2 Preliminaries

We let $\mathbb{N}$ denote the set of natural numbers. Let $A$ be a set. Then, $\overline {{A}}$ is the set of finite sequences of elements of $A$ , which includes the empty sequence, denoted as $\mathsf{e}$ . We use the delimiters $\langle$ and $\rangle$ for writing elements of $\overline {{A}}$ and juxtaposition to denote the concatenation operation, for example, $\left \langle {a_0,a_1}\right \rangle \left \langle {a_2,a_3}\right \rangle = \left \langle {a_0,a_1,a_2,a_3}\right \rangle$ . We generally denote elements of $\overline {{A}}$ using lowercase letters with an overline, for example, $\overline {{a}}$ .

2.4 Logic programing

We refer to Apt (Reference Apt1997) for the basics of logic programing. To simplify our presentation, we place ourselves in the general framework of term reduction systems, that is, we do not distinguish predicate/function symbols, terms/atoms… and we do not always use the standard terminology and notations of logic programing (e.g., rule instead of clause).

Definition 1. A program is a subset of $T(\Sigma , X) \times {\overline {{T(\Sigma , X)}}}$ , every element of which is called a rule. A rule $(u,{\overline {{v}}})$ is binary if $\overline {{v}}$ is empty or is a singleton. We let $\Re$ denote the set of binary rules. For the sake of readability, we omit the delimiters $\langle$ and $\rangle$ in the right-hand side of a binary rule, which amounts to considering that $\Re \subseteq T(\Sigma , X) \times (T(\Sigma , X) \cup \{{\mathsf{e}}\})$ .

Given a rule $(u,{\overline {{v}}})$ , we let $\left [(u,{\overline {{v}}})\right ] = \left \{(u\gamma ,{\overline {{v}}}\gamma ) \mid \gamma \text{ is a renaming}\right \}$ denote its equivalence class modulo renaming. For all sets of rules $U$ , we let $[U] = \bigcup _{r \in U} [r]$ . Moreover, for all rules or sequences of terms $S$ , we write ${\overline {{r}}} \ll _S U$ to denote that $\overline {{r}}$ is a sequence of elements of $U$ variable disjoint from $S$ and from each other.

The rules of a program allow one to rewrite finite sequences of terms. This is formalized by the following binary relation, which corresponds to the operational semantics of logic programing with the leftmost selection rule.

Definition 2. For all programs $P$ , we let $\mathop {\Rightarrow }_P = \bigcup \left \{\mathop {\Rightarrow }_r \;\middle \vert \; r\in P \right \}$ where, for all $r \in P$ ,

\begin{align*} \mathop {\Rightarrow }_r =& \left \{ \big (\left \langle {s}\right \rangle {\overline {{s}}}, ({\overline {{v}}}\,{\overline {{s}}})\theta \big ) \in {\overline {{T(\Sigma , X)}}}^2 \;\Bigg \vert \; \begin{array}{l} \left \langle {(u, {\overline {{v}}})}\right \rangle \ll _{\left \langle {s}\right \rangle {\overline {{s}}}} [r], \ \theta \in \operatorname {\mathit{mgu}}(u, s) \end{array}\right \} \end{align*}

For all $s \in T(\Sigma , X)$ , $\operatorname {\mathit{calls}}_P(s) = \{t \in T(\Sigma , X) \mid \left \langle {s}\right \rangle \mathop {\Rightarrow }_P^+ \left \langle {t,\ldots }\right \rangle \} \cup \{{\mathsf{e}} \mid \left \langle {s}\right \rangle \mathop {\Rightarrow }_P^+ {\mathsf{e}}\}$ is the set of calls in the $\mathop {\Rightarrow }_P$ -chains that start from $s$ .

2.5 Binary unfolding

The binary unfolding of a program $P$ (see the paper by Codish and Taboch (Reference Codish and Taboch1999)) is a set of binary rules, denoted as $\operatorname {\mathit{binunf}}(P)$ , that captures call patterns of $P$ . It corresponds to the transitive closure of a binary relation which relates consecutive calls selected in a computation (Prop. 1 below). Non-termination for a specific sequence of terms implies the existence of a corresponding infinite chain in this relation (Thm. 1 below).

More precisely, $\operatorname {\mathit{binunf}}(P)$ is defined as the least fixed point of a function $T_P^{\beta }$ on the power set of $\Re$ . For all $U \subseteq \Re$ , $T_P^{\beta }(U)$ is constructed by unfolding prefixes of right-hand sides of rules from $P$ using $U$ . Let $(u,\left \langle {v_1,\ldots ,v_m}\right \rangle ) \in P$ :

1. (i) for each $1 \leq i \leq m$ , one unfolds $v_1,\ldots ,v_{i-1}$ with $(u_1,{\mathsf{e}}),\ldots ,(u_{i-1},{\mathsf{e}})$ from $U$ to obtain a corresponding instance of $(u,v_i)$ ,

2. (ii) for each $1 \leq i \leq m$ , one unfolds $v_1,\ldots ,v_{i-1}$ with $(u_1,{\mathsf{e}}),\ldots ,(u_{i-1},{\mathsf{e}})$ from $U$ and $v_i$ with $(u_i,v)$ from $U$ to obtain a corresponding instance of $(u,v)$ ,

3. (iii) one unfolds $v_1,\ldots ,v_m$ with $(u_1,{\mathsf{e}}),\ldots ,(u_m,{\mathsf{e}})$ from $U$ to obtain a corresponding instance of $(u,{\mathsf{e}})$ .

This is formally expressed as follows, using the set $\operatorname {\mathit{id}}$ of identity binary rules, which consists of every pair $(\operatorname {\mathsf{f}}(x_1,\ldots ,x_m),\operatorname {\mathsf{f}}(x_1,\ldots ,x_m))$ where $\operatorname {\mathsf{f}} \in \Sigma ^{(m)}$ and $x_1,\ldots ,x_m$ are distinct variables. The use of $\operatorname {\mathit{id}}$ allows one to cover case (i) above.

Definition 3. For all programs $P$ and all $U \subseteq \Re$ , we let

\begin{equation*}T_P^{\beta }(U) = \big [(u,{\mathsf{e}}) \in P\big ] \cup \left [\left (u\theta , v\theta \right ) \; \middle | \; \begin{array}{l} r = (u,\left \langle {v_1,\ldots ,v_m}\right \rangle ) \in P,\ 1 \leq i \leq m \\ \left \langle {(u_1,{\mathsf{e}}), \ldots , (u_{i-1},{\mathsf{e}}), (u_i,v)}\right \rangle \ll _r U \cup {\operatorname {\mathit{id}}} \\ \text{if $i \lt m$ then $v \neq {\mathsf{e}}$} \\ \theta \in \operatorname {\mathit{mgu}}\big (\left \langle {u_1,\ldots ,u_i}\right \rangle , \left \langle {v_1,\ldots ,v_i}\right \rangle \big ) \end{array} \right ]\end{equation*}

The binary unfolding of $P$ is the set of binary rules $\operatorname {\mathit{binunf}}(P) = \big (T_P^{\beta }\big )^*(\emptyset )$ .

Intuitively, each $(u,v) \in \operatorname {\mathit{binunf}}(P)$ specifies that some instance of $v$ belongs to $\operatorname {\mathit{calls}}_P(u)$ . More generally, we have:

Proposition 1. Let $P$ be a program, $(u,v) \in \operatorname {\mathit{binunf}}(P)$ and $\sigma \in S(\Sigma , X)$ . Then, for some $\theta \in S(\Sigma , X)$ , we have $v\theta \in \operatorname {\mathit{calls}}_P(u\sigma )$ .

