1 Introduction

Reduction orders are a fundamental tool in termination analysis of term rewrite systems, and they also underlie completion-based automated theorem proving. Weighted path orders (WPOs) [27] are known as a versatile class of reduction orders; WPOs can simulate (generalized) Knuth–Bendix orders [7, 13, 16] and lexicographic path orders [12], depending on the choice of parameters, namely simple monotone algebras and precedences. In fact, weighted path orders are so powerful that they characterize simple termination of term rewrite systems [20, Definition 6.3.7], that is, a term rewrite system is simply terminating if and only if it admits a compatible WPO. Besides automated termination analysis [14, 26], WPOs are used in reachability analysis [25], and automated theorem proving [11, 18].

Another well-known class of reduction orders is the class of monotonic semantic path orders (MSPOs) [4, 5], which are a monotonic version of semantic path orders (SPOs) [12]. MSPOs take triples of orders (called reduction triples) as parameters, and provide a complete characterization of terminating term rewrite systems: A term rewrite system is terminating if and only if it admits a compatible MSPO. However, the relationship between WPOs and MSPOs has not been known [24].

In this paper, we give a solution to the open problem, demonstrating an effective construction of an MSPO from the algebra and the precedence of a given WPO. The key of the proof lies in finding a suitable new variant of MSPOs, which is described as follows: First, the variant uses lexicographic comparison [4, Definition 4.5.1], as the original WPOs [27, Definition 5] are based on this comparison strategy. Second, the variant employs reduction triples [4, Definition 4.1.19] because an example shows that a variant based on (quasi-)reduction pairs [5, Definition 4] leads to an invalid construction.

The obtained simulation result leads to a generalization of WPOs that does not impose simplicity on their underlying algebras. The generalization can show termination of term rewrite systems that are not simply terminating. This is a sharp contrast to the termination proving power of WPOs. In addition, upgrading WPOs to GWPOs can be done with little implementation effort, so we anticipate that tools which employ WPOs as reduction orders (e.g. [11, 14, 18, 23, 25, 26]) may benefit from power of GWPOs.

The remaining part of the paper is organized as follows. After recalling notions and notations for term rewriting and WPOs in Sect. 2, we introduce a slightly modified version of semantic path orders that employs order pairs in Sect. 3. In Sect. 4 we show that weighted path orders are instances of semantic path orders. Using this fact, we introduce a generalization of WPOs in Sect. 5. In Sect. 6 experimental data for (generalized) weighted path orders are reported. As in the case of MSPOs [5, Section 5.2], GWPOs are capable of simulating a basic version of the dependency pair method [1]. This is discussed in Sect. 7. The paper is concluded by stating related work in Sect. 8.

2 Preliminaries

Throughout the paper, we assume familiarity with term rewriting [3, 20]. First we briefly recall basic notions for term rewriting and reduction orders, and then introduce weighted path orders.

2.1 Term Rewriting

Let \(\mathcal {F}\) be a signature and \(\mathcal {V}\) a countable set of variables with \(\mathcal {F}\cap \mathcal {V}= \varnothing \). The set of all terms built from \(\mathcal {F}\) and \(\mathcal {V}\) is referred to as \(\mathcal {T}(\mathcal {F},\mathcal {V})\), or just as \(\mathcal {T}\) when \(\mathcal {F}\) and \(\mathcal {V}\) are clear from the context. When we need to indicate the arity n of a function symbol f, we write \(f^{(n)}\) for f. Quasi-orders on the signature are called (quasi-)precedences. A quasi-precedence \(\succsim \) is called well-founded if its strict part \(\succ \) is well-founded. The size |t| of a term t is the number of function symbols and variables occurring in t. Let \(\Box \) be a constant with \(\Box \notin \mathcal {F}\). Contexts are terms over \(\mathcal {F}\cup \{ \Box \}\) that contain exactly one \(\Box \). The term resulting from replacing \(\Box \) in a context C by a term t is denoted by C[t]. We write if there is a context C with \(s = C[t]\). The strict part of is denoted by \(\rhd \). A substitution is a mapping \(\sigma \) from variables to terms such that \(\{ x \in \mathcal {V}\mid \sigma (x) \ne x \}\) is finite. The application \(t\sigma \) of a substitution \(\sigma \) to a term t is inductively defined as follows: \(t\sigma = \sigma (t)\) if t is a variable, and \(t\sigma = f(t_1\sigma ,\ldots ,t_n\sigma )\) if \(t = f({t_1},\dots ,{t_{n}})\).

A pair \((\ell ,r)\) of terms is said to be a rewrite rule if \(\ell \) is not a variable and every variable in r occurs in \(\ell \). Rewrite rules \((\ell ,r)\) are written by \(\ell \rightarrow r\). A set of rewrite rules is called a term rewrite system (TRS). Let \(\mathcal {R}\) be a TRS. We write \(\mathcal {D}_\mathcal {R}\) for the set of defined symbols \(\{ f \mid f({\ell _1},\dots ,{\ell _{n}}) \rightarrow r \in \mathcal {R}\}\). The relation \(\rightarrow _\mathcal {R}\) is defined on terms as follows: \(s \rightarrow _\mathcal {R}t\) if there exist a rewrite rule \(\ell \rightarrow r \in \mathcal {R}\), a context C, and a substitution \(\sigma \) such that \(s = C[\ell \sigma ]\) and \(t = C[r\sigma ]\) hold. The TRS \(\mathcal {R}\) is said to be terminating if there is no infinite sequence \(t_1 \rightarrow _\mathcal {R}t_2 \rightarrow _\mathcal {R}\cdots \). A relation \(\leadsto \) on terms is closed under contexts if \(C[s] \leadsto C[t]\) holds whenever \(s \leadsto t\) and C is a context, and it is called closed under substitutions or just stable if \(s\sigma \leadsto t\sigma \) holds whenever \(s \leadsto t\) and \(\sigma \) is a substitution. We say \(\leadsto \) has the subterm property if \(s \leadsto t\) for all terms st satisfying \(s \rhd t\). Relations closed under contexts and substitutions are called rewrite relations.

Termination is often shown by using orders. We say that a rewrite relation is a rewrite preorder or reduction order if it is a preorder or a well-founded order, respectively. A TRS \(\mathcal {R}\) is compatible with a strict order > if \(\mathcal {R}\subseteq {>}\).

Proposition 1

A TRS \(\mathcal {R}\) is terminating if \(\mathcal {R}\) is compatible with some reduction order >.

