Subsumption lattice

A subsumption lattice is a mathematical structure used in the theoretical background of automated theorem proving and other symbolic computation applications.

Pic. 1: Non-modular sublattice N5 in subsumption lattice

Definition

A term t1 is said to subsume a term t2 if a substitution σ exists such that σ applied to t1 yields t2. In this case, t1 is also called more general than t2, and t2 is called more specific than t1, or an instance of t1.

The set of all (first-order) terms over a given signature can be made a lattice over the partial ordering relation "... is more specific than ..." as follows:

  • consider two terms equal if they differ only in their variable naming,[1]
  • add an artificial minimal element Ω (the overspecified term), which is considered to be more specific than any other term.

This lattice is called the subsumption lattice. Two terms are said to be unifiable if their meet differs from Ω.

Properties
Pic. 2: Part of the subsumption lattice showing that the terms f(a,x), f(x,x), and f(x,c) are pairwise unifiable, but not simultaneously. (f omitted for brevity.)

The join and the meet operation in this lattice are called anti-unification and unification, respectively. A variable x and the artificial element Ω are the top and the bottom element of the lattice, respectively. Each ground term, i.e. each term without variables, is an atom of the lattice. The lattice has infinite descending chains, e.g. x, g(x), g(g(x)), g(g(g(x))), ..., but no infinite ascending ones.

If f is a binary fuction symbol, g is a unary function symbol, and x and y denote variables, then the terms f(x,y), f(g(x),y), f(g(x),g(y)), f(x,x), and f(g(x),g(x)) form the minimal non-modular lattice N5 (see Pic. 1); its appearance prevents the subsumption lattice from being modular and hence also from being distributive.

The set of terms unifiable with a given term need not be closed with respect to meet; Pic. 2 shows a counter-example.

Denoting by Gnd(t) the set of all ground instances of a term t, the following properties hold:[2]

  • t equals the join of all members of Gnd(t), modulo renaming,
  • t1 is an instance of t2 if and only if Gnd(t1) is a subset of Gnd(t2),
  • terms with the same set of ground instances are equal modulo renaming,
  • if t is the meet of t1 and t2, then Gnd(t) is the intersection of Gnd(t1) and Gnd(t2),
  • if t is the join of t1 and t2, then Gnd(t) includes the union of Gnd(t1) and Gnd(t2).

'Sublattice' of linear terms
Pic. 5: Distributive sublattice of linear terms
Pic. 4: M3 built from linear terms
Pic. 3: N5 built from linear terms

The set of linear terms, that is of terms without multiple occurrences of a variable, is a sub-poset of the subsumption lattice, and is itself a lattice. This lattice, too, includes N5 and the minimal non-distributive lattice M3 as sublattices (see Pic. 3 and Pic. 4) and is hence not modular, let alone distributive.

The meet operation yields always the same result in the lattice of all terms as in the lattice of linear terms. The join operation in the all terms lattice yields always an instance of the join in the linear terms lattice; for example, the (ground) terms f(a,a) and f(b,b) have the join f(x,x) and f(x,y) in the all terms lattice and in the linear terms lattice, respectively. As the join operations do not in general agree, the linear terms lattice is not properly speaking a sublattice of the all terms lattice.

Join and meet of two proper[3] linear terms, i.e. their anti-unification and unification, corresponds to intersection and union of their path sets, respectively. Therefore, every sublattice of the lattice of linear terms that does not contain Ω is isomorphic to a set lattice, and hence distributive (see Pic. 5).

Origin

Apparently, the subsumption lattice was first investigated by Gordon D. Plotkin, in 1970.[4]

gollark: Without computers lots of stuff will break. If you consider microcontrollers to be computers that is. Otherwise somewhat less stuff.
gollark: The designs for computers and computer bits are all stored on computers.
gollark: Without computers, transport and global supply chains will fail.
gollark: All the tooling for making computers... uses computers.
gollark: Much, much longer.

References
  1. formally: factorize the set of all terms by the equivalence relation "... is a renaming of ..."; for example, the term f(x,y) is a renaming of f(y,x), but not of f(x,x)
  2. Reynolds, John C. (1970). Meltzer, B.; Michie, D. (eds.). "Transformational Systems and the Algebraic Structure of Atomic Formulas" (PDF). Machine Intelligence. Edinburgh University Press. 5: 135–151.
  3. i.e. different from Ω
  4. Plotkin, Gordon D. (Jun 1970). Lattice Theoretic Properties of Subsumption. Edinburgh: Univ., Dept. of Machine Intell. and Perception.
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