Example 1. Let $P$ be the program which consists of the rules

\begin{equation*} \begin{array}{ccl ccl} r_1 & = & {\left (\operatorname {\mathsf{while}}(x, y), \left \langle {{\operatorname {\mathsf{gt}}}(x, y), \operatorname {\mathsf{add}}(x, y, z), \operatorname {\mathsf{while}}(z, \operatorname {\mathsf{s}}(y))}\right \rangle \right )} \\ r_2 & = & \left ({\operatorname {\mathsf{gt}}}(\operatorname {\mathsf{s}}(x), {\mathsf{0}}), {\mathsf{e}}\right ) & r_3 & = & \left ({\operatorname {\mathsf{gt}}}(\operatorname {\mathsf{s}}(x), \operatorname {\mathsf{s}}(y)), {\operatorname {\mathsf{gt}}}(x, y)\right ) \\ r_4 & = & \left (\operatorname {\mathsf{add}}(x, {\mathsf{0}}, x),{\mathsf{e}}\right ) & r_5 & = & \left (\operatorname {\mathsf{add}}(x, \operatorname {\mathsf{s}}(y), \operatorname {\mathsf{s}}(z)), \operatorname {\mathsf{add}}(x, y, z)\right ) \\ r_6 & = & \left (\operatorname {\mathsf{while}}(x, y), \operatorname {\mathsf{le}}(x, y)\right ) & & & \\ r_7 & = & \left (\operatorname {\mathsf{le}}({\mathsf{0}}, x), {\mathsf{e}}\right ) & r_8 & = & \left (\operatorname {\mathsf{le}}(\operatorname {\mathsf{s}}(x), \operatorname {\mathsf{s}}(y)), \operatorname {\mathsf{le}}(x, y)\right ) \\ \end{array}\end{equation*}

and which corresponds to the imperative program fragment

$$while(x{\rm{ > }}\;y)\{ x = x + y;y = y + 1;\} $$

Rule $r_1$ is used to continue the loop and $r_6$ is used to stop it. Note that this imperative fragment does not terminate if it is run from integers $x,y$ such that $x \gt y \gt 0$ .

Let us compute some elements of $\operatorname {\mathit{binunf}}(P)$ by applying Def. 3 .

• • Obviously, we have $[r_2] \cup [r_4] \subseteq T_P^{\beta }(\emptyset )$ .

• • Let us unfold the whole right-hand side of $r_1$ , that is let us consider $i = m = 3$ . We have

  \begin{align*} \big \langle \left ({\operatorname {\mathsf{gt}}}(\operatorname {\mathsf{s}}(x_1), {\mathsf{0}}), {\mathsf{e}}\right)\!, & \left (\operatorname {\mathsf{add}}(x_2, {\mathsf{0}}, x_2), {\mathsf{e}}\right ), \\ & \left (\operatorname {\mathsf{while}}(x_3, y_3), \operatorname {\mathsf{while}}(x_3, y_3)\right ) \big \rangle \ll _{r_1} [r_2] \cup [r_4] \cup {\operatorname {\mathit{id}}} \end{align*}
  and $\theta = \{x \mapsto \operatorname {\mathsf{s}}(x_1), y \mapsto {\mathsf{0}}, z \mapsto \operatorname {\mathsf{s}}(x_1), x_2 \mapsto \operatorname {\mathsf{s}}(x_1), x_3 \mapsto \operatorname {\mathsf{s}}(x_1), y_3 \mapsto \operatorname {\mathsf{s}}({\mathsf{0}})\}$ is the mgu of $\left \langle {{\operatorname {\mathsf{gt}}}(\operatorname {\mathsf{s}}(x_1), {\mathsf{0}}), \operatorname {\mathsf{add}}(x_2, {\mathsf{0}}, x_2), \operatorname {\mathsf{while}}(x_3, y_3)}\right \rangle$ and $\langle {\operatorname {\mathsf{gt}}}(x, y), \operatorname {\mathsf{add}}(x, y, z), $ $\operatorname {\mathsf{while}}(z, \operatorname {\mathsf{s}}(y)) \rangle$ . Consequently, the set $(T_P^{\beta })^2(\emptyset )$ contains the binary rule $\left (\operatorname {\mathsf{while}}(x, y)\theta , \operatorname {\mathsf{while}}(x_3, y_3)\theta \right )$ $= \left (\operatorname {\mathsf{while}}\left (\operatorname {\mathsf{s}}(x_1), {\mathsf{0}}\right ), \operatorname {\mathsf{while}}\left (\operatorname {\mathsf{s}}(x_1), \operatorname {\mathsf{s}}({\mathsf{0}})\right )\right )$ .

• • More generally, we have $[r'_n \mid n \in \mathbb{N}] \subseteq \operatorname {\mathit{binunf}}(P)$ where, for all $n \in \mathbb{N}$ , $r'_n = \left (\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+1}(x),\operatorname {\mathsf{s}}^n({\mathsf{0}})), \operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{2n+1}(x),\operatorname {\mathsf{s}}^{n+1}({\mathsf{0}})) \right )$ .

The binary unfolding exhibits the termination properties of a program:

Theorem 1 (see the paper by Codish and Taboch (Reference Codish and Taboch1999)). Let $P$ be a program and $\overline {{s}}$ be a sequence of terms. Then, there is an infinite $\mathop {\Rightarrow }_P$ -chain that starts from $\overline {{s}}$ if and only if there is an infinite $\mathop {\Rightarrow }_{\operatorname {\mathit{binunf}}(P)}$ -chain that starts from $\overline {{s}}$ .

Example 2 (Ex. 1 cont.). For all $n \gt m \gt 0$ , we have the infinite $\mathop {\Rightarrow }_P$ -chain

\begin{align*} \left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^n({\mathsf{0}}), \operatorname {\mathsf{s}}^m({\mathsf{0}}))}\right \rangle & \mathop {\big (\mathop {\Rightarrow }_{r_1} \circ \mathop {\Rightarrow }_{r_3}^m \circ \mathop {\Rightarrow }_{r_2} \circ \mathop {\Rightarrow }_{r_5}^m \circ \mathop {\Rightarrow }_{r_4}\big )} \left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+m}(0), \operatorname {\mathsf{s}}^{m+1}(0))}\right \rangle \\ & \mathop {\big (\mathop {\Rightarrow }_{r_1} \circ \mathop {\Rightarrow }_{r_3}^{m+1} \circ \mathop {\Rightarrow }_{r_2} \circ \mathop {\Rightarrow }_{r_5}^{m+1} \circ \mathop {\Rightarrow }_{r_4}\big )} \cdots \end{align*}

and also the infinite $\mathop {\Rightarrow }_{\operatorname {\mathit{binunf}}(P)}$ -chain

\begin{equation*}\left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^n({\mathsf{0}}), \operatorname {\mathsf{s}}^m({\mathsf{0}}))}\right \rangle \mathop {\mathop {\Rightarrow }_{r'_m}} \left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+m}(0), \operatorname {\mathsf{s}}^{m+1}(0))}\right \rangle \mathop {\mathop {\Rightarrow }_{r'_{m+1}}} \cdots \end{equation*}

We note that none of these chains embeds a loop: in the $\mathop {\Rightarrow }_P$ -chain, the number of applications of $r_3$ and $r_5$ gradually increases and, in the $\mathop {\Rightarrow }_{\operatorname {\mathit{binunf}}(P)}$ -chain, a new binary rule (not occurring before) is used at each step.