An ordered \(\mathcal {F}\)-algebra is a triple \((A, \{f_\mathcal {A}\}_{f \in \mathcal {F}}, {>})\), where A is a set called a carrier, \(f_\mathcal {A}\) is an n-ary function on A (called an interpretation function) associated with each \(f^{(n)} \in \mathcal {F}\), and > is a strict order on A. Let \(\mathcal {A}= (A, \{f_\mathcal {A}\}_{f \in \mathcal {F}}, {>})\) be an ordered algebra. A mapping from \(\mathcal {V}\) to A is called an assignment for \(\mathcal {A}\). The interpretation \([\alpha ]_\mathcal {A}(t)\) of a term t under an assignment \(\alpha \) is inductively defined as follows: \([\alpha ]_\mathcal {A}(t) = \alpha (t)\) if t is a variable, and \([\alpha ]_\mathcal {A}(t) = f_\mathcal {A}([\alpha ]_\mathcal {A}(t_1),\ldots ,[\alpha ]_\mathcal {A}(t_n))\) if \(t = f({t_1},\dots ,{t_{n}})\). We write \(s >_\mathcal {A}t\) if \([\alpha ]_\mathcal {A}(s) > [\alpha ]_\mathcal {A}(t)\) for all assignments \(\alpha \). The relation \(>_\mathcal {A}\) is a strict order. Similarly we write \(s \geqslant _\mathcal {A}t\) if \([\alpha ]_\mathcal {A}(s) \geqslant [\alpha ]_\mathcal {A}(t)\) holds for all assignments \(\alpha \), where \(\geqslant \) stands for the reflexive closure of >. The relation \(\geqslant _\mathcal {A}\) is a quasi-order, and satisfies \(\geqslant _\mathcal {A}\cdot>_\mathcal {A}\cdot \geqslant _\mathcal {A}{\subseteq } >_\mathcal {A}\). We say that the ordered algebra \(\mathcal {A}\) is

  • simple if \(f_\mathcal {A}(a_1, \ldots , a_i, \ldots , a_n) \geqslant a_i\) for all \(f^{(n)} \in \mathcal {F}\), \(1 \leqslant i \leqslant n\), and \({a_1},\dots ,{a_{n}} \in A\);

  • weakly monotone if \( f_\mathcal {A}(a_1,\ldots ,a_i,\ldots ,a_n) \geqslant f_\mathcal {A}(a_1,\ldots ,b,\ldots ,a_n) \) for all \(f^{(n)} \in \mathcal {F}\), argument positions \(1 \leqslant i \leqslant n\), and \({a_1},\dots ,{a_{n}}, b \in A\) with \(a_i > b\);

  • simple monotone if it is simple and weakly monotone; and

  • well-founded if > is well-founded.

If > is well-founded, so is \(>_\mathcal {A}\). If \(\mathcal {A}\) is a weakly monotone algebra, \(\geqslant _\mathcal {A}\) is a rewrite preorder. If in addition \(\mathcal {A}\) is simple, \(\geqslant _\mathcal {A}\) has the subterm property \({\rhd } \subseteq {\geqslant _\mathcal {A}}\).

2.2 Weighted Path Orders

Weighted path orders (WPOs) are reduction orders introduced by Yamada et al. [27]. The definition of WPOs is based on the pair of an ordered algebra \(\mathcal {A}\) and a precedences \(\succsim \). A WPO compares terms st as a generalized KBO does: First the terms are compared by \(s >_\mathcal {A}t\). If only weak inequality \(s \geqslant _\mathcal {A}t\) holds then their root symbols, say f and g, are compared by the precedence \(\succsim \). If again only weak inequality \(f \succsim g\) holds, arguments are compared lexicographically.

Lexicographic comparison is formalized as follows. Let > be a strict order on a set A and let \(A^*\) denote the set of all strings (tuples) over A. The lexicographic extension \(>^\textsf{lex}\) of > is defined on \(A^*\) as follows: \((a_1, \ldots , a_n) >^{\textsf{lex}} (b_1, \ldots , b_m)\) if there is a natural number \(k < n\) such that

  • \(a_j = b_j\) for all \(1 \leqslant j \leqslant k\), and

  • either \(k = m\), or \(k < m\) and \(a_{k+1} > b_{k+1}\).

It is known that \(>^{\textsf{lex}}\) is a strict order on \(A^*\).

Definition 1

([27]). Let \(\mathcal {A}\) be an ordered \(\mathcal {F}\)-algebra and \(\succsim \) a precedence. The weighted path order \(>_\textsf{wpo}\) is defined on terms over \(\mathcal {F}\) as follows: \(s >_\textsf{wpo}t\) if

  1. 1.

    \(s >_\mathcal {A}t\), or

  2. 2.

    \(s \geqslant _\mathcal {A}t\), \(s = f({s_1},\dots ,{s_{m}})\), and one of the following conditions holds.

    1. a.

      \(s_i \geqslant _\textsf{wpo}t\) for some \(1 \leqslant i \leqslant m\).

    2. b.

      \(t = g({t_1},\dots ,{t_{n}})\) and \(s >_\textsf{wpo}t_j\) for all \(1 \leqslant j \leqslant n\), and moreover

      1. (i)

        \(f \succ g\), or

      2. (ii)

        \(f \succsim g\) and \(({s_1},\dots ,{s_{m}}) >_\textsf{wpo}^\textsf{lex}({t_1},\dots ,{t_{n}})\).

Here \(\geqslant _\textsf{wpo}\) denotes the reflexive closure of \(>_\textsf{wpo}\).

Theorem 1

([27]). Suppose that the signature is finite. For every simple monotone well-founded algebra and well-founded precedence the induced relation \(>_\textsf{wpo}\) is a reduction order with the subterm property.

Example 1

Consider the following TRS \(\mathcal {R}\) taken from [27, Example 9]:

$$\begin{aligned} \textsf{f}(\textsf{g}(x))&\rightarrow \textsf{g}(\textsf{f}(\textsf{f}(x)))&\textsf{f}(\textsf{h}(x))&\rightarrow \textsf{h}(\textsf{h}(\textsf{f}(x))) \end{aligned}$$

Let \(\mathcal {A}\) be the simple monotone algebra on \(\mathbb {N}\) with \(\textsf{f}_\mathcal {A}(x) = \textsf{h}_\mathcal {A}(x) = x\) and \(\textsf{g}_\mathcal {A}(x) = x + 1\). Take a precedence \(\succsim \) with \(\textsf{f} \succ \textsf{g} \succ \textsf{h}\). The relation \(\textsf{f}(\textsf{g}(x)) >_\textsf{wpo}\textsf{g}(\textsf{f}(\textsf{f}(x)))\) is verified by the following derivation:

figure c

Here WPO 1 and WPO 2b(i) indicate the corresponding conditions in Definition 1. Similarly, one can verify \(\textsf{f}(\textsf{h}(x)) >_\textsf{wpo}\textsf{h}(\textsf{h}(\textsf{f}(x)))\). Therefore, \(\mathcal {R}\subseteq {>_\textsf{wpo}}\) follows. Hence, we conclude that \(\mathcal {R}\) is terminating.

The following example shows that the simplicity condition cannot be dropped from Theorem 1.

Example 2

Any WPO \(>_\textsf{wpo}\) induced by the weakly monotone but non-simple algebra \(\mathcal {A}\) on \(\mathbb {N}\) with \(\textsf{a}_\mathcal {A}= 1\) and \(\textsf{f}_\mathcal {A}(x) = 0\) lacks well-foundedness as it admits the cyclic sequence \(\textsf{f}(\textsf{a})>_\textsf{wpo}\textsf{f}(\textsf{f}(\textsf{a})) >_\textsf{wpo}\textsf{f}(\textsf{a})\).

3 Semantic Path Orders Based on Order Pairs

Borralleras [4, Definition 4.1.19] introduced a variant of SPO that employs a pair of a quasi-order and a strict order. This variant compares arguments of terms by a multiset order. In order to simulate WPOs which compare arguments in a lexicographic manner, we introduce another variant of SPO.

