An affine function h -> a * h + b.

Composition of affine functions: (f2 . f1)(h) = f2(f1(h)).

>>> let f1 = Aff 2 3; f2 = Aff 5 7 in affComp f2 f1
Aff {affA = 10.0, affB = 22.0}

Because (f2 . f1)(h) = 5*(2*h+3)+7 = 10*h + 22.

Sequential scan of a scalar SSM from an initial state.

>>> seqSSM 0 [Aff 1 1, Aff 1 2, Aff 1 3]
[1.0,3.0,6.0]

Associative scan on affine coefficients. Returns the list of composed affine functions [f1, f2 . f1, f3 . f2 . f1, ...].

The operator is reversed composition because scanl' threads the accumulator on the left, but we want the rightmost function applied first.

Apply the associatively-scanned affine functions to an initial state.

Tree-shaped associative scan with a given chunk size.

Each chunk is reduced to a summary affine function, the summaries are prefix-composed, and the result is expanded back to a per-step prefix. If affComp is associative, this must agree with assocScan for every chunk size. This is the genuine parallelisation principle behind S4 Mamba linear attention.

Elementwise affine function on a harpie 1-D array: h_i -> a_i * h_i + b_i. This is the diagonal-matrix special case of the full matrix SSM; it already exercises the associative-scan law on arrays.

Elementwise composition of diagonal affine functions.

Sequential scan of a vector SSM.

Associative scan on vector affine coefficients. The shape is inherited from the first input.

Apply the associatively-scanned vector affine functions to an initial state.

Run a deterministic System with a monomial interface over a list of inputs. This is the same semantics as scan, but stated directly on System.

Vector SSM as a 'System (->)' with harpie state, input AffVec, and full state observation.

Two independent vector SSM heads packaged as a single System over the Dirichlet tensor Tensor (Mono AffVec (Array Double)) (Mono AffVec (Array Double)).

The two heads share the same input direction type (AffVec) but receive independent direction values. This is the right polynomial for parallel layers: both heads fire on the same tick, each with its own input. A cartesian Prod would force a choice between heads via Either directions.

Run the multi-head SSM with independent per-head inputs.

Returns the pair of head outputs at each step and the final pair of states.

Run the multi-head SSM with the same input supplied to both heads.

This is the diagonal shared-input case; building it from independent inputs requires an explicit copy of the input direction.

A Process whose state is the hidden state h and whose output is h. Input is the affine coefficient pair (a_t, b_t).

This is the first-input-seeded presentation with h0 = 0. Use ssmSystem with systemToProcess when you need a non-zero seed.

A System whose state is the hidden state h and whose output is h. Input is the affine coefficient pair (a_t, b_t); the initial state h0 is supplied when converting to a Process or running directly.

Two-headed SSM with a cross-head coupling: head 1 reads head 2's state.

This breaks premonoidal centrality: the order in which the two heads are threaded through the shared medium matters, because head 1's update depends on head 2's state. It is the "flip" oracle for the multi-head centrality pair.