3 Patterns

In this section, we describe our adaptation to logic programing of the pattern approach introduced by Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012). Our main idea is similar to that of Payet and Mesnard (Reference Payet and Mesnard2006), that is, unfold the program and try to prove its non-termination from the resulting set. To this end, based on the binary unfolding mentioned previously (Def. 3), we introduce a new unfolding technique that produces patterns of rules (Def. 10) and a sufficient condition to non-termination that we apply to the generated patterns (Thm. 3).

First, we recall the definition of pattern term and pattern rule, that we formulate differently from Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012) to fit our needs. In particular, we introduce the concept of pattern substitution.

For instance, if $\sigma = \{x \mapsto \operatorname {\mathsf{s}}(x), y \mapsto \operatorname {\mathsf{s}}(y)\}$ and $\mu = \{x \mapsto \operatorname {\mathsf{s}}(x), y\mapsto {\mathsf{0}}\}$ then $\theta = {{\sigma } \star {\mu }}$ is a pattern substitution. For all $n \in \mathbb{N}$ , we have $\theta (n) = \sigma ^n\mu = \{x\mapsto \operatorname {\mathsf{s}}^{n+1}(x), y\mapsto \operatorname {\mathsf{s}}^n({\mathsf{0}})\}$ .

From pattern substitutions, we define pattern terms.

For instance, $p = {{{\operatorname {\mathsf{gt}}}(x,y)} \star {\{x \mapsto \operatorname {\mathsf{s}}(x), y \mapsto \operatorname {\mathsf{s}}(y)\}}} \star {\{x \mapsto \operatorname {\mathsf{s}}(x), y\mapsto {\mathsf{0}}\}}$ is a pattern term. For all $n \in \mathbb{N}$ , we have $p(n) = {\operatorname {\mathsf{gt}}}(\operatorname {\mathsf{s}}^{n+1}(x),\operatorname {\mathsf{s}}^n({\mathsf{0}}))$ .

Then, from pattern terms one can define pattern rules.

Example 3 (Ex. 1 cont.). Let $u = \operatorname {\mathsf{while}}(x,y)$ be the left-hand side of $r_1$ , $\sigma = \{x\mapsto \operatorname {\mathsf{s}}(x), y\mapsto \operatorname {\mathsf{s}}(y)\}$ , $\sigma ' = \{x\mapsto \operatorname {\mathsf{s}}(x)\}$ and $\mu = \{x\mapsto \operatorname {\mathsf{s}}(x), y\mapsto {\mathsf{0}}\}$ . The pattern terms

\begin{align*} p &= {{u\sigma } \star {\sigma } \star {\mu }} = {{\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}(x),\operatorname {\mathsf{s}}(y))} \star {\sigma } \star {\mu }} \\ q &= {{u\sigma ^2} \star {\sigma \sigma '} \star {\mu }} = {{\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^2(x),\operatorname {\mathsf{s}}^2(y))} \star { } } {\{x\mapsto \operatorname {\mathsf{s}}^2(x), y\mapsto \operatorname {\mathsf{s}}(y)\}} \star { } {\mu } \end{align*}

respectively describe the sets of terms $\left \{p(n) = \operatorname {\mathsf{while}}\left (\operatorname {\mathsf{s}}^{n+2}(x),\operatorname {\mathsf{s}}^{n+1}({\mathsf{0}})\right ) \;\middle \vert \; n \in \mathbb{N}\right \}$ and $\left \{q(n) = \operatorname {\mathsf{while}}\left (\operatorname {\mathsf{s}}^{2n+3}(x),\operatorname {\mathsf{s}}^{n+2}({\mathsf{0}})\right ) \;\middle \vert \; n \in \mathbb{N}\right \}$ . Moreover,

\begin{align*} \operatorname {\mathit{rules}}((p,q)) &= \left \{\left (\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+2}(x),\operatorname {\mathsf{s}}^{n+1}({\mathsf{0}})), \operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{2n+3}(x),\operatorname {\mathsf{s}}^{n+2}({\mathsf{0}}))\right ) \;\middle \vert \; n \in \mathbb{N}\right \} \\ &= \left \{r'_n \mid n \gt 0\right \} \subseteq \left \{r'_n \mid n \in \mathbb{N}\right \} \subseteq \operatorname {\mathit{binunf}}(P) \text{ (see Ex. 1)} \end{align*}

The notion of correctness of a pattern rule is defined by Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012) in the context of term rewriting. We reformulate it as follows in logic programing.

So, if a pattern rule $(p,q)$ is correct w.r.t. $P$ then, for all $n \in \mathbb{N}$ , we have $(p(n),q(n)) \in \operatorname {\mathit{binunf}}(P)$ , that is, by Prop. 1, $\left \langle {p(n)}\right \rangle \mathop {\Rightarrow }^+_P \left \langle {q(n)\theta ,\ldots }\right \rangle$ for some $\theta \in S(\Sigma , X)$ . Intuitively, this means that for all $n \in \mathbb{N}$ , a call to $p(n)$ necessarily leads to a call to $q(n)$ . For instance, in Ex. 3, we have $\operatorname {\mathit{rules}}((p,q)) \subseteq \operatorname {\mathit{binunf}}(P)$ , hence $(p,q)$ is correct w.r.t. $P$ and we have $\left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+2}(x),\operatorname {\mathsf{s}}^{n+1}({\mathsf{0}}))}\right \rangle \mathop {\Rightarrow }_P^+ \left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{2n+3}(x),\operatorname {\mathsf{s}}^{n+2}({\mathsf{0}}))}\right \rangle$ for all $n \in \mathbb{N}$ .

The next result allows one to infer correct pattern rules from a program. It considers pairs of rules that have the same form as $(r_2,r_3)$ , $(r_4,r_5)$ and $(r_7,r_8)$ in Ex. 1. It uses the set of contexts $\chi ^{(1)}$ (see Sect. 2.3).

Proposition 2. Suppose that a program $P$ contains two binary rules $r = (u,v)$ and $r' = (u', {\mathsf{e}})$ such that

• • $u = c(c_1(x_1),\ldots ,c_m(x_m))$ , $v = c(x_1,\ldots ,x_m)$ and $u' = c(t_1,\ldots ,t_m)$ ,

• • $\{c_1,\ldots ,c_m\} \subseteq \chi ^{(1)}$ and $c$ is an $m$ -context with $\operatorname {\mathit{Var}}(c) = \emptyset$ ,

• • $x_1,\ldots ,x_m$ are distinct variables and $t_1,\ldots ,t_m$ are terms.

