Algebraic Effects

Operation, EffectSignature, Handler, FreeMonad, Plotkin-style effect signatures with handler-based interpretation.

algebraic

Algebraic effects + handlers.

Following Plotkin & Power and Bauer & Pretnar, exposes a free-monad- over-signature construction so any user-defined effect signature induces a Monad instance automatically. Handlers translate a free-monad computation into a target monad's effects.

The free-monad carrier Free_Σ^{≤d}(A) is realised as a flat FinSet whose flat-index layout is:

• Indices [0, |A|) represent leaf nodes leaf(a).
• For each operation op_i (in declared order), a contiguous block of |op_i.parameter| · |Free^{d-1}(A)|^{|op_i.result|} indices represents operation nodes op_i(p, k) where p is a parameter and k is a deterministic continuation op_i.result → Free^{d-1}(A).

The flat-FinSet representation avoids the CoproductSet auto-flattening collision that would otherwise dissolve the recursive leaf-vs-operation distinction when A is itself a coproduct. Structural decomposition is recovered from signature + depth rather than from the carrier object.

Handler interpretation walks the flat-index decomposition in post-order: leaves are routed through return_clause and operation nodes through operation_clauses.

References

• Plotkin, G. and Power, J. (2003). Algebraic operations and generic effects. Applied Categorical Structures, 11(1), 69–94. doi:10.1023/A:1023064908962.
• Bauer, A. and Pretnar, M. (2015). Programming with algebraic effects and handlers. Journal of Logical and Algebraic Methods in Programming, 84(1), 108–123. doi:10.1016/j.jlamp.2014.02.001.

Operation

Bases: Model

One operation in an effect signature.

ATTRIBUTE	DESCRIPTION
name	Operation name, used as the key in handler clauses. TYPE: str
parameter	Operation parameter type. TYPE: SetObject
result	Operation result type given to the continuation. TYPE: SetObject

EffectSignature

Bases: Model

A signature of effect operations.

Realisable both as a Python value (a tuple of operations) and as a panproto.Theory whose sorts are SetObject and whose operations are the listed Operation instances.

ATTRIBUTE	DESCRIPTION
name	Signature name; appears in panproto theory naming. TYPE: str
operations	The operations comprising the signature. TYPE: tuple of Operation

to_theory

to_theory() -> object

Realise this signature as a panproto Theory.

RETURNS	DESCRIPTION
Theory	The theory whose single sort is Carrier, whose operations are one per operations entry with input parameter * (result -> Carrier) and output Carrier, and whose equations are the three monad laws (left/right unit, associativity).

Source code in src/quivers/monadic/algebraic.py

def to_theory(self) -> object:
    """Realise this signature as a panproto Theory.

    Returns
    -------
    panproto.Theory
        The theory whose single sort is ``Carrier``, whose
        operations are one per `operations` entry with
        input ``parameter * (result -> Carrier)`` and output
        ``Carrier``, and whose equations are the three monad
        laws (left/right unit, associativity).
    """
    import panproto

    op_specs: list[dict] = []
    for op in self.operations:
        op_specs.append(
            {
                "name": op.name,
                "inputs": [
                    ("parameter", "Carrier"),
                    ("continuation", "Carrier"),
                ],
                "output": "Carrier",
            }
        )
    spec = {
        "id": f"quivers.effect.{self.name}",
        "description": f"Free monad over the {self.name} signature.",
        "theories": [
            {
                "theory": f"Th{self.name}",
                "sorts": [{"name": "Carrier"}],
                "ops": op_specs,
            }
        ],
    }
    return panproto.create_theory(spec)

Handler

Bases: Model

Interpretation of a free-monad computation in a target monad.

A handler supplies a clause for each operation in its signature plus a return-clause that lifts plain values into the target monad. run folds a bounded-depth signature tree by interpreting each leaf through return_clause and each operation node through operation_clauses.

Operation-clause shape: for an operation op whose result feeds a continuation k : op.result → target(B), the clause has signature op.parameter × [op.result → target(B)] → target(B). The implementation translates each free-monad operation node's continuation (a function-space flat index into the inner-tree carrier) into the corresponding target-side function-space continuation by recursive descent, then routes through the clause.

ATTRIBUTE	DESCRIPTION
signature	The effect signature this handler interprets. TYPE: EffectSignature
target	The target monad in which the operations are realised. TYPE: Monad
return_clause	A → target(A): lifts plain values into the target monad. TYPE: Morphism
operation_clauses	For each signature.operations entry, a morphism parameter × [result → target(B)] → target(B) (encoded with the function-space FinSet). TYPE: dict[str, Morphism]

as_theory_morphism

as_theory_morphism() -> object

Realise this handler as a panproto theory morphism.

The morphism's domain is signature.to_theory(); its codomain is the target monad's panproto theory.

Source code in src/quivers/monadic/algebraic.py

def as_theory_morphism(self) -> object:
    """Realise this handler as a panproto theory morphism.