We say that the pair \(({\gtrsim },{>})\) of a quasi-order \(\gtrsim \) and a strict order > is an order pair if \(\gtrsim \cdot > \cdot \gtrsim {\subseteq }>\). The inclusion is referred to as compatibility. We say that an order pair on terms is stable if both and \(\sqsupset \) are stable.

Definition 2

Let be a stable order pair on \(\mathcal {T}\setminus \mathcal {V}\).Footnote 1 The semantic path order \(>_\textsf{spo}\) (SPO) is defined on terms as follows: \(s >_\textsf{spo}t\) if \(s = f({s_1},\dots ,{s_{m}})\) and one of the following conditions hold:

  1. 1.

    \(s_i \geqslant _\textsf{spo}t\) for some \(1 \leqslant i \leqslant m\).

  2. 2.

    \(t = g({t_1},\dots ,{t_{n}})\) and \(s >_\textsf{spo}t_j\) for all \(1 \leqslant j \leqslant n\), and moreover

    1. a.

      \(s \sqsupset t\), or

    2. b.

      and \(({s_1},\dots ,{s_{m}}) >_\textsf{spo}^{\textsf{lex}} ({t_1},\dots ,{t_{n}})\).

Here \(\geqslant _\textsf{spo}\) denotes the reflexive closure of \(>_\textsf{spo}\).

Remark 1

The standard definitions of SPOs ( [12] and [4, Definition 4.1.19]) use the multiset extension of \(>_\textsf{spo}\) in SPO 2b instead of the lexicographic extension. The lexicographic version of SPOs, introduced by Borralleras [4, Definition 4.5.1], can be obtained by setting \(\sqsupset \) to the strict part of in Definition 2.

Example 3

Lexicographic path orders (LPOs) are special instances of SPOs. Let \(\succsim \) be a precedence. Define by \(f \succsim g\), and let \(\sqsupset \) be the strict part of . The semantic path order induced by is the lexicographic path order induced by \(\succsim \).

Let be a stable order pair on \(\mathcal {T}\setminus \mathcal {V}\) and let \(>_\textsf{spo}\) be the semantic path order induced by . The transitivity, reflexivity, and stability of \(>_\textsf{spo}\) are straightforward. A small remark is that the compatibility   is used in the proof of the transitivity.

Lemma 1

The SPO \(>_\textsf{spo}\) is a stable strict order.    \(\square \)

When the signature is infinite, the lexicographic version of SPOs is not well-founded in general even if \(\sqsupset \) is well-founded. This forms a contrast to the multiset versions of SPOs mentioned in Remark 1.

Example 4

Consider the signature consisting of \(\textsf{a}^{(0)}\), \(\textsf{b}^{(0)}\), and \(\textsf{f}_i^{(i)}\) for all numbers \(i \in \mathbb {N}\). Let \(\succsim \) be a well-founded precedence satisfying \(\textsf{a} \succ \textsf{b}\) and \(\textsf{f}_i \succsim \textsf{f}_j\) for all \(i, j \in \mathbb {N}\). The pair defined as in Example 3 is an order pair with \(\sqsupset \) well-founded, but the SPO \(>_\textsf{spo}\) induced from admits the infinite chain:

$$ \textsf{f}_1(\textsf{a})>_\textsf{spo}\textsf{f}_2(\textsf{b},\textsf{a})>_\textsf{spo}\textsf{f}_3(\textsf{b},\textsf{b},\textsf{a}) >_\textsf{spo}\cdots $$

See [22, Section 3] and [19, Section 3] for related discussions.

Well-foundedness of \(>_\textsf{spo}\) is restored by assuming existence of an upper bound of arities. We refer to this property as boundedness of the signature. Needless to say, a signature is bounded whenever it is finite.

Hereafter we assume that \(\sqsupset \) is well-founded and \(\mathcal {F}\) is bounded. For showing that \(>_\textsf{spo}\) is well-founded, we adopt Buchholz’s method [6]. One can find a similar proof in [27, Lemma 8]. We write \(\textsf{SN}(>_\textsf{spo})\) for the set of all terms t such that there is no infinite descending sequence \(t>_\textsf{spo}t_1>_\textsf{spo}t_2 >_\textsf{spo}\cdots \) starting from t.Footnote 2 The following properties are immediate:

  • The restriction of \(>_\textsf{spo}\) to \(\textsf{SN}(>_\textsf{spo})\) is a well-founded order on \(\textsf{SN}(>_\textsf{spo})\).

  • \(t \in \textsf{SN}(>_\textsf{spo})\) if \(u \in \textsf{SN}(>_\textsf{spo})\) for all terms u with \(t >_\textsf{spo}u\).

Buchholz’s method proves well-foundedness by well-founded induction. To express our well-founded order for induction, we recall the notion of the lexicographic product of order pairs. Let \(({\gtrsim _1},{>_1}), \ldots ,({\gtrsim _n},{>_n})\) be n order pairs on sets \(A_1, \ldots , A_n\), respectively. The lexicographic product \( ({\gtrsim _1},{>_1}) \otimes \cdots \otimes ({\gtrsim _n},{>_n}) \) is the strict order > defined on \(A_1 \times \cdots \times A_n\) as follows: \(({a_1},\dots ,{a_{n}}) > ({b_1},\dots ,{b_{n}})\) if there exists an index \(k \in \{ 1, \ldots , n \}\) such that \(a_k >_k b_k\) and \(a_j \gtrsim _j b_j\) for all \(1 \leqslant j < k\). Note that the lexicographic product > is well-founded if every \(>_i\) is well-founded.

Given a set A, we write \(A^{\leqslant k}\) for the union of \(A^{i}\) for all \(i \leqslant k\). If a strict order > on A is well-founded, then the restriction of \(>^\textsf{lex}\) to \(A^{\leqslant k}\) is also well-founded, see [19, Section 3]. Thus, the lexicographic product \(\gg \) given by

figure q

is a well-founded order on \((\mathcal {T}\setminus \mathcal {V}) \times \mathcal {T}^{\leqslant M} \times \mathcal {T}\). Here M stands for the maximum arity in the signature \(\mathcal {F}\), and \(\geqslant _\textsf{spo}^\textsf{lex}\) for the reflexive closure of \(>_\textsf{spo}^\textsf{lex}\).

Lemma 2

The term u belongs to \(\textsf{SN}(>_\textsf{spo})\) whenever \(t = f({t_1},\dots ,{t_{n}}) >_\textsf{spo}u\) and \({t_1},\dots ,{t_{n}} \in \textsf{SN}(>_\textsf{spo})\).

Proof

We show the claim by well-founded induction on \((t, ({t_1},\dots ,{t_{n}}), u)\) with respect to \(\gg \). Here we proceed by analyzing the derivation of \(t >_\textsf{spo}u\). If \(t >_\textsf{spo}u\) is derived from SPO 1 then \(t_i \geqslant _\textsf{spo}u\) for some \(i \in \{ 1, \ldots , n \}\). In this case \(u \in \textsf{SN}(>_\textsf{spo})\) trivially follows from \(t_i \in \textsf{SN}(>_\textsf{spo})\). If \(t >_\textsf{spo}u\) is derived from SPO 2a or SPO 2b, then u is of the form \(g({u_1},\dots ,{u_{m}})\) and \(t >_\textsf{spo}u_j\) for all \(j \in \{ 1, \ldots , m \}\). From \(u \rhd u_j\) we have \((t, ({t_1},\dots ,{t_{n}}), u) \gg (t, ({t_1},\dots ,{t_{n}}), u_j)\). So from the induction hypothesis \(u_i \in \textsf{SN}(>_\textsf{spo})\) for each j. For showing our goal \(u \in \textsf{SN}(>_\textsf{spo})\) fix an arbitrary term v with \(u >_\textsf{spo}v\). We further distinguish the case of SPO 2a and that of SPO 2b.

