Variants in the Infinitary Unification Wonderland

Abstract

So far, results about variants, the finite variant property (FVP), and variant unification have been developed for equational theories $E \cup B$ where B is a set of axioms having a finitary unification algorithm, and the equations E, oriented as rewrite rules $\vec {E}$, are convergent modulo B. The extension to the case when B has an infinitary unification algorithm, for example because of non-commutative symbols having associative axioms, seems undeveloped. This paper takes a first step in developing such an extension. In particular, the relationships between the FVP and the boundedness properties, the identification of conditions on $E \cup B$ ensuring FVP, and the effective computation of variants and variant unifiers are explored in detail. The extension from the finitary to the infinitary case includes both surprises and opportunities.

A Proof of Theorem 6

We just need to show that for each $\mathbf{M} \in \mathcal {C}$ and sort s in $\mathbf{M}$, $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$ is FB. But: (i) FB is a rule-local property: it holds iff it does for each rewrite rule; (ii) up to renaming to ensure symbol disjointness, the rewrite rules in $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$ are just the disjoint union of those in $\vec {\mathbf {St}}[X]$ and those in $\mathbf{M}$; (iii) by the construction of $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$, the rewrite rules in $\mathbf{M}$ apply to the exact same terms in both $\mathbf{M}$ and in $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$, and, since both the rules and the terms they apply to do not involve any operators in $\vec {\mathbf {St}}[X]$, narrowing with those rules modulo the entire set of axioms $A \uplus B_{\mathbf {M}}$, where A is the associativity axiom in $\vec {\mathbf {St}}[X]$, is identical with narrowing with such rules modulo $B_{\mathbf {M}}$ only. Therefore, the rules from $\mathbf{M}$ are FB in $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$. Thus, all we need to check is that the rewrite rules in $\vec {\mathbf {St}}[X]$ are FB in $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$. First of all, note that the axioms $A \uplus B_{\mathbf {M}}$ in $\vec {\mathbf {St}}[\vec {\mathbf {M}},X \mapsto s]$ only involve associativity and/or commutativity axioms. Since FB is a property satisfied by each rule, we just reason one rule at a time. I prove the FB property for the rules: (1) $u \; \varepsilon \rightarrow u$ (2) $first (x)\rightarrow x$, (3) $first (x \;q)\rightarrow x$, (4) $rest (x)\rightarrow \varepsilon$, and (5) $rest (x \; q)\rightarrow q$.

Case (1). Up to A-equivalence and disregarding parentheses, the $A \uplus B_{\mathbf {M}}$-unification problem $u \; \varepsilon = w$, where w is a non-variable term in $\vec {\mathbf {St}}[ s ]$ not sharing any variables with $u \; \varepsilon$ is just the $A \uplus B_{\mathbf {M}}$-unification problem $u \; \varepsilon = w_{1} \; \ldots \; w_{n}$ with $n \geqslant 1$ where each $w_{i}$ is either: (i) a variable of sort $St$ or (ii) a so-called A-alien subterm or constant of the form $f(t_1 , \ldots , t_n)$ with f different from $\_\;\_$. When $n=1$, the only possible $A \uplus B_{\mathbf {M}}$-unifier exists when $w_{1}$ is a variable v of sort $St$ and is the unifier $\{v \mapsto u \; \varepsilon \}$. When $n\geqslant 2$, the only possible $A \uplus B_{\mathbf {M}}$-unifiers exist when either: (i) $w_{n} = \varepsilon$, with unifier $\{u \mapsto w_{1} \; \ldots \; w_{n-1}\}$, or (ii) $w_{n}$ is a variable v of sort $St$, with unifier $\{u \mapsto w_{1} \; \ldots \; w_{n-1}, v \mapsto \varepsilon \}$.

Case (2). Up to A-equivalence and disregarding parentheses, the $A \uplus B_{\mathbf {M}}$-unification problem $first (x) = w$ can only be solved if w is a term of the form $first (w_{1} \; \ldots \; w_{n})$ with $w_{y}$ a variable of sort $NeSt$ or less, or an A-alien subterm of sort s or less, and has a solution only when $n=1$ and either: (i) $w_{1}$ is a variable $q'$ of sort $NeSt$, yielding the $A \uplus B_{\mathbf {M}}$-unifier $\{q' \mapsto x\}$, or (ii) $w_{1}$ is a variable y of sort s or less or an A-alien subterm of sort s or less, yielding the $A \uplus B_{\mathbf {M}}$-unifier $\{x \mapsto w_{1}\}$.

Case (3). Up to A-equivalence and disregarding parentheses, the $A \uplus B_{\mathbf {M}}$-unification problem $first (x \; q) = w$ can only be solved if w is a term of the form $first (w_{1} \; \ldots \; w_{n})$ with $w_{y}$ a variable of sort $NeSt$ or less, or an A-alien subterm of sort s or less, and has a solution only when: (i) $n=1$ and $w_{1}$ is a variable $q'$ of sort $NeSt$, yielding the $A \uplus B_{\mathbf {M}}$-unifier $\{q' \mapsto x \; q\}$, or (ii) $n \geqslant 2$ and either (ii).1 $w_{1}$ is a variable $q'$ of sort $NeSt$, yielding the $A \uplus B_{\mathbf {M}}$-unifier $\{q' \mapsto x \; q \mapsto w_{2} \; \ldots \; w_{n}\}$, or (ii).2 $w_{1}$ is either a variable of sort s or less, or an A-alien subterm of sort s or less, yielding the $A \uplus B_{\mathbf {M}}$-unifier $\{x \mapsto w_{1}, q \mapsto w_{2} \; \ldots \; w_{n}\}$.

Cases (4), resp. (5), have proofs entirely analogous to Cases (2), resp. (3).

This finishes the proof of Theorem 6.    $\Box$