Then, $(p,{{\mathsf{e}}}^{\star })$ and $(q,{v}^{\star })$ are correct w.r.t. $P$ where $p = {{v} \star {\sigma } \star {\mu }}$ , $q = {{u} \star {\sigma } \star {\emptyset }}$ and

\begin{align*} \sigma &= \{x_k \mapsto c_k(x_k) \mid 1 \leq k \leq m,\ c_k(x_k) \neq x_k \} \\ \mu &= \{x_k \mapsto t_k \mid 1 \leq k \leq m,\ t_k \neq x_k\} \end{align*}

Unification for pattern terms is not considered by Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012). As we need it in our development (see Def. 10 below), we define it here.

for example, if $p = {{\operatorname {\mathsf{f}}(x,y)} \star {} {\{x \mapsto \operatorname {\mathsf{s}}(x)\}}}\star {\{x \mapsto {\mathsf{0}}\}}$ and $q = {{\operatorname {\mathsf{f}}(x,y)} {} \star {\{y \mapsto \operatorname {\mathsf{s}}(y)\}}}\star {\{y \mapsto {\mathsf{1}}\}}$ , then $\theta = {{\{x \mapsto \operatorname {\mathsf{s}}(x), y \mapsto \operatorname {\mathsf{s}}(y)\}} \star { }}{\{x \mapsto {\mathsf{0}},y \mapsto {\mathsf{1}}\}}$ is a unifier of $p$ and $q$ . Indeed, for all $n \in \mathbb{N}$ , $\theta (n) = \{x \mapsto \operatorname {\mathsf{s}}^n({\mathsf{0}}), y \mapsto \operatorname {\mathsf{s}}^n({\mathsf{1}})\}$ is a unifier of $p(n) = \operatorname {\mathsf{f}}(\operatorname {\mathsf{s}}^n({\mathsf{0}}),y)$ and $p(n) = \operatorname {\mathsf{f}}(x,\operatorname {\mathsf{s}}^n({\mathsf{1}}))$ . All these notions are naturally extended to finite sequences of pattern terms.

We also need to adapt the notion of equivalence class modulo renaming (see Sect. 2.4).

Hence, $[r]$ consists of all pattern rules $r'$ that describe a subset of $[\operatorname {\mathit{rules}}(r)]$ . For instance, if $p = {{\operatorname {\mathsf{f}}(x,y)} {} \star {\{x \mapsto \operatorname {\mathsf{s}}(x)\}}}\star {\{x \mapsto {\mathsf{0}}\}}$ and $p' = {{\operatorname {\mathsf{f}}(\operatorname {\mathsf{s}}(x),y')} {} \star {\{x \mapsto \operatorname {\mathsf{s}}(x)\}}}\star {\{x \mapsto {\mathsf{0}}\}}$ , then $(p',{{\mathsf{e}}}^{\star }) \in [(p,{{\mathsf{e}}}^{\star })]$ . Indeed, we have $\operatorname {\mathit{rules}}((p',{{\mathsf{e}}}^{\star })) = \{(\operatorname {\mathsf{f}}(\operatorname {\mathsf{s}}^{n+1}({\mathsf{0}}),y'),{\mathsf{e}}) \mid n \in \mathbb{N}\} \subseteq [(\operatorname {\mathsf{f}}(\operatorname {\mathsf{s}}^n({\mathsf{0}}),y),{\mathsf{e}}) \mid n \in \mathbb{N}] = [\operatorname {\mathit{rules}}((p,{{\mathsf{e}}}^{\star }))]$ .

Now we provide a counterpart of Def. 3 (binary unfolding) for pattern rules. A notable difference, however, is the use of an arbitrary set $B$ instead of $E = \left \{\left ({u}^{\star },{{\mathsf{e}}}^{\star }\right ) \;\middle \vert \; (u,{\mathsf{e}}) \in P \right \}$ . The set $B$ plays a similar role to the pattern creation inference rules of Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012). Using suitable sets $B$ ’s (as those consisting of rules provided by Prop. 2), we get an approach that computes pattern rules finitely describing infinite subsets of $\operatorname {\mathit{binunf}}(P)$ , that is, an approach that unfolds “faster” (see Ex. 5). We let $\operatorname {\mathit{patid}}$ denote the set of all pairs $({\operatorname {\mathsf{f}}(x_1,\ldots ,x_m)}^{\star },{\operatorname {\mathsf{f}}(x_1,\ldots ,x_m)}^{\star })$ where $\operatorname {\mathsf{f}} \in \Sigma ^{(m)}$ and $x_1,\ldots ,x_m$ are distinct variables. For all rules $r$ , the notation $\ll _r$ is naturally extended to sets of pattern rules, according to the following definitions: the set of variables of a pattern term $p = {{s} \star {\sigma } \star {\mu }}$ is $\operatorname {\mathit{Var}}(p) = \operatorname {\mathit{Var}}(s) \cup \operatorname {\mathit{Var}}(\sigma ) \cup \operatorname {\mathit{Var}}(\mu )$ and that of a pattern rule $r = (p,q)$ is $\operatorname {\mathit{Var}}(r) = \operatorname {\mathit{Var}}(p) \cup \operatorname {\mathit{Var}}(q)$ .

Definition 10. For all programs $P$ and all $B,U \subseteq \Im$ , we let

\begin{equation*}T_{P,B}^{\pi }(U) = [B] \cup \left [\begin{array}{l} ({{u} \star {\sigma } \star {\mu }}, \\ \ {{v} \star {\sigma _i\sigma } \star {\mu _i\mu }} ) \end{array} \; \middle | \; \begin{array}{l} r = (u,\left \langle {v_1,\ldots ,v_m}\right \rangle ) \in P,\ 1 \leq i \leq m \\ \left \langle {(p_1,{{\mathsf{e}}}^{\star }), \ldots , (p_{i-1},{{\mathsf{e}}}^{\star }), (p_i,{{v} \star {\sigma _i} \star {\mu _i}})}\right \rangle \\ \quad \qquad \ll _r U \cup \operatorname {\mathit{patid}} \\ \text{if $i \lt m$ then $v \neq {\mathsf{e}}$} \\ {{\sigma } \star {\mu }} \in \operatorname {\mathit{mgu}}\left (\left \langle {p_1,\ldots ,p_i}\right \rangle , \left \langle {{v_1}^{\star },\ldots ,{v_i}^{\star }}\right \rangle \right )\\ \text{$\sigma $ commutes with $\sigma _i$ and $\mu _i$} \end{array}\right ]\end{equation*}

The pattern unfolding of $P$ using $B$ is the set $\operatorname {\mathit{patunf}}(P,B) = \big (T_{P,B}^{\pi }\big )^*(\emptyset )$ .

Example 5. Let $B$ be the set consisting of the rules $(p_1,{{\mathsf{e}}}^{\star })$ and $(p_2,{{\mathsf{e}}}^{\star })$ of Ex. 4 . Let us compute some elements of $\operatorname {\mathit{patunf}}(P,B)$ by applying Def. 10 . We have $\{(p'_1,{{\mathsf{e}}}^{\star }), (p'_2,{{\mathsf{e}}}^{\star })\} \subseteq [B] \subseteq T_{P,B}^{\pi }(\emptyset )$ where $p'_1$ and $p'_2$ are renamed versions of $p_1$ and $p_2$ respectively:

\begin{align*} p'_1 &= {{{\operatorname {\mathsf{gt}}}(x_1,y_1)} \star { } } {\left \{x_1 \mapsto \operatorname {\mathsf{s}}(x_1), y_1 \mapsto \operatorname {\mathsf{s}}(y_1)\right \}} \star { } {\left \{x_1 \mapsto \operatorname {\mathsf{s}}(x_1), y_1 \mapsto {\mathsf{0}}\right \}} \\ p'_2 &= {{\operatorname {\mathsf{add}}(x_2,y_2,z_2)} \star { }} {\left \{ y_2 \mapsto \operatorname {\mathsf{s}}(y_2), z_2 \mapsto \operatorname {\mathsf{s}}(z_2)\right \}} \star { } {\left \{ y_2 \mapsto {\mathsf{0}}, z_2 \mapsto x_2\right \}} \end{align*}

