}=\star$ for a multiple $i$ of $k+2$ and $s_{i}\in T^{\prime}_{k,n}$ for the others.

Then we define $T_{k,n}$ by $T_{k,n}=\left\{\langle t_{0},~{}t_{1},~{}\ldots,~{}t_{k+1}\rangle\ \middle|\ t_{0},\ t_{1},\ldots,t_{k+1}\in T^{\prime}_{k,n}\right\}$. It is obvious that $(B^{k}B)^{(k+2)n}\in T_{k,n}$. Now we shall prove that $T_{k,n}$ is closed under right application of $(B^{k}B)^{(k+2)n}$.

Lemma 4.4

If $X\in T_{k,n}$ then $X~{}(B^{k}B)^{(k+2)n}\in T_{k,n}$.

Proof.  From the definition of $T_{k,n}$, if $X\in T_{k,n}$ then $X$ can be written in the form $\langle t_{0},\ t_{1},\ \dots,\ t_{k+1}\rangle$ for some $t_{0},\ldots,t_{k+1}\in T^{\prime}_{k,n}$. In the case where $t_{0}=\star$, we have $X~{}(B^{k}B)^{(k+2)n}=\langle t_{1},~{}\ldots,~{}t_{k+1},~{}\langle\underbrace{\star,~{}\ldots,~{}\star}_{(k+2)n}\rangle\rangle\in T_{k,n}$. In the case where $t_{0}$ has the form of 2 in the definition of $T^{\prime}_{k,n}$, then $X=\langle\star,~{}s_{1},~{}\ldots,~{}s_{(k+2)n},~{}t_{1},~{}\ldots,~{}t_{k+1}\rangle$ with $s_{i}\in T^{\prime}_{k,n}$ and so

[TABLE]

We can easily see $s_{1},\ \ldots,\ s_{k+1}$, and $\langle s_{k+2},\ \ldots,\ s_{(k+2)n},\ t_{1},\ \ldots,\ t_{k+1},\ \star\rangle$ are in $T^{\prime}_{k,n}$. $\Box$

From the definition of $T_{k,n}$, we can compute that $a(X)$ equals $k+1$ or $(k+2)n+k+1$. Particularly, we get the following:

Lemma 4.5

For any $X\in T_{k,n}$, $a(X)\leq(k+2)n+k+1=l((B^{k}B)^{(k+2)n})-1$.

This lemma is crucial to show that the number of variables in $((B^{k}B)^{(k+2)n})_{(i)}$ is monotonically non-decreasing. Put $Z=(B^{k}B)^{(k+2)n}$ for short. Since $Z\in T_{k,n}$, we have $\{Z_{(i)}\mid i\geq 1\}\subset T_{k,n}$ by Lemma 4.4. Using Lemma 4.5, we can simplify Lemma 4.3 in the case where $X=Z_{(i)}$ and $X^{\prime}=Z$ as follows:

[TABLE]

By (5) and Lemma 4.5, we get $l(Z_{(i+1)})\geq l(Z_{(i)})$.

To prove that $Z$ does not have the $\rho$-property, it suffices to show the following:

Lemma 4.6

For any $i\geq 1$, there exists $j>i$ that satisfies $l(Z_{(j)})>l(Z_{(i)})$.

Proof.  Suppose that there exists $i\geq 1$ that satisfies $l(Z_{(i)})=l(Z_{(j)})$ for any $j>i$. We get $a(Z_{(j)})=(k+2)n+k+1$ by (5) and then $a(N_{1}(Z_{(j)}))=(k+2)n$ by (6). Therefore $N_{1}(Z_{(j)})$ is not a variable for any $j>i$ and from (7), we obtain $N_{1}(Z_{(j)})=N_{1}(N_{1}(Z_{(j-1)}))=\cdots=\underbrace{N_{1}(\cdots N_{1}(}_{j-i+1}Z_{(i)})\cdots)$ for any $j>i$. However, this implies that $Z_{(i)}$ has infinitely many variables and it contradicts. $\Box$

Now, we get the desired result:

Theorem 4.7

For any $k\geq 0$ and $n>0$, $(B^{k}B)^{(k+2)n}$ does not have the $\rho$-property.

The key fact which enables to show the anti-$\rho$-property of $(B^{k}B)^{(k+2)n}$ is the existence of the set $T_{k,n}\supset\left\{((B^{k}B)^{(k+2)n})_{(i)}\ \middle|\ i\geq 1\right\}$ which satisfies Lemma 4.5. In the same way to above, we can show the anti-$\rho$-property of a $B$-term which has such a “good” set. That is,

Theorem 4.8

Let $X$ be a $B$-term and $T$ be a set of $B$-terms. If $\left\{X_{(i)}\ \middle|\ i\geq 1\right\}\subset T$ and $l(X)\geq a(X^{\prime})+1$ for any $X^{\prime}\in T$, then $X$ does not have the $\rho$-property.

Here is an example of the $B$-terms which satisfies the condition in Theorem 4.8 with some set $T$. Consider $X=(B^{2}B)^{2}\circ(BB)^{2}\circ B^{2}=\langle\star,\ \langle\star,\ \langle\star,\ \langle\star,\ \star,\ \star\rangle,\ \star\rangle,\ \star\rangle\rangle$. We inductively define $T^{\prime}$ as follows:

$\star\in T^{\prime}$ 2. 2.