  1. a.

    If \(t >_\textsf{spo}u\) is derived from SPO 2a then \(t \sqsupset u\). Thus, \((t, ({t_1},\dots ,{t_{n}}), u) \gg (u, ({u_1},\dots ,{u_{m}}), v)\), and the induction hypothesis yields \(v \in \textsf{SN}(>_\textsf{spo})\).

  2. b.

    If \(t >_\textsf{spo}u\) is derived from SPO 2b then we additionally have and \(({t_1},\dots ,{t_{n}}) >_\textsf{spo}^{\textsf{lex}} ({u_1},\dots ,{u_{m}})\). Thus, \(({t}, {({t_1},\dots ,{t_{n}})}, {u}) \gg ({u}, {({u_1},\dots ,{u_{m}})}, {v})\) holds. So from the induction hypothesis we obtain \(v \in \textsf{SN}(>_\textsf{spo})\).

In either case \(v \in \textsf{SN}(>_\textsf{spo})\). So we conclude \(u \in \textsf{SN}(>_\textsf{spo})\).    \(\square \)

Lemma 3

The relation \(>_\textsf{spo}\) is well-founded.

Proof

We show that \(t \in \textsf{SN}(>_\textsf{spo})\) by induction on |t|. If t is a variable trivially \(t \in \textsf{SN}(>_\textsf{spo})\). Otherwise, Lemma 2 applies.    \(\square \)

Theorem 2

Every semantic path order is a stable well-founded order, provided that the signature is bounded.    \(\square \)

In general, semantic path orders are not closed under contexts. For a remedy, Borralleras et al. [5] propose the use of another preorder with the harmony property. This results in monotonic semantic path orders.

Definition 3

([4, Definition 4.1.20]). A triple is a reduction triple if \(\gtrsim \) is a rewrite preorder on terms,   is a stable order pair on \(\mathcal {T}\setminus \mathcal {V}\) with \(\sqsupset \) well-founded, and \(\gtrsim \) and have the harmony property, meaning that for every \(f^{(n)} \in \mathcal {F}\) the implication

figure v

holds for all terms \({s_1},\dots ,{s_{n}},t\) and argument positions \(1 \leqslant i \leqslant n\).

Definition 4

Let be a reduction triple, and let \(>_\textsf{spo}\) be the semantic path order induced from . The monotonic semantic path order \(s >_\textsf{mspo}t\) (MSPO) is defined as \(s \gtrsim t\) and \(s >_\textsf{spo}t\).

Theorem 3

Every monotonic semantic path order is a reduction order, provided that the signature is bounded.

Proof

The proof due to Borralleras et al. [5, Theorem 2] goes through.    \(\square \)

4 Simulating WPOs by SPOs

We show that WPOs are instances of SPOs by constructing a suitable order pair from a weakly monotone well-founded algebra \(\mathcal {A}\) and a well-founded precedence \(\succsim \). For terms \(s = f({s_1},\dots ,{s_{m}}), t = g({t_1},\dots ,{t_{n}})\) we write if \(s >_\mathcal {A}t\), or both \(s \geqslant _\mathcal {A}t\) and \(f \succsim g\). Similarly, we define \(s \sqsupset t\) if \(s >_\mathcal {A}t\), or both \(s \geqslant _\mathcal {A}t\) and \(f \succ g\). It is worth noting that the proof of [27, Lemma 8] also combines the interpretation order and precedence in a lexicographic manner.

Lemma 4

The pair is a stable order pair with \(\sqsupset \) well-founded.    \(\square \)

In the remaining part of the section we consider the WPO \(>_\textsf{wpo}\) induced by \(\mathcal {A}\) and \(\succsim \), and the SPO \(>_\textsf{spo}\) induced by the corresponding order pair . Note that \(\sqsupset \) is not a strict part of in general, as \(>_\mathcal {A}\) is not necessarily the strict part of \(\geqslant _\mathcal {A}\). This is why we decoupled \(\sqsupset \) from in Definition 2; see also Remark 1.

Example 5

Let the signature \(\mathcal {F}= \{ \textsf{f}^{(1)} \}\). Consider the trivial precedence \(\textsf{f} \succsim \textsf{f}\) and the algebra \(\mathcal {A}\) over the carrier \(\mathbb {N}\) with the interpretation \(\textsf{f}_\mathcal {A}(x) = 2 x\). On the one hand we have   from \(\textsf{f}(\textsf{f}(x)) \geqslant _\mathcal {A}\textsf{f}(x)\) but not as \(\textsf{f}(x) \ngeqslant _\mathcal {A}\textsf{f}(\textsf{f}(x))\). On the other hand \(\textsf{f}(\textsf{f}(x)) \sqsupset \textsf{f}(x)\) does not hold.

We illustrate how the derivation of \(>_\textsf{wpo}\) in Example 1 is simulated by the semantic path order.

Example 6 (continued from Example 1)

From \(\textsf{f}(\textsf{g}(x)) \geqslant _\mathcal {A}\textsf{g}(\textsf{f}(\textsf{f}(x)))\) and \(\textsf{f} \succ \textsf{g}\) the inequality \(\textsf{f}(\textsf{g}(x)) \sqsupset \textsf{g}(\textsf{f}(\textsf{f}(x)))\) is obtained. Moreover, we have \(\textsf{f}(\textsf{g}(x)) >_\mathcal {A}\textsf{f}(\textsf{f}(x))\). Since \(\geqslant _\mathcal {A}\) has the subterm property, the subterm \(\textsf{f}(x)\) of \(\textsf{f}(\textsf{f}(x))\) also satisfies \(\textsf{f}(\textsf{g}(x)) >_\mathcal {A}\textsf{f}(x)\). Thus we obtain \(\textsf{f}(\textsf{g}(x)) \sqsupset \textsf{f}(\textsf{f}(x)), \textsf{f}(x)\). Therefore, \(\textsf{f}(\textsf{g}(x)) >_\textsf{spo}\textsf{g}(\textsf{f}(\textsf{f}(x)))\) is verified as follows:

figure ag

Similarly, \(\textsf{f}(\textsf{h}(x)) >_\textsf{spo}\textsf{h}(\textsf{h}(\textsf{f}(x)))\) can be verified. Hence, the inclusion \(\mathcal {R}\subseteq {>_\textsf{spo}}\) holds. Observe that the use of WPO 1 in Example 1 is replaced by successive application of SPO 1 and SPO 2a.

As shown in the example, the subterm property of \(\geqslant _\mathcal {A}\) is a key for filling in the gap between \(>_\textsf{spo}\) and \(>_\textsf{wpo}\).

Lemma 5

Suppose that \(\mathcal {A}\) is simple. If \(s >_\textsf{wpo}t\) then \(s >_\textsf{spo}t\).