Let us unfold the whole right-hand side of $r_1 \in P$ , i.e., let us consider $i = m = 3$ . We have $\left \langle {(p'_1,{{\mathsf{e}}}^{\star }),(p'_2,{{\mathsf{e}}}^{\star }),(p'_3,p'_3)}\right \rangle \ll _{r_1} T_{P,B}^{\pi }(\emptyset ) \cup \operatorname {\mathit{patid}}$ where $p'_3 = {\operatorname {\mathsf{while}}(x_3,y_3)}^{\star }$ . The right-hand side of $r_1$ is $\left \langle {v_1, v_2, v_3}\right \rangle = \left \langle {{\operatorname {\mathsf{gt}}}(x, y), \operatorname {\mathsf{add}}(x, y, z), \operatorname {\mathsf{while}}(z, \operatorname {\mathsf{s}}(y))}\right \rangle$ . Let $S = \left \langle {{v_1}^{\star },{v_2}^{\star },{v_3}^{\star }}\right \rangle$ and $S' = \left \langle {p'_1,p'_2,p'_3}\right \rangle$ . We show in Ex. 11 that ${{\rho } \star {\nu }} \in \operatorname {\mathit{mgu}}(S,S')$ where

\begin{align*} \rho &= \big \{ x \mapsto \operatorname {\mathsf{s}}(x), \ y \mapsto \operatorname {\mathsf{s}}(y),\ z \mapsto \operatorname {\mathsf{s}}^2(z),\ x_2 \mapsto \operatorname {\mathsf{s}}(x_2),\ x_3 \mapsto \operatorname {\mathsf{s}}^2(x_3),\ y_3 \mapsto \operatorname {\mathsf{s}}(y_3)\big \} \\ \nu &= \big \{ x \mapsto \operatorname {\mathsf{s}}(x_1), \ y \mapsto {\mathsf{0}},\ z \mapsto \operatorname {\mathsf{s}}(x_1),\ x_2 \mapsto \operatorname {\mathsf{s}}(x_1),\ x_3 \mapsto \operatorname {\mathsf{s}}(x_1),\ y_3 \mapsto \operatorname {\mathsf{s}}({\mathsf{0}})\big \} \end{align*}

So, $r'' = \left ({{u} \star {\rho } \star {\nu }}, {{\operatorname {\mathsf{while}}(x_3,y_3)} \star {\rho } \star {\nu }}\right ) \in (T_{P,B}^{\pi })^2(\emptyset )$ where $u = \operatorname {\mathsf{while}}(x, y)$ is the left-hand side of $r_1$ . It describes the set of binary rules

\begin{equation*}\left \{r''_n = \left (\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+1}(x_1),\operatorname {\mathsf{s}}^n({\mathsf{0}})), \operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{2n+1}(x_1),\operatorname {\mathsf{s}}^{n+1}({\mathsf{0}})) \right ) \;\middle \vert \; n \in \mathbb{N}\right \}\end{equation*}

and we have $[r''_n \mid n \in \mathbb{N}] = [r'_n \mid n \in \mathbb{N}]$ (see Ex. 1 ).

The following result corresponds to the Soundness Thm. 7 of Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012).

Finally, we adapt the non-termination criterion of Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012) to our setting.

Example 6. Let us regard the pattern rule $(p,q)$ of Ex. 3 . As $rules((p,q)) \subseteq \left \{r'_n \mid n \in \mathbb{N}\right \}$ with $\left \{r'_n \mid n \in \mathbb{N}\right \} \subseteq \left [r'_n \mid n \in \mathbb{N}\right ] = \left [{r''}_n \mid n \in \mathbb{N}\right ] = [\operatorname {\mathit{rules}}(r'')]$ (see Ex. 5 ), we have $(p,q) \in [r''] \subseteq \operatorname {\mathit{patunf}}(P,B)$ . Moreover, $B$ is correct w.r.t. $P$ (see Ex. 4 ) and $(p,q) = ({{u\sigma } \star {\sigma } \star {\mu }}, {{(u\sigma )\sigma } \star {\sigma \sigma '} \star {\mu }})$ (see Ex. 3 ) where $\sigma '$ commutes with $\sigma$ and $\mu$ . By Thm. 3 , for all $m,n \in \mathbb{N}$ and $\theta = \{x \mapsto \operatorname {\mathsf{s}}^n({\mathsf{0}})\}$ , the sequence $\left \langle {(u\sigma )\sigma ^m\mu \theta }\right \rangle$ starts an infinite $\mathop {\Rightarrow }_P$ -chain, with

\begin{equation*}\left \langle {(u\sigma )\sigma ^m\mu \theta }\right \rangle = \left \langle {u\sigma ^{m+1}\mu \theta }\right \rangle = \left \langle {\operatorname {\mathsf{while}}\left (\operatorname {\mathsf{s}}^{(n+1)+(m+1)}({\mathsf{0}}), \operatorname {\mathsf{s}}^{m+1}({\mathsf{0}})\right )}\right \rangle \end{equation*}

Hence, for all $n \gt m \gt 0$ , the sequence $\left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^n({\mathsf{0}}), \operatorname {\mathsf{s}}^m({\mathsf{0}}))}\right \rangle$ starts an infinite $\mathop {\Rightarrow }_P$ -chain. This had already been observed in Ex. 2 .

4 Simple patterns

In practice, to implement the approach presented in the previous section, one has to find a way to compute mgu’s of pattern terms and to check the non-termination condition of Thm. 3. In this section, we introduce a class of pattern terms of a special form, called simple, that is more restrictive but for which we provide a unification algorithm as well as a non-termination criterion that is easier to check than that of Thm. 3. We describe them using a new signature that consists of unary symbols only:

\begin{equation*}\Upsilon = \left \{ c^{a,b} : \text{a unary symbol} \;\middle \vert \; c \in \chi ^{(1)}, (a,b) \in \mathbb{N}^2\right \}\end{equation*}

Any symbol $c^{a,b} \in \Upsilon$ represents all successive embeddings of $c$ into itself of the form $c^{a \times n + b}$ where $n \in \mathbb{N}$ . Hence, for all $u \in T(\Sigma \cup \Upsilon , X)$ and all $n \in \mathbb{N}$ , we let $u(n)$ be the element of $T(\Sigma , X)$ obtained from $u$ by replacing every $c^{a,b} \in \Upsilon$ by $c^{a \times n + b}$ . Moreover, for all $\theta \in S(\Sigma \cup \Upsilon , X)$ and all $n \in \mathbb{N}$ , we let $\theta (n)$ be the element of $S(\Sigma , X)$ defined as: for all $x \in X$ , $(\theta (n))(x) = (\theta (x))(n)$ . In the rest of this section, we consider elements of $T(\Sigma \cup \Upsilon , X)$ modulo the following equivalence relation.

The following straightforward result can be used to simplify $(\Sigma \cup \Upsilon )$ -terms.