For any $t\in T^{\prime}$, $\langle\star,\ t,\ \star\rangle\in T^{\prime}$ 3. 3.

For any $t_{1},t_{2}\in T^{\prime}$, $\langle\star,\ t_{1},\ \star,\ \langle\star,\ t_{2},\ \star\rangle,\ \star\rangle\in T^{\prime}$

Then $T=\left\{\langle t_{1},\ \langle\star,\ t_{2},\ \star\rangle\rangle~{}\middle|~{}t_{1},t_{2}\in T^{\prime}\right\}$ satisfies the condition in Theorem 4.8. It can be checked simply by case analysis. Thus

Theorem 4.9

$(B^{2}B)^{2}\circ(BB)^{2}\circ B^{2}$* does not have the $\rho$-property.*

Theorem 4.8 gives a possible technique to prove the monotonicity with respect to $l(X_{(i)})$, or, the anti-$\rho$-property of $X$, for some $B$-term $X$. Moreover, we can consider another problem on $B$-terms: “Give a necessary and sufficient condition to have the monotonicity for $B$-terms.”

5 Concluding remark

We have investigated the $\rho$-properties of $B$-terms in particular forms so far. While the $B$-terms equivalent to $B^{n}B$ with $n\leq 6$ have the $\rho$-property, the $B$-terms $(B^{k}B)^{(k+2)n}$ with $k\geq 0$ and $n>0$ and $(B^{2}B)^{2}\circ(BB)^{2}\circ B^{2}$ do not. In this section, remaining problems related to these results are introduced and possible approaches to illustrate them are discussed.

5.1 Remaining problems

The $\rho$-property is defined any combinatory terms (and closed $\lambda$-terms). We investigates it only for $B$-terms as a simple but interesting instance in the present paper. From his observation on repetitive right applications for several $B$-terms, Nakano [6] has conjectured as follows.

Conjecture 5.1

$B$-term $e$ has the $\rho$-property if and only if $e$ is a monomial, i.e., $e$ is equivalent to $B^{n}B$ with $n\geq 0$.

The if-part for $n\leq 6$ has been shown by computation and the only-if-part for $(B^{k}B)^{(k+2)n}~{}\allowbreak(k\geq 0,n>0)$ and $(B^{2}B)^{2}\circ(BB)^{2}\circ B^{2}$ has been shown by Theorem 4.7. This conjecture implies that the $\rho$-property of $B$-terms is decidable. We surmise that the $\rho$-property of even $BCK$- and $BCI$-terms is decidable. The decidability for the $\rho$-property of $S$-terms and $L$-terms can also be considered. Waldmann’s work on a ration representation of normalizable $S$-terms may be helpful to solve it. We expect that none of $S$-terms have the $\rho$-property as $S$ itself does not, though. Regarding to $L$-terms, Statman’s work [10] may be helpful where equivalence of $L$-terms is shown decidable up to a congruence relation induced by $L~{}e_{1}~{}e_{2}\to e_{1}~{}(e_{2}~{}e_{2})$. It would be interesting to investigate the $\rho$-property of $L$-terms in this setting.

5.2 Possible approaches

The present paper introduces a canonical representation to make equivalence check of $B$-terms easier. The idea of the representation is based on that we can lift all $\circ$’s (2-argument $B$) to outside of $B$ (1-argument $B$) by equation (B2’). One may consider it the other way around. Using the equation, we can lift all $B$’s (1-argument $B$) to outside of $\circ$ (2-argument $B$). Then one of the arguments of $\circ$ becomes $B$. By equation (B3’), we can move all $B$’s to right. Thereby we find another canonical representation for $B$-terms given by

[TABLE]

whose uniqueness could be easily proved in a way similar to Theorem 3.9.

Waldmann [11] suggests that the $\rho$-property of $B^{n}B$ may be checked even without converting $B$-terms into canonical forms. He simply defines $B$-terms by

[TABLE]

and regards $B^{k}$ as a constant which has a rewrite rule $B^{k}~{}e_{1}~{}e_{2}~{}\dots~{}e_{k+2}\to e_{1}~{}(e_{2}~{}\dots~{}e_{k+2})$. He implemented a check program in Haskell to confirm the $\rho$-property. Even in the restriction on rewriting rules, he found that $(B^{0}B)_{\left(9\right)}=(B^{0}B)_{\left(13\right)}$, $(B^{1}B)_{\left(36\right)}=(B^{1}B)_{\left(56\right)}$, $(B^{2}B)_{\left(274\right)}=(B^{2}B)_{\left(310\right)}$ and $(B^{3}B)_{\left(4267\right)}=(B^{3}B)_{\left(10063\right)}$, in which it requires a bit more right applications to find the $\rho$-property than the case of a canonical representation. If the $\rho$-property of $B^{n}B$ for any $n\geq 0$ is shown under the restricted equivalence given by rewriting rules, then we can conclude the if-part of Conjecture 5.1.

Another possible approach is to observe the change of (principal) types by right repetitive application. Although there are many distinct $\lambda$-terms of the same type, we can consider a desirable subset of typed $\lambda$ terms. As shown by Hirokawa [3], each $BCK$-term can be characterized by its type, that is, any two $\lambda$-terms in $\mathbf{CL}(BCK)$ of the same principal type are identical up to $\beta$-equivalence. This approach may require to observe unification between types in a clever way.