Displacement Calculus

Origin. Morrill (2010). Discontinuity in Lambek. Intercalation and wrap. String-tuple semantics. Foundation for discontinuous constituency.

Models. Sorted strings with placeholders. Hypersequents. Intercalation: insert at marked position. Phonological discontinuity.

Formalism.

Sorts: 0: continuous (strings) 1: one discontinuity (string pairs) n: n discontinuities

Sort 0 types: Standard Lambek: A • B, A \ B, A / B

Sort 1 types: A ⊙ B: discontinuous product A ↑ B: right wrap A ↓ B: left wrap

Discontinuous product: ⟨s₁, s₂⟩ : A ⊙ B s₁ and s₂ separated.

Wrap: ⟨s₁, s₂⟩ : A ↑ B s₁ "wraps around" s₂. Result: s₁ + s₂_before + s₂_after

Intercalation: ⟨s₁, s₂⟩ • t = s₁ + t + s₂ Fill the gap.

Hypersequent: Γ ⊢ᵢ A (sort i sequent) Track discontinuities.

Linguistics: "the person... that I saw" "give... up": particle verbs Extraction: ⟨wh, gap⟩

Focusing: Structured proof search. Complexity control.

Symbols.

Symbol	Unicode	Name	Meaning
⊙	U+2299	D-product	Discontinuous
↑	U+2191	Up	Right wrap
↓	U+2193	Down	Left wrap
⟨,⟩	—	Pair	String with gap

Metatheory. Decidable. Mildly context-sensitive. Polynomial parsing. Cut elimination.

Applies to. Linguistic discontinuity. Wh-movement. Particle verbs. Quantifier scope. Typological variation.

Limitations. Complex type system. Parsing efficiency. Linguistic coverage debates. Multiple sorts.