Definition 12. A pattern term $p = {{s} \star {\sigma } \star {\mu }}$ is called simple if, for all $x \in \operatorname {\mathit{Var}}(s)$ , $\sigma (x) = c^a(x)$ and $\mu (x) = c^b(t)$ for some $c \in \chi ^{(1)}$ , $a,b \in \mathbb{N}$ and $t \in T(\Sigma , X)$ . Then, we let $\upsilon (p) = [s\theta _p]$ where

\begin{equation*}\theta _p = \left \{x \mapsto u \;\middle \vert \; \begin{array}{l} x \in \operatorname {\mathit{Var}}(s),\ \sigma (x) = c^a(x),\ \mu (x) = c^b(t) \\ \text{if $\sigma (x) = x$ then $u = \mu (x)$ else $u = c^{a,b}(t)$} \end{array}\right \}\end{equation*}

A pattern rule $(p,q)$ is called simple if $p$ and $q$ are simple.

The next result follows from Def. 5 and Def. 12.

We note that the pattern rules $(p,{{\mathsf{e}}}^{\star })$ and $(q,{v}^{\star })$ produced from Prop. 2 are simple. In Ex. 4, $\upsilon (p_1) = [{\operatorname {\mathsf{gt}}}(c_1^{1,1}(x), c_1^{1,0}({\mathsf{0}}))]$ and $\upsilon (p_2) = [\operatorname {\mathsf{add}}(x, c_1^{1,0}({\mathsf{0}}), c_1^{1,0}(x))]$ where $c_1 = \operatorname {\mathsf{s}}(\square _1)$ .

Example 9. We illustrate the fact that non-termination detection with simple pattern terms is more restrictive than with the full class of pattern terms. Let $P$ be the program consisting of

\begin{align*} r_1 &= \left (\operatorname {\mathsf{while}}(x,y), \left \langle {\operatorname {\mathsf{isList}}(y), \operatorname {\mathsf{while}}(x,\operatorname {\mathsf{cons}}(x,y))}\right \rangle \right ) \\ r_2 & = \left (\operatorname {\mathsf{isList}}(\operatorname {\mathsf{nil}}),{\mathsf{e}}\right ) \\ r_3 &= \left (\operatorname {\mathsf{isList}}(\operatorname {\mathsf{cons}}(x,y)),\operatorname {\mathsf{isList}}(y)\right ) \end{align*}

Let $c = \operatorname {\mathsf{cons}}(x,\square _1)$ . Then, $\left \langle {\operatorname {\mathsf{while}}(x,c^n(\operatorname {\mathsf{nil}}))}\right \rangle \mathop {(\mathop {\Rightarrow }_{r_1} \circ \mathop {\Rightarrow }^n_{r_3} \circ \mathop {\Rightarrow }_{r_2})} \left \langle {\operatorname {\mathsf{while}}(x,c^{n+1}(\operatorname {\mathsf{nil}}))}\right \rangle$ holds for all $n \in \mathbb{N}$ . Hence, for all $n \in \mathbb{N}$ , $\left \langle {\operatorname {\mathsf{while}}(x,c^n(\operatorname {\mathsf{nil}}))}\right \rangle$ starts an infinite $\mathop {\Rightarrow }_P$ -chain. To detect that, one would need an unfolded pattern rule of the form $({{\operatorname {\mathsf{while}}(x,y)} \star {\sigma } \star {\mu }}, {{\operatorname {\mathsf{while}}(x,y)\sigma } \star {\sigma } \star {\mu }})$ where $\sigma = \{y \mapsto c(y)\}$ and $\mu = \{y \mapsto \operatorname {\mathsf{nil}}\}$ . Such a rule satisfies the condition of Thm. 3 but is not simple because $c \not \in \chi ^{(1)}$ ( $\operatorname {\mathit{Var}}(c) \neq \emptyset$ ).

We also define simple substitutions.

The next result follows from Def. 4 and Def. 13.

4.1 Unification of simple pattern terms

For all sequences $S = \left \langle {p_1,\ldots ,p_m}\right \rangle$ of simple pattern terms, we define $\upsilon (S) = \left \{\left \langle {u_1,\ldots ,u_m}\right \rangle \mid u_1 \in \upsilon (p_1), \ldots , u_m \in \upsilon (p_m) \right \}$ .

Unification Algorithm 1

Let $S$ and $S'$ be sequences of simple pattern terms. Let $S_1 \in \upsilon (S)$ and $S'_1 \in \upsilon (S')$ .

• • If $\operatorname {\mathit{mgu}}(S_1,S'_1)$ contains a simple substitution $\theta$ then return $\upsilon ^{-1}(\theta )$

• • else halt with failure.

Partial correctness follows from the next theorem.

Theorem 4. If the unification algorithm successfully terminates then it produces a pattern substitution which is an mgu of the input sequences.

In practice, as $S_1$ and $S'_1$ are sequences of elements of $T(\Sigma \cup \Upsilon , X)$ , one can use any classical unification algorithm (Robinson, Martelli-Montanari…) to compute $\theta \in \operatorname {\mathit{mgu}}(S_1,S'_1)$ . Then, it suffices to check whether $\theta$ is simple, for instance using Lem. 1.

Example 11 (Related to Ex. 5). Consider the sequences of simple pattern terms $S = \left \langle {{v_1}^{\star },{v_2}^{\star },{v_3}^{\star }}\right \rangle$ and $S' = \left \langle {p'_1,p'_2,p'_3}\right \rangle$ where $v_1 = {\operatorname {\mathsf{gt}}}(x,y)$ , $v_2 = \operatorname {\mathsf{add}}(x,y,z)$ , $v_3 = \operatorname {\mathsf{while}}(z, \operatorname {\mathsf{s}}(y))$ and

\begin{align*} p'_1 &= {{{\operatorname {\mathsf{gt}}}(x_1,y_1)} \star { } } {\left \{x_1 \mapsto \operatorname {\mathsf{s}}(x_1), y_1 \mapsto \operatorname {\mathsf{s}}(y_1)\right \}} \star { } {\left \{x_1 \mapsto \operatorname {\mathsf{s}}(x_1), y_1 \mapsto {\mathsf{0}}\right \}} \\ p'_2 &= {{\operatorname {\mathsf{add}}(x_2,y_2,z_2)} \star { }} {\left \{ y_2 \mapsto \operatorname {\mathsf{s}}(y_2), z_2 \mapsto \operatorname {\mathsf{s}}(z_2)\right \}} \star { } {\left \{ y_2 \mapsto {\mathsf{0}}, z_2 \mapsto x_2\right \}} \\ p'_3 &= {\operatorname {\mathsf{while}}(x_3, y_3)}^{\star } \end{align*}

Let $c = \operatorname {\mathsf{s}}(\square _1)$ , $S_1 = \left \langle {{\operatorname {\mathsf{gt}}}(x,y), \operatorname {\mathsf{add}}(x,y,z), \operatorname {\mathsf{while}}(z, c(y))}\right \rangle$ and

\begin{equation*}S'_1 = \big \langle { {\operatorname {\mathsf{gt}}}\big (c^{1,1}(x_1),c^{1,0}({\mathsf{0}})\big ), \operatorname {\mathsf{add}}\big (x_2,c^{1,0}({\mathsf{0}}),c^{1,0}(x_2)\big ), \operatorname {\mathsf{while}}(x_3, y_3)}\big \rangle \end{equation*}