Proof

We prove the claim by induction on \(|s| + |t|\). Let \(s = f({s_1},\dots ,{s_{m}}) >_\textsf{wpo}t\). Depending on the derivation of \(s >_\textsf{wpo}t\), we distinguish five cases.

  • Suppose that t is a variable and \(s >_\textsf{wpo}t\) is derived from \(s >_\mathcal {A}t\). One can verify that t occurs in s. Because s is not a variable, \(s \rhd t\) follows. By the subterm property of \(>_\textsf{spo}\) we obtain \(s >_\textsf{spo}t\).

  • Suppose that \(t = g({t_1},\dots ,{t_{n}})\) and \(s >_\textsf{wpo}t\) is derived from \(s >_\mathcal {A}t\). From \(s >_\mathcal {A}t\) we obtain \(s \sqsupset t\). Since \(\mathcal {A}\) is simple, for every \(1 \leqslant j \leqslant n\) we have \(s >_\mathcal {A}t \geqslant _\mathcal {A}t_j\), which leads to \(s >_\mathcal {A}t_j\). Hence, \(s >_\textsf{spo}t\) is derived as follows:

    figure ah
  • Suppose that \(s >_\textsf{wpo}t\) is derived as follows:

    figure ai

    By the induction hypothesis we have \(s_i \geqslant _\textsf{spo}t\) for some i, and thus \(s >_\textsf{spo}t\).

  • Suppose that \(s >_\textsf{wpo}t\) is derived as follows:

    figure aj

    From \(s \geqslant _\mathcal {A}t\) and \(f \succ g\) we obtain \(s \sqsupset t\). Thus, we have:

    figure ak
  • Suppose that \(s >_\textsf{wpo}t\) is derived as follows:

    figure al

    From \(s \geqslant _\mathcal {A}t\) we obtain . Thus, we have:

    figure an

In any case we have \(s >_\textsf{spo}t\).    \(\square \)

Next we prove the converse direction of Lemma 5. The next lemma is a basic property of WPOs.

Lemma 6

If \(s >_\textsf{wpo}t\) then \(s \geqslant _\mathcal {A}t\).    \(\square \)

Lemma 7

Suppose that \(\mathcal {A}\) is simple. If \(s >_\textsf{spo}t\) then \(s >_\textsf{wpo}t\).

Proof

We prove the claim by induction on \(|s| + |t|\). We distinguish three cases, depending on the derivation of \(s >_\textsf{spo}t\).

  • Suppose that \(s >_\textsf{spo}t\) is derived as follows:

    figure ao

    The induction hypothesis yields \(s_i \geqslant _\textsf{wpo}t\) for some i. By Lemma 6 and the subterm property of \(\geqslant _\mathcal {A}\) we have \(s \geqslant _\mathcal {A}t\). Thus, we obtain the following derivation of \(s >_\textsf{wpo}t\):

    figure ap
  • Suppose that \(s >_\textsf{spo}t\) is derived as follows:

    figure aq

    According to the definition of \(s \sqsupset t\), we further distinguish two subcases. If \(s >_\mathcal {A}t\) then \(s >_\textsf{wpo}t\) is immediate. Otherwise, \(s \geqslant _\mathcal {A}t\) and \(f \succ g\) hold. In this case we derive \(s >_\textsf{wpo}t\) as follows:

    figure ar
  • Suppose that \(s >_\textsf{spo}t\) is derived as follows:

    figure as

    Because of , we have \(s >_\mathcal {A}t\) or both \(s \geqslant _\mathcal {A}t\) and \(f \succsim g\). In the former case \(s >_\textsf{wpo}t\) is immediate. In the latter case \(s >_\textsf{wpo}t\) is derived by WPO 2b(ii) as follows:

    figure au

In any case we have \(s >_\textsf{wpo}t\).    \(\square \)

As a consequence, \(>_\textsf{wpo}\) and \(>_\textsf{spo}\) coincide, provided that \(\mathcal {A}\) is simple. This result can be extended to monotonic semantic path orders.

Lemma 8

The triple is a reduction triple.    \(\square \)

Let \(>_\textsf{mspo}\) denote the monotonic semantic path order induced from \(\geqslant _\mathcal {A}\) and \(>_\textsf{spo}\). Since \(s >_\textsf{wpo}t\) implies \(s \geqslant _\mathcal {A}t\) (Lemma 6), \(s >_\textsf{mspo}t\) is equivalent to \(s >_\textsf{spo}t\). By using this equivalence together with Lemmata 5 and 7, we obtain the following result.

Theorem 4

The three orders \(>_\textsf{wpo}\), \(>_\textsf{spo}\), and \(>_\textsf{mspo}\) coincide, provided that \(\mathcal {A}\) is simple.    \(\square \)

5 Generalized Weighted Path Orders

According to Theorem 4, weighted path orders can be defined as monotonic semantic path orders. Moreover, Lemma 8 reveals that even for non-simple algebras the construction of reduction triples is valid. This observation suggests a generalization of weighted path orders, which does not impose simplicity on algebras. Besides, we exploit the fact that stable order pairs need not be closed under contexts, marking root symbols of function applications; see [1] and [5, Definition 5].

Let \(\mathcal {F}\) be a signature. For each \(f \in \mathcal {F}\) we associate a marked function symbol \(f^\sharp \notin \mathcal {F}\) of the same arity. The set \(\{ f^{\sharp } \mid f \in \mathcal {F}\}\) is denoted by \(\mathcal {F}^\sharp \). For each term \(t = f({t_1},\dots ,{t_{n}}) \in \mathcal {T}(\mathcal {F},\mathcal {V})\) we denote \(f^\sharp ({t_1},\dots ,{t_{n}})\) by \(t^\sharp \). Let \(\mathcal {A}\) be a weakly monotone well-founded \((\mathcal {F}\cup \mathcal {F}^\sharp )\)-algebra and \(\succsim \) a well-founded precedence on \(\mathcal {F}\). The pair of relations on \(\mathcal {T}(\mathcal {F}, \mathcal {V}) \setminus \mathcal {V}\) is defined as follows: Let \(s = f({s_1},\dots ,{s_{n}})\) and \(t = g({t_1},\dots ,{t_{m}})\). We write if \(s^\sharp >_{\mathcal {A}} t^\sharp \), or \(s^\sharp \geqslant _{\mathcal {A}} t^\sharp \) and \(f \succsim g\). Similarly, we write \(s \sqsupset ^\sharp t\) if \(s^\sharp >_{\mathcal {A}} t^\sharp \), or \(s^\sharp \geqslant _{\mathcal {A}} t^\sharp \) and \(f \succ g\). The relation \(\gtrsim \) is defined as the restriction of \(\geqslant _\mathcal {A}\) to \(\mathcal {T}(\mathcal {F}, \mathcal {V})\).

Proposition 2

The triple is a reduction triple on \(\mathcal {T}(\mathcal {F}, \mathcal {V})\).    \(\square \)

Definition 5

The generalized weighted path order (GWPO) \(>_\textsf{gwpo}\) induced from \(\mathcal {A}\) and \(\succsim \) is the monotonic semantic path order induced from   .

Corollary 1

Every generalized weighted path order is a reduction order, provided that the signature is bounded.    \(\square \)

For convenience, we reformulate the definition of \(>_\textsf{gwpo}\) in the style of Definition 1.

