Unification with Expansion Variables: Preliminary Results and Problems

Expansion generalises substitution. An expansion is a special term whose leaves can be substitutions. Substitutions map term variables to ordinary terms and expansion variables to expansions. Expansions (resp., ordinary terms) may contain expansion variables, each applied to an argument expansion (resp., ordinary term). Instances of the unification problem in this setting are constraint sets, where constraints are pairs of ordinary terms, and unifiers are expansions. This problem offers many interesting challenges. The theory of unification with expansion variables was first considered in relation to the study of systems of intersection types for the λ-calculus. Solving constraint sets, under appropriate conditions, corresponds to type inference for lambda-terms in these systems. We explain expansions and present a simple rewrite system for unification with expansion variables where ordinary terms uses the intersection type constructors. The simple rewrite system lacks some important properties. We indicate how it can be adapted to: simulate β-reduction, and intersection typing, of λ-terms; be a complete semi-decision procedure for unification; be confluent; produce most-general unifiers. Every constraint set has a trivial unifier. However, finding a single most-general unifier is often impossible. We study the concept of mostgeneral unifiers and introduce principal unifiers, which are easier to construct. Most-general unifiers exist for the unification problem formed by a certain restriction of substitutions, and we give an incomplete variant of simple unification to that finds them. A second variant system addresses completeness and principality, producing covering substitution-unifier sets for constraints (every substitutionunifier is an instance of a set member, and all expansion-unifiers can be obtained from the set). For covering unifier sets we modify the problem to a form of E-unification where the constant ω is the unit of the intersection constructor. 1 Background and Motivation The study of unification with expansion variables has both practical and theoretical ramifications. The motivation comes from type systems for the λ-calculus ? Work partly funded by NSF grant CCR-0113193 Implementing Modular Program Analysis via Intersection and Union Types. and functional programming languages. The theoretical framework gives rise to a host of problems whose solutions require appropriately adapted algebraic and combinatorial techniques. The key novelty in our framework is the concept of an expansion variable. We start right off with a brief tutorial on expansion concepts in Section 1.1. The connection with type systems is briefly discussed in Section 1.2. The scope and organization of the paper are presented in Sections 1.3 and 1.4. 1.1 Expansion concepts An expansion generalizes the notion of substitution. It is a term whose leaves can be substitutions. For example, an expansion called E1 may be defined by E1 = ({a1 7→ a1 ⊗ a1} ⊗ a2)⊗ {a1 7→ a3} where ⊗ is a binary term constructor and each ai is a term variable (T-variable). Replacing each substitution leaf in E1 by the identity substitution, we obtain the term part of E1, namely ({}⊗a2)⊗{}. Applying the expansion E1 to a T-variable, say a1, written [E1] a1, results in the term part of E1 with each substitution leaf S replaced by S(a1). Thus, [E1] a1 = ((a1 ⊗ a1)⊗ a2)⊗ a3. Applying the expansion E1 to a plain term τ1 (without expansion variables) results in the term part of E1 with each substitution leaf S replaced by S(τ1). For example, if τ1 = (a1 ⊗ a1), then [E1] τ1 = [E1] (a1 ⊗ a1) = (((a1 ⊗ a1)⊗ (a1 ⊗ a1))⊗ a2)⊗ (a3 ⊗ a3). An expansion variable (E-variable) e, then, is a kind of function variable. Occurrences in terms are always applied to one argument. We say that E-variable e wraps its argument; the namespace of a subterm is the sequence of E-variables encountered on the path to it from the root of the term; and e occurs outermost if its namespace is empty. For example, in τ2 = e1 (a1 ⊗ e2 a1), E-variable e1 occurs outermost; e2 is in the namespace e1 and T-variable a1 has occurrences in the namespaces e1 and e1 · e2. Substitutions are extended to map E-variables to expansions as well as T-variables to terms. Substitution application is defined such that only the outermost variables of the argument are affected. For a more involved example, consider the following expansion E2: E2 = ({} ⊗ S1)⊗ S2 where S1 = {a1 7→ a4, e2 7→ E1}, S2 = {a1 7→ a5, e2 7→ e3 {a1 7→ a6}} 1 The notation {a1 7→ a1 ⊗ a1} defines the support of a total function f ; i.e. f(x) = x for all x except a1 where f(a1) = a1 ⊗ a1. The identity substitution is therefore the function whose support is {}, the empty set.