We have $S_1 \in \upsilon (S)$ and $S'_1 \in \upsilon (S')$ . Moreover, $\theta \in \operatorname {\mathit{mgu}}(S_1,S'_1)$ where

\begin{align*} \theta = \big \{ & x \mapsto c^{1,1}(x_1),\ y \mapsto c^{1,0}({\mathsf{0}}),\ z \mapsto c^{1,0}(c^{1,1}(x_1)), \\ & x_2 \mapsto c^{1,1}(x_1),\ x_3 \mapsto c^{1,0}(c^{1,1}(x_1)),\ y_3 \mapsto c(c^{1,0}({\mathsf{0}}))\big \} \end{align*}

We note that $\theta (x) \in [c^{1,1}(x_1)]$ , $\theta (y) \in [c^{1,0}({\mathsf{0}})]$ , $\theta (z) \in [c^{2,1}(x_1)]$ , $\theta (x_2) \in [c^{1,1}(x_1)]$ , $\theta (x_3) \in [c^{2,1}(x_1)]$ and $\theta (y_3) \in [c^{1,1}({\mathsf{0}})]$ . So, $\theta$ is a simple substitution and the algorithm produces the pattern substitution $\upsilon ^{-1}(\theta ) = {{\rho } \star {\nu }}$ where

\begin{align*} \rho &= \big \{ x \mapsto \operatorname {\mathsf{s}}(x), \ y \mapsto \operatorname {\mathsf{s}}(y),\ z \mapsto \operatorname {\mathsf{s}}^2(z),\ x_2 \mapsto \operatorname {\mathsf{s}}(x_2),\ x_3 \mapsto \operatorname {\mathsf{s}}^2(x_3),\ y_3 \mapsto \operatorname {\mathsf{s}}(y_3)\big \} \\ \nu &= \big \{ x \mapsto \operatorname {\mathsf{s}}(x_1), \ y \mapsto {\mathsf{0}},\ z \mapsto \operatorname {\mathsf{s}}(x_1),\ x_2 \mapsto \operatorname {\mathsf{s}}(x_1),\ x_3 \mapsto \operatorname {\mathsf{s}}(x_1),\ y_3 \mapsto \operatorname {\mathsf{s}}({\mathsf{0}})\big \} \end{align*}

By Thm. 4 , ${{\rho } \star {\nu }} \in \operatorname {\mathit{mgu}}(S,S')$ .

A natural choice for $S_1$ and $S'_1$ in our unification algorithm is to consider, for all $p = {{s} \star {\sigma } \star {\mu }}$ in $S \cup S'$ , the term $s\theta _p \in [s\theta _p]$ (see Def. 12). But this leads to an incomplete approach, that is, an approach that may fail to find a unifier even if one exists.

Example 12. Let $p = {{s} \star {\{x \mapsto c(x)\}} \star {\emptyset }}$ and $q = {{s} \star {\{x \mapsto c_2(x)\}} \star {\{x \mapsto y\}}}$ where $s = \operatorname {\mathsf{f}}(x)$ , $c = \operatorname {\mathsf{s}}(\square _1)$ and $c_2 = c^2$ . Then, $p$ and $q$ are simple. Let $\theta = {{\{x \mapsto c(x)\}} \star {\{x \mapsto y\}}}$ . For all $n \in \mathbb{N}$ , we have $p(n) = \operatorname {\mathsf{f}}(c^n(x))$ and $q(n) = \operatorname {\mathsf{f}}(c^{2n}(y))$ , hence $\theta (n) = \{x \mapsto c^n(y)\}$ is a unifier of $p(n)$ and $q(n)$ . Therefore, $\theta$ is a unifier of $p$ and $q$ . On the other hand, we have $s\theta _p = \operatorname {\mathsf{f}}(c^{1,0}(x))$ and $s\theta _q = \operatorname {\mathsf{f}}(c_2^{1,0}(y))$ . As $c^{1,0}$ and $c_2^{1,0}$ are different symbols, $\operatorname {\mathit{mgu}}(s\theta _p, s\theta _q) = \emptyset$ . So, from $s\theta _p$ and $s\theta _q$ , the unification algorithm fails to find a unifier for $p$ and $q$ . Now, let us choose the term $u = \operatorname {\mathsf{f}}(c^{1,0}(c^{1,0}(y)))$ in $[s\theta _q]$ . The substitution $\eta = \{x \mapsto c^{1,0}(y)\}$ is simple and belongs to $\operatorname {\mathit{mgu}}(s\theta _p, u)$ . So, the unification algorithm succeeds and returns $\upsilon ^{-1}(\eta ) = \theta$ .

4.2 A non-termination criterion

Now, we provide a non-termination criterion that is simpler to implement than that of Thm. 3. It relies on pattern rules of the following form, which is easy to check in practice.

Definition 14. We say that a pattern rule $r = (p,q)$ is special if it is simple and there exists

\begin{align*} c\big (c_1^{a_1,b_1}(t_1),\ldots ,c_m^{a_m,b_m}(t_m)\big ) \in \upsilon (p) \quad \text{and}\quad c\big (c_1^{a'_1,b'_1}(t_1\rho ),\ldots ,c_m^{a'_m,b'_m}(t_m\rho )\big ) \in \upsilon (q) \end{align*}

such that $c$ is an $m$ -context with $\operatorname {\mathit{Var}}(c) = \emptyset$ , $\rho \in S(\Sigma , X)$ and

1. 1. $\forall i: (t_i \in X) \lor (t_i \in T(\Sigma , X) \land \operatorname {\mathit{Var}}(t_i) = \emptyset )$ ,

2. 2. $\forall i,j: (t_i \in X \land t_i = t_j) \Rightarrow c_i = c_j$ ,

3. 3. $\{(a_i,a'_i) \mid \operatorname {\mathit{Var}}(t_i) = \emptyset \} = \{(e,e)\}$ with $0 \lt e$ , $\{(a_i,a'_i) \mid t_i \in X\} = \{(a,a')\}$ with $a \leq a'$ ,

4. 4. $\{(b_i,b'_i) \mid \operatorname {\mathit{Var}}(t_i) = \emptyset \} = \{(b,b')\}$ with $b \leq b'$ , $\{(b_i,b'_i) \mid t_i \in X\} = \{(d,d')\}$ ,

5. 5. $k = (b' - b) / e \in \mathbb{N}$ and $a = a' \Rightarrow 0 \leq (d' - d) - a \times k$ .

Then, we let $\alpha (r) = 0$ if $a = a'$ and $\alpha (r) = \frac {a \times k - (d' - d)}{a' - a}$ otherwise.

The existence of a special pattern rule implies non-termination:

Theorem 5. Let $P$ be a program and $B \subseteq \Im$ be correct w.r.t. $P$ . Suppose that $\operatorname {\mathit{patunf}}(P,B)$ contains a special pattern rule $r = (p,q)$ . Then, for all $n \in \mathbb{N}$ such that $n \geq \alpha (r)$ and all $\theta \in S(\Sigma , X)$ , there is an infinite $\mathop {\Rightarrow }_P$ -chain that starts from $\left \langle {p(n)\theta }\right \rangle$ .

