Semantic nominal terms

Nominal terms use a characteristic combination of features: Two levels of variable: atoms a and unknowns X. Freshness conditions a#X and permutations π. An abstraction [a]r with a distinctive α-equivalence based on freshness and permutation. The first statement above is rendered in nominal terms as the equality judgement b#X ` [a]X =α [b](b a) ·X. a and b denote atoms, which represent the ‘x’ and ‘y’; X denotes an unknown, it represents the ‘t’; b#X is a freshness side-condition, it represents the ‘y 6∈ fv(t)’; (b a) is a permutation meaning ‘map a to b and b to a’, it represents the ‘[y/x]’ (it can do this, since we assume y 6∈ fv(t)). The second statement above is rendered as the nominal unification problem {λ[a]λ[b](b X) ?=? λ[a]λ[a](a Y )}. Nominal terms display some curious properties: −The native notion of equivalence of first-order syntax is syntactic identity. For nominal terms, a notion of derivable α-equality is also needed. We cannot ‘just quotient terms by α-equality’ because α-equality is not a property of terms but of terms-in-freshness-contexts. Thus [a]X = [b](b a) ·X has no meaning, only, for example, ∅ ` [a]X = [b](b a) ·X and b#X ` [a]X = [b](b a) ·X have meaning (the former is not derivable, the latter is). −With first-order and higher-order terms, it is always possible to ‘just pick a fresh name’, and therefore we can ‘always α-convert’. In nominal terms it may not be possible to ‘just pick a fresh atom’ or ‘just αconvert’. For example in the empty freshness context ∅, there is no a such that a#X, and there is no atom b such that we can α-convert a to b in ∅ ` [a]X. 1 Nominal terms also raise a question which has puzzled us for some time. Atoms in nominal terms are denoted by themselves in the cumulative hierarchy (Definition 2.1, or [GP01]). Unknowns take denotation using a valuation (the standard first-order semantics for variables). There is nothing wrong with this but we find it inelegant because it is asymmetric with the denotation of atoms (and also, using functions to give semantics to unknowns is a circularity we want to avoid; one motivation of nominal terms is to axiomatise function binding). We want an alternative