Definition 6

The relation \(>_\mathsf {wpo'}\) is defined on terms as follows: \(s >_\mathsf {wpo'}t\) if \(s = f({s_1},\dots ,{s_{m}})\) and one of the following conditions hold.

  1. 1.

    \(s_i \geqslant _\mathsf {wpo'}t\) for some \(1 \leqslant i \leqslant m\).

  2. 2.

    \(t = g({t_1},\dots ,{t_{n}})\), \(s^\sharp \geqslant _{\mathcal {A}} t^\sharp \), and \(s >_\mathsf {wpo'}t_j\) for all \(1 \leqslant j \leqslant n\), and moreover

    1. a.

      \(s^\sharp >_{\mathcal {A}} t^\sharp \),

    2. b.

      \(f \succ g\), or

    3. c.

      \(f \succsim g\) and \(({s_1},\dots ,{s_{m}}) >_\mathsf {wpo'}^\textsf{lex}({t_1},\dots ,{t_{n}})\).

Proposition 3

The SPO \(>_\textsf{spo}\) induced from coincides with \(>_\mathsf {wpo'}\). For all terms s and t the relation \(s >_\textsf{gwpo}t\) is equivalent to \(s \geqslant _\mathcal {A}t\) and \(s >_\mathsf {wpo'}t\).    \(\square \)

Corollary 2

The relations \(>_\textsf{gwpo}\) and \(>_\textsf{wpo}\) coincide, provided that \(\mathcal {A}\) is simple and \(f_\mathcal {A}({x_1},\dots ,{x_{n}}) = f^\sharp _\mathcal {A}({x_1},\dots ,{x_{n}})\) for all \(f^{(n)} \in \mathcal {F}\).    \(\square \)

Since polynomial interpretation orders [15] and Knuth–Bendix orders [13] as well as LPOs are simulated by WPOs [27], they are also subsumed by GWPOs. We demonstrate termination proofs by GWPOs with a few examples. All examples are not handled by WPOs.

Example 7

Consider the TRS \(\mathcal {R}\) for round-up division:

$$\begin{aligned} \textsf{p}(\textsf{0})&\rightarrow \textsf{0}&x - \textsf{0}&\rightarrow x&\textsf{0} \div \textsf{s}(y)&\rightarrow \textsf{0} \\ \textsf{p}(\textsf{s}(x))&\rightarrow x&x - \textsf{s}(y)&\rightarrow \textsf{p}(x) - y&\textsf{s}(x) \div \textsf{s}(y)&\rightarrow \textsf{s}((x - y) \div \textsf{s}(y)) \end{aligned}$$

Let \(\mathcal {A}\) be the weakly monotone algebra on \(\mathbb {N}\) with the interpretations

$$\begin{aligned} \textsf{0}_\mathcal {A}&= 0&\textsf{s}_A(x)&= x + 1&\textsf{p}_\mathcal {A}(x)&= x&x -_\mathcal {A}y&= x&x \div _\mathcal {A}y&= x \\ \textsf{0}^\sharp _\mathcal {A}&= 0&\textsf{s}^\sharp _\mathcal {A}(x)&= 0&\textsf{p}^\sharp _\mathcal {A}(x)&= 0&x -^\sharp _\mathcal {A}y&= y&x \div ^\sharp _\mathcal {A}y&= x + y \end{aligned}$$

and let \(\succsim \) be an arbitrary precedence. The GWPO induced from \(\mathcal {A}\) and \(\succsim \) orients all rules in \(\mathcal {R}\). In particular, \(x - \textsf{s}(y) >_\mathsf {wpo'}\textsf{p}(x) - y\) is derived from the inequalities \(x -^\sharp \textsf{s}(y) >_\mathcal {A}\textsf{p}(x) -^\sharp y\) and \(x -^\sharp \textsf{s}(y) >_\mathcal {A}\textsf{p}^\sharp (x)\).

Example 8

Consider the TRS \(\mathcal {R}\) taken from [2, Example 4.28], which computes the bit length of a natural number:

figure bb

Let \(\mathcal {A}\) be the weakly monotone algebra on \(\mathbb {N}\) with:

$$\begin{aligned} \textsf{0}_\mathcal {A}&= 0&\textsf{s}_\mathcal {A}(x)&= x + 1&\textsf{half}_\mathcal {A}(x)&= \max \{ 0, x - 1 \}&\textsf{bits}_\mathcal {A}(x)&= x \\ \textsf{0}^\sharp _\mathcal {A}&= 0&\textsf{s}^\sharp _\mathcal {A}(x)&= x + 1&\textsf{half}^\sharp _\mathcal {A}(x)&= \max \{ 0, x - 1 \}&\textsf{bits}^\sharp _\mathcal {A}(x)&= x \end{aligned}$$

The GWPO \(>_\textsf{gwpo}\) induced by \(\mathcal {A}\) and a precedence \(\succsim \) with \(\textsf{half}, \textsf{bits} \succ \textsf{s}\) satisfies \(\mathcal {R}\subseteq {>_\textsf{gwpo}}\) as \(\ell \geqslant _\mathcal {A}r\) and \(\ell >_\mathsf {wpo'}r\) for all rules \(\ell \rightarrow r \in \mathcal {R}\). In particular, \(\textsf{bits}(\textsf{s}(x)) >_\mathsf {wpo'}\textsf{s}(\textsf{bits}(\textsf{half}(\textsf{s}(x))))\) is derived as follows. The inequality \(\textsf{bits}(\textsf{s}(x)) >_\mathsf {wpo'}\textsf{bits}(\textsf{half}(\textsf{s}(x)))\) is derived from repeated application of WPO\('\) 2a:

figure bc

Thus, \(\textsf{bits}(\textsf{s}(x)) >_\mathsf {wpo'}\textsf{s}(\textsf{bits}(\textsf{half}(\textsf{s}(x))))\) follows from WPO\('\) 2b with \(\textsf{bits} \succ \textsf{s}\) and \(\textsf{bits}^\sharp (\textsf{s}(x)) \geqslant _\mathcal {A}\textsf{s}^\sharp (\textsf{bits}(\textsf{half}(\textsf{s}(x))))\).

6 Experimental Results

In order to evaluate GWPOs in termination analysis we implemented a prototype termination tool based on Proposition 1 and Corollary 1. Following the automation techniques of WPO [27], we search a suitable weakly monotone well-founded algebra from two classes of algebras over \(\mathbb {N}\): One is linear interpretation and the other is max/plus interpretation. Since simplicity of algebras is not required for GWPOs, we may use more general forms of interpretations.

Linear Interpretations. Algebras \(\mathcal {A}\) of this class use linear polynomials over \(\mathbb {N}\) like Example 7. For each \(f^{(n)} \in \mathcal {F}\cup \mathcal {F}^\sharp \) its interpretation is of the form \(f_\mathcal {A}(x_1, \ldots , x_n) = c_0 + c_1 x_1 + \cdots + c_n x_n\) where \(c_0 \in \mathbb {N}\) and \(c_1, \ldots , c_n \in \{ 0, 1 \}\). Simple monotone algebras for WPOsFootnote 3 are obtained by setting \(c_1 = \cdots = c_n = 1, f_\mathcal {A}= f^\sharp _\mathcal {A}\) for all \(f^{(n)} \in \mathcal {F}\), and those for Knuth–Bendix orders (KBOs) are obtained by further restriction for admissibility, see [27]. Comparison of linear polynomials is reduced to that of coefficients by using the following trivial fact:

Proposition 4

Let \(f({x_1},\dots ,{x_{n}}) = c_0 + c_1 x_1 + \cdots + c_n x_n\) and \(g({x_1},\dots ,{x_{n}}) = d_0 + d_1 x_1 + \cdots + d_n x_n\) be linear polynomials over \(\mathbb {N}\). The next statements hold.