Example 13. In Ex. 5 , the set $\operatorname {\mathit{patunf}}(P,B)$ contains the pattern rule $r'' = (p,q)$ and we have

\begin{align*} \operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{1,1}(x_1),\operatorname {\mathsf{s}}^{1,0}({\mathsf{0}})) &= c\left (c_1^{a_1,b_1}(t_1), c_2^{a_2,b_2}(t_2)\right ) \in \upsilon (p) \\ \operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{2,1}(x_1),\operatorname {\mathsf{s}}^{1,1}({\mathsf{0}})) &= c\left (c_1^{a'_1,b'_1}(t_1), c_2^{a'_2,b'_2}({\mathsf{0}})\right ) \in \upsilon (q) \end{align*}

(with a slight abuse of notation when writing $\operatorname {\mathsf{s}}^{1,1}$ , $\operatorname {\mathsf{s}}^{1,0}$ and $\operatorname {\mathsf{s}}^{2,1}$ ). Moreover,

• • $\{(a_i,a'_i) \mid \operatorname {\mathit{Var}}(t_i) = \emptyset \} = \{(a_2,a'_2)\} = \{(1,1)\} = \{(e,e)\}$ with $0 \lt e$ ,

• • $\{(a_i,a'_i) \mid t_i \in X\} = \{(a_1,a'_1)\} = \{(1,2)\} = \{(a,a')\}$ with $a \lt a'$ ,

• • $\{(b_i,b'_i) \mid \operatorname {\mathit{Var}}(t_i) = \emptyset \} = \{(b_2,b'_2)\} = \{(0,1)\} = \{(b,b')\}$ with $b \leq b'$ ,

• • $\{(b_i,b'_i) \mid t_i \in X\} = \{(1,1)\} = \{(d,d')\}$ ,

• • $k = (b' - b) / e = (1 - 0) / 1 = 1 \in \mathbb{N}$ .

So, $\alpha (r) = \frac {1 \times 1 - (1 - 1)}{2 - 1} = 1$ . Then, by Thm. 5 , for all $n \in \mathbb{N}$ such that $n \geq 1$ and all $\theta \in S(\Sigma , X)$ , there is an infinite $\mathop {\Rightarrow }_P$ -chain that starts from $\left \langle {p(n)\theta }\right \rangle = \left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^{n+1}(x_1),\operatorname {\mathsf{s}}^n({\mathsf{0}}))\theta }\right \rangle$ . This corresponds to what we observed in Ex. 2 and Ex. 6 . For instance, from $n = 1$ and $\theta = \{x_1 \mapsto {\mathsf{0}}\}$ , we get: there is an infinite $\mathop {\Rightarrow }_P$ -chain that starts from $\left \langle {\operatorname {\mathsf{while}}(\operatorname {\mathsf{s}}^2({\mathsf{0}}),\operatorname {\mathsf{s}}({\mathsf{0}}))}\right \rangle$ .

Def. 14 requires that $A_1 = \{i \mid \operatorname {\mathit{Var}}(t_i) = \emptyset \}$ and $A_2 = \{i \mid t_i \in X\}$ are not empty. This can be lifted as follows. If $A_1 = \emptyset$ or $A_2 = \emptyset$ then we demand that $a_1 = \ldots = a_m = a$ , $b_1 = \ldots = b_m = b$ , $a'_1 = \ldots = a'_m = a'$ and $b'_1 = \ldots = b'_m = b'$ . Moreover:

• • if $A_1 = \emptyset$ then we replace 3–5 in Def. 14 by $a \leq a'$ and $a = a' \Rightarrow b \leq b'$ ; we also let $\alpha (r) = 0$ if $a = a'$ and $\alpha (r) = \frac {b - b'}{a' - a}$ otherwise;

• • if $A_2 = \emptyset$ then we demand that $a = a'$ and we replace 3–5 in Def. 14 by $0 \lt a$ and $k = (b' - b) / a \in \mathbb{N}$ ; we also let $\alpha (r) = 0$ .

5 Experimental evaluation

We have implemented the approach of Sect. 4 in our tool NTI, which is the only tool participating in the International Termination Competition Footnote ² capable of disproving termination of logic programs (LPs). We used the natural choice for $S_1$ and $S'_1$ in the unification algorithm, even if it leads to an incomplete approach (see end of Sect. 4.1). We ran NTI on 41 LPs obtained by translating term rewrite systems (TRSs) of the Termination Problem Data Base Footnote ³ (TPDB) that are known to be non-looping non-terminating. These LPs are small but they are representative of the kind of non-termination that we want to capture. We used the following configuration: MacBook Pro 2020 with Apple M1 chip, 16 GB RAM, macOS Sequoia 15.4.1. Table 1 shows the results for 7 LPs obtained from directory AProVE_10 (which consists of 14 TRSs but we discarded those that translate to LPs that are not in the scope of our technique, i.e., LPs that terminate or involve 1-contexts which are not elements of $\chi ^{(1)}$ , see Ex. 9). Table 2 shows the results for 34 LPs obtained from directory EEG_IJCAR_12, originally proposed to evaluate the approach of Emmes et al. (Reference Emmes, Enger, GIESL, Gramlich, Miller and Sattler2012) (it consists of 49 TRSs but, again, we discarded those that translate to LPs that are out of scope). The tables have the following structure: column “Program” gives the name of the program together with its number of rules and relations, “Mode” gives the mode of interest (i means input, i.e., a term with no variable), “NTI” gives the non-terminating term provided by NTI, “#unf” gives the number of generated unfolded rules and “Time(ms)” gives the time in milliseconds (we used a time-out of 10 s). The 4 programs marked with $\dagger$ are LP translations of TRSs that have not been proven non-terminating by any TRS analyzer participating in the competition until 2024. The results show that our approach succeeds on them. On the other hand, our approach fails on 5 programs. Our results can be reproduced using our tool and the benchmarks available at https://github.com/etiennepayet/nti.

Table 1. Logic programs obtained from TPDB/TRS_Standard/AProVE_10

Table 2. Logic programs obtained from TPDB/TRS_Standard/EEG_IJCAR_12

6 Related work

The only other approach we are aware of for proving non-looping non-termination of logic programs is that of Payet (Reference Payet2024). Roughly, it detects infinite chains of the form ${\overline {{s}}}_0 \mathop {(\mathop {\Rightarrow }^*_{r_1} \circ \mathop {\Rightarrow }_{r_2})} {\overline {{s}}}_1 \mathop {(\mathop {\Rightarrow }^*_{r_1} \circ \mathop {\Rightarrow }_{r_2})} \cdots$ where $(r_1,r_2)$ is a recurrent pair of binary rules (we note that the infinite $\mathop {\Rightarrow }_P$ -chain of Ex. 2 does not have this form). This approach seems to address another class of non-loopingness (compared to that of this paper): it is not able to disprove termination of the programs of Tables 1 and 2 and, on the other hand, it is able to disprove termination of programsFootnote ⁴ on which our approach fails.

Loop checking (see, e.g., the paper by Bol et al. (Reference Bol, Apt and KLOP1991)) is also related to our work. It attempts to prune infinite rewrites at runtime using necessary conditions for the existence of infinite chains (hence, there is a risk of pruning a finite rewrite). In contrast, our approach uses a sufficient condition (see Thm. 3 and Thm. 5) to prove the existence of atomic goals that start a non-terminating chain.

Tabling (see, e.g., the paper by Sagonas et al. (Reference Sagonas, Swift, WARREN, Snodgrass and Winslett1994)) is another related technique that avoids infinite rewrites by storing intermediate results in a table and reusing them when needed. Only loops of a particular simple form are detected (i.e., when a variant occurs in the evaluation of a subgoal). We are not aware of any tabling-based approach that can capture non-looping non-termination.

Another related technique is that of Payet and Mesnard (Reference Payet and Mesnard2006) which proves the existence of loops using neutral argument positions of predicate symbols.