Minimal Search as a weighting of Merge

Marcolli, Chomsky and Berwick implement Minimal Search by weighting the terms of the coproduct (their §1.5). Extracting an accessible term at distance d from its root costs ε^d, the costs of simultaneous extractions multiply, and Merge pays the weight of the terms it grafts, their M^ε_{S,S'} = ⊔ ∘ (B^ε ⊗ id) ∘ δ_{S,S'} ∘ Δ of (1.5.1) and (1.5.2).

The weighted coproduct comulEps ε scales a tree by ε for each level it is moved down on its way to the left channel. At ε = 1 it is the pruning coproduct, so the weighted Merge mergeOpEps 1 is Merge at the pruning cuts. At ε = 0 only whole components reach the left channel and the coproduct is primitive, so mergeOpEps 0 merges two components of the workspace and nothing else. At every ε, weighted Merge agrees with its value at ε = 0 up to a multiple of ε, and External and Internal Merge have weight 1 (Proposition 1.5.1): they cost nothing, every Sideward term costs at least one, and in the limit only External and Internal Merge remain.

Main definitions

• Minimalist.Merge.comulEps: the depth-weighted coproduct.
• Minimalist.Merge.mergeOpEps: Merge weighted by Minimal Search.

Main results

• Minimalist.Merge.comulEps_one, comulEps_zero: at ε = 1 the weighted coproduct is the pruning coproduct, and at ε = 0 it is the primitive coproduct.
• Minimalist.Merge.mergeOpEps_pair, mergeOpEps_im_composition: External and Internal Merge have weight 1 at every ε.
• Minimalist.Merge.mergeOpEps_eq_add_smul, mergeOpEps_of'_eq_smul_of_not_le: weighted Merge is its value at ε = 0 plus a multiple of ε, so every form of Sideward Merge is a multiple of ε.
• Minimalist.Merge.mergeOpEps_zero_of'_of_not_le, mergeOpEps_zero_pair_add: at ε = 0, Merge vanishes unless both objects are components, and External Merge survives beside any spectators that do not repeat its objects.

Implementation notes

• The book weights the left channel by ε^d and the right channel by ε^{-d}, and lets B^ε read the left weight. In a tensor product over R the two weights would cancel, so the weight is the coefficient of the left channel only. The book uses the right weight only to compute the cost d - d = 0 of Internal Merge, which here is the cost 0 of merging two components.
• The unit stage M_{β,1} grafts nothing and costs nothing (the book's rule 4), so Internal Merge is mergeOpEps ε lbl Q β ∘ mergeOpUnitG cutSummandsN β, as the book writes it in proving Proposition 1.5.1.
• The weights are put on the pruning coproduct Δ^ρ. The book's Δ is Δ^d, which differs from Δ^ρ only by contracting the unary vertices of the right channel (mergeOp_eq_comp_comulAlgHomN), the channel the weights do not read.
• The weights are polynomial in ε, and the limit ε → 0 is evaluation at ε = 0, the book's M^ε_{S,S'}|_{ε=0}.

References

• Marcolli et al. 2025, Mathematical Structure of Syntactic Merge: An Algebraic Model for Generative Linguistics formalized