  • \(f \geqslant g\) if and only if \(c_0 \geqslant d_0\) and \(c_i \geqslant d_i\) for all \(1 \leqslant i \leqslant n\).

  • \(f > g\) if and only if \(c_0 > d_0\) and \(c_i \geqslant d_i\) for all \(1 \leqslant i \leqslant n\).

Here \(f \geqslant g\) (\(f > g\)) means that \(f({a_1},\dots ,{a_{n}}) \geqslant g({a_1},\dots ,{a_{n}})\) (\(f({a_1},\dots ,{a_{n}}) > g({a_1},\dots ,{a_{n}})\)) for all \({a_1},\dots ,{a_{n}} \in \mathbb {N}\).

Max/plus Interpretations. Algebras \(\mathcal {A}\) of this class use a combination of \(+\) and \(\max \) like Example 8. For each \(f^{(n)} \in \mathcal {F}\cup \mathcal {F}^\sharp \) its interpretation is of the form \(f_\mathcal {A}(x_1, \ldots , x_n) = \max \{c_0, c_1 + c'_1 x_1, \cdots , c_n + c'_n x_n\}\) where \(c_0 \in \mathbb {N}, c_1, \ldots , c_n \in \mathbb {Z}\) and \(c'_1, \ldots , c'_n \in \{ 0, 1 \}\). Simple monotone algebras for WPOs are obtained by imposing \(c_1, \ldots , c_n \in \mathbb {N}, c'_1 = \cdots = c'_n = 1, f_\mathcal {A}= f^\sharp _\mathcal {A}\) for all \(f^{(n)} \in \mathcal {F}\), and algebras for lexicographic path orders (LPOs) are obtained by additionally setting \(c_0 = c_1 = \cdots = c_n = 0\) for all \(f^{(n)} \in \mathcal {F}\) as in [27]. The restriction \(c_1, \ldots , c_n \in \mathbb {N}\) is necessary for WPOs because allowing \(c_1, \ldots , c_n < 0\) results in non-simple interpretations such as \(\max \{0, x -1\}\). Under this form of algebras, an interpretation of a term is flattened to the form of \(\max \{g_1, \ldots , g_m\}\) where \(g_1, \ldots , g_m\) are linear polynomials over \(\mathbb {N}\). So comparison of max/plus interpretation is reduced to that of coefficients, using the following trivial fact and Proposition 4 in turn:

Proposition 5

Let G and H be non-empty sets of linear polynomials over \(\mathbb {N}\). The next statements hold.

  • \(\max G \geqslant \max H\) if and only if for every \(h \in H\) there exists a linear polynomial \(g \in G\) with \(g \geqslant h\).

  • \(\max G > \max H\) if and only if for every \(h \in H\) there exists a linear polynomial \(g \in G\) with \(g > h\).

Since precedence constraints can be regarded as inequalities on natural numbers [28], searching a suitable combination of a precedence and an interpretation is done by solving linear arithmetic constraints (with if-then-else expressions).

The problem set for experiments consists of 1511 term rewrite systems from version 11.3 of the Termination Problem Database (TPDB) [21]. The reference implementation uses the SMT solver Z3 [17] as an external tool for solving linear constraints. The experiments were run on a PC with Intel Core i7-1065G7 CPU (1.30 GHz) and 16 GB memory.

Now let us discuss the experimental results.Footnote 4 Table 1 shows that, as a whole, use of non-simple algebras substantially improves termination analysis, at the small cost of extra running time. In particular, in the case of linear interpretation, GWPOs significantly outperform WPOs. As a matter of fact, linear WPOs are unable to orient variable duplicating rules \(\ell \rightarrow r\) such as \(\textsf{f}(x) \rightarrow \textsf{g}(x, x)\) since \(\ell \geqslant _\mathcal {A}r\) cannot be satisfied, but this does not apply to GWPOs based on linear interpretations with \(\{ 0, 1 \}\)-coefficients. In the case of max/plus interpretations there are two TRSs (with over 100 rules) that are proved to be terminating by WPOs, but not by GWPOs due to the time limit. This indicates that using non-simple algebras for max/plus interpretation can result in increase of search space. This is not the case for linear interpretations.

Table 1. Experiments on 1511 TRSs from TPDB 11.3.

7 Simulating Dependency Pairs by GWPOs

The powerfulness of GWPOs revealed in Sect. 6 can partly be explained by the fact that GWPO is capable of simulating a basic result of the dependency pair method [1]. To show the fact, we recall the dependency pair method. The set \(\textsf{DP}(\mathcal {R})\) of dependency pairs of a TRS \(\mathcal {R}\) is defined as follows:

figure bd

An order pair \(({\geqslant }, {\sqsupset })\) on terms is a reduction pair if \(\geqslant \) is a rewrite preorder and \(\sqsupset \) is a well-founded stable order. The following theorem states a basic result of the dependency pair method.

Theorem 5

([1]). A TRS \(\mathcal {R}\) is terminating if \(\mathcal {R}\subseteq {\geqslant }\) and \(\textsf{DP}(\mathcal {R}) \subseteq {\sqsupset }\) for some reduction pair \(({\geqslant }, {\sqsupset })\).

We illustrate Theorem 5, using the fact that every weakly monotone algebra \(\mathcal {A}\) on \(\mathbb {N}\) induces the reduction pair \(({\geqslant _\mathcal {A}}, {>_\mathcal {A}})\).

Example 9

Consider the TRS \( \mathcal {R}= \{ \textsf{f}(\textsf{f}(x)) \rightarrow \textsf{f}(\textsf{g}(\textsf{f}(x))), \textsf{f}(x) \rightarrow \textsf{g}(x) \} \). We show the termination of \(\mathcal {R}\) using Theorem 5. The set \(\textsf{DP}(\mathcal {R})\) consists of the two dependency pairs:

$$\begin{aligned} \textsf{f}^\sharp (\textsf{f}(x))&\rightarrow \textsf{f}^\sharp (\textsf{g}(\textsf{f}(x)))&\textsf{f}^\sharp (\textsf{f}(x))&\rightarrow \textsf{f}^\sharp (x) \end{aligned}$$

By taking the \(\{\textsf{f},\textsf{g}, \textsf{f}^\sharp ,\textsf{g}^\sharp \}\)-algebra \(\mathcal {A}\) with the interpretations

$$\begin{aligned} \textsf{f}_\mathcal {A}(x)&= x + 1&\textsf{g}_\mathcal {A}(x)&= 0&\textsf{f}^\sharp _\mathcal {A}(x)&= x&\textsf{g}^\sharp _\mathcal {A}(x)&= 1 \end{aligned}$$

the inclusions \(\mathcal {R}\subseteq {\geqslant _\mathcal {A}}\) and \(\textsf{DP}(\mathcal {R}) \subseteq {>_\mathcal {A}}\) hold. Hence, \(\mathcal {R}\) is terminating.

We show that every termination proof by Theorem 5 with a weakly monotone algebra on \(\mathbb {N}\) can be simulated by a GWPO. This class of algebras include linear polynomial interpretations and max/plus interpretations described in Sect. 6. Let \(\mathcal {R}\) be a TRS and \(\mathcal {A}\) a weakly monotone \((\mathcal {F}\cup \mathcal {F}^\sharp )\)-algebra on \(\mathbb {N}\) satisfying \(\mathcal {R}\subseteq {\geqslant _\mathcal {A}}\) and \(\textsf{DP}(\mathcal {R}) \subseteq {>_\mathcal {A}}\). Define the \((\mathcal {F}\cup \mathcal {F}^\sharp )\)-algebra \(\mathcal {B}\) on \(\mathbb {N}\) by

$$\begin{aligned} f_\mathcal {B}({a_1},\dots ,{a_{n}})&= f_\mathcal {A}({a_1},\dots ,{a_{n}}) \\ f^{\sharp }_\mathcal {B}({a_1},\dots ,{a_{n}})&= {\left\{ \begin{array}{ll} f^\sharp _\mathcal {A}({a_1},\dots ,{a_{n}}) + 1 &{} \text {if } f \in \mathcal {D}_\mathcal {R}\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

for each \(f^{(n)} \in \mathcal {F}\). Let \(>_\textsf{gwpo}\) and \(>_\mathsf {wpo'}\) denote the orders induced from \(\mathcal {B}\) and an arbitrary but fixed precedence. First, let us see that \(\mathcal {R}\subseteq {>_\textsf{gwpo}}\) holds for the last example.

Example 10 (continued from Example 9)

The corresponding algebra \(\mathcal {B}\) is:

$$\begin{aligned} \textsf{f}_\mathcal {B}(x)&= x + 1&\textsf{g}_\mathcal {B}(x)&= 0&\textsf{f}^\sharp _\mathcal {B}(x)&= x + 1&\textsf{g}^\sharp _\mathcal {B}(x)&= 0 \end{aligned}$$

We have \(\mathcal {R}\subseteq {\geqslant _\mathcal {B}}\) by construction. The inequality \(\textsf{f}(\textsf{f}(x)) >_\mathsf {wpo'}\textsf{f}(\textsf{g}(\textsf{f}(x)))\) is derived by successive application of WPO\('\) 2a as follows:

figure be

The inequality \(\textsf{f}(x) >_\mathsf {wpo'}\textsf{g}(x)\) follows from \(\textsf{f}^\sharp (x) >_\mathcal {B}\textsf{g}^\sharp (x)\). Hence \(\mathcal {R}\subseteq {>_\textsf{gwpo}}\). Note that neither \(\textsf{f}^\sharp (\textsf{f}(x)) >_\mathcal {A}\textsf{g}^\sharp (\textsf{f}(x))\) nor \(\textsf{f}^\sharp (x) >_\mathcal {A}\textsf{g}^\sharp (x)\) holds.

Now we verify that \(\mathcal {R}\subseteq {>_\textsf{gwpo}}\) holds in general. By construction \(\mathcal {R}\subseteq {\geqslant _\mathcal {B}}\) is immediate from \(\mathcal {R}\subseteq {\geqslant _\mathcal {A}}\). So it remains to show \(\mathcal {R}\subseteq {>_\mathsf {wpo'}}\). We prove the following stronger property.

Lemma 9

Let \(\ell \rightarrow r \in \mathcal {R}\). For every subterm t of r the relation \(\ell >_\mathsf {wpo'}t\) holds.

Proof

We use structural induction on t. If t is a variable, then x must be a subterm of \(\ell \), and thus \(\ell >_\mathsf {wpo'}t\). Otherwise, t is in the form of \(g({t_1},\dots ,{t_{n}})\). The induction hypothesis yields \(\ell >_\mathsf {wpo'}t_j\) for all \(1 \leqslant j \leqslant n\). We claim \(\ell ^\sharp >_\mathcal {B}t^\sharp \), from which the desired inequality \(\ell >_\mathsf {wpo'}t\) follows by WPO\('\) 2a. To show the claim, consider an arbitrary assignment \(\alpha \) for \(\mathcal {B}\). Depending on g, we distinguish two cases.

  • If \(g \notin \mathcal {D}_\mathcal {R}\) then \( [\alpha ]_\mathcal {B}(\ell ^\sharp ) = [\alpha ]_\mathcal {A}(\ell ^\sharp ) + 1 > 0 = [\alpha ]_\mathcal {B}(t^\sharp ) \).

  • If \(g \in \mathcal {D}_\mathcal {R}\) then \(\ell ^\sharp \rightarrow t^\sharp \in \textsf{DP}(\mathcal {R})\). The assumption \(\textsf{DP}(\mathcal {R}) \subseteq {>_\mathcal {A}}\) yields the inequality \(\ell ^\sharp >_\mathcal {A}t^\sharp \). Thus, \( [\alpha ]_\mathcal {B}(\ell ^\sharp ) = [\alpha ]_\mathcal {A}(\ell ^\sharp ) + 1 > [\alpha ]_\mathcal {A}(t^\sharp ) + 1 = [\alpha ]_\mathcal {B}(t^\sharp ) \).

In either case \([\alpha ]_\mathcal {B}(\ell ^\sharp ) > [\alpha ]_\mathcal {B}(t^\sharp )\) is obtained. Hence, \(\ell ^\sharp >_\mathcal {B}t^\sharp \) holds.    \(\square \)

Theorem 6

The inclusion \(\mathcal {R}\subseteq {>_\textsf{gwpo}}\) holds.    \(\square \)

8 Conclusion

We have shown that weighted path orders can be simulated by a suitable variant of SPOs based on order pairs, and introduced a generalization of WPOs whose termination proving power goes beyond the realm of simple termination. To conclude the paper, we discuss related work and future work.

Simulating KBOs by SPOs. A key observation for simulating WPOs by SPOs is that weight comparison can be simulated by successive application of SPO 1 and SPO 2a as observed in Example 6. Another observation is that the SPOs are already reduction orders without a help of harmonious rewrite preorders. These two observations owe to Geser’s work [9, Theorem 5], where it is shown that extended KBOs [7, Sect. 5] can be simulated by SPOs. Unifying our result and Geser’s result is future work.

General Path Orders. In this paper the lexicographic versions of path orders were investigated. However, it is very likely that the same result can be obtained even if we adopt multiset comparison or status functions. General path orders (GPOs) [8, 10] are a unifying framework for such extensions, parameterizing the way to compare arguments. It is worth investigating simulation results between GPOs and WPOs by extending the parameters of GPOs so as to take order pairs.

Reduction Pairs Based on WPOs. In order to build reduction pairs from WPOs Yamada et al. [27, Sect. 4] extended the definition of WPOs by the notion of partial status function \(\pi \). The extension allows us to specify argument positions \(\pi (f) = [{i_1},\dots ,{i_{m}}]\) compared in WPO 2b and WPO 2b(ii) for each function symbol \(f^{(n)} \in \mathcal {F}\cup \mathcal {F}^\sharp \). We anticipate that partial status functions can also be integrated into GWPOs and the thus-obtained version characterizes the reduction pair version of WPOs.