    The morphism's domain is ``signature.to_theory()``; its
    codomain is the target monad's panproto theory.
    """
    import panproto

    spec = {
        "id": f"quivers.handler.{self.signature.name}",
        "description": (
            f"Handler from {self.signature.name} into the registered target monad."
        ),
        "theories": [
            {
                "theory": f"Th{self.signature.name}Target",
                "sorts": [{"name": "Carrier"}],
                "ops": [
                    {
                        "name": "return_clause",
                        "inputs": [("value", "Carrier")],
                        "output": "Carrier",
                    },
                    *[
                        {
                            "name": op.name,
                            "inputs": [
                                ("parameter", "Carrier"),
                                ("continuation", "Carrier"),
                            ],
                            "output": "Carrier",
                        }
                        for op in self.signature.operations
                    ],
                ],
            }
        ],
    }
    return panproto.create_theory(spec)

run

run(A: SetObject, depth: int = 3) -> Morphism

Apply this handler to a free-monad computation up to depth.

Returns a morphism Free_Σ^{≤depth}(A) → target(A).

The construction proceeds by induction on the tree depth: leaves are routed through return_clause; each operation node is decomposed into its parameter and continuation components, the continuation is recursively interpreted into the target monad, and the result is fed through the corresponding operation-clause morphism.

Source code in src/quivers/monadic/algebraic.py

def run(self, A: SetObject, depth: int = 3) -> Morphism:
    """Apply this handler to a free-monad computation up to ``depth``.

    Returns a morphism ``Free_Σ^{≤depth}(A) → target(A)``.

    The construction proceeds by induction on the tree depth:
    leaves are routed through `return_clause`; each
    operation node is decomposed into its parameter and
    continuation components, the continuation is recursively
    interpreted into the target monad, and the result is fed
    through the corresponding operation-clause morphism.
    """
    target_A = self.target.fmap_obj(A)
    carrier = _tree_carrier(self.signature, A, depth)
    leaf_size = A.size
    if depth == 0 or not self.signature.operations:
        # The carrier coincides with A; the result is the
        # return-clause itself.
        return self.return_clause
    # Recursive case: build the per-summand interpretations and
    # combine them through a single tensor.
    sub_morph = self.run(A, depth - 1)
    sub_size = _carrier_size(self.signature, leaf_size, depth - 1)
    target_A_size = target_A.size
    data = torch.full((carrier.size, target_A_size), PRODUCT_FUZZY.zero)
    # Leaf branch.
    return_t = self.return_clause.tensor.reshape(leaf_size, target_A_size)
    for a_flat in range(leaf_size):
        data[a_flat] = return_t[a_flat]
    # Operation branches.
    offset = leaf_size
    for op in self.signature.operations:
        clause = self.operation_clauses[op.name]
        # Clause has source = op.parameter × [op.result → target(A)]
        # and codomain = target(A). Encode the target-side function-
        # space cardinality:
        target_cont_card = target_A_size**op.result.size
        target_cont_fs = FinSet(
            name=f"_target_cont_{op.name}",
            cardinality=target_cont_card,
        )
        clause_src = ProductSet(components=(op.parameter, target_cont_fs))
        if clause.domain != clause_src or clause.codomain != target_A:
            raise TypeError(
                f"handler clause {op.name!r} has shape "
                f"{clause.domain!r} → {clause.codomain!r}; expected "
                f"{clause_src!r} → {target_A!r}"
            )
        clause_t = clause.tensor.reshape(
            op.parameter.size, target_cont_card, target_A_size
        )
        sub_t = sub_morph.tensor.reshape(sub_size, target_A_size)
        cont_card = sub_size**op.result.size
        block = op.parameter.size * cont_card
        for p_idx in range(op.parameter.size):
            for cont_flat in range(cont_card):
                outer_flat = offset + p_idx * cont_card + cont_flat
                # Decode the continuation as a deterministic function
                # op.result → sub-carrier; translate each output
                # through sub_morph to obtain a distribution over
                # target(A); enumerate the joint cartesian product
                # across result slots and translate back to the
                # target_cont function-space encoding.
                cont_outs = _decode_continuation(
                    cont_flat, sub_size, op.result.size
                )
                per_slot: list[list[tuple[int, float]]] = []
                for s in cont_outs:
                    nonzero = [
                        (j, float(sub_t[s, j].item()))
                        for j in range(target_A_size)
                        if sub_t[s, j].item() > 0
                    ]
                    per_slot.append(nonzero)
                if any(not slot for slot in per_slot):
                    continue
                # Enumerate cartesian product across slots.
                stack: list[tuple[int, float, list[int]]] = [(0, 1.0, [])]
                while stack:
                    slot_idx, w, choices = stack.pop()
                    if slot_idx == op.result.size:
                        tgt_cont_flat = _encode_continuation(
                            tuple(choices), target_A_size
                        )
                        # Route through the clause.
                        row = clause_t[p_idx, tgt_cont_flat]
                        for b in range(target_A_size):
                            bw = w * float(row[b].item())
                            if bw > 0:
                                cur = data[outer_flat, b].item()
                                # Algebra join: noisy-OR-like
                                # aggregation across cartesian
                                # branches.
                                data[outer_flat, b] = max(cur, bw)
                        continue
                    for j, jw in per_slot[slot_idx]:
                        stack.append((slot_idx + 1, w * jw, choices + [j]))
        offset += block
    return observed(carrier, target_A, data)

FreeMonad

Bases: Model

The free monad over an effect signature with a depth bound.

FreeMonad(sig, depth) carries the bounded-depth signature-tree construction. Its typeclass instance methods are concrete V-relation morphisms over the flat-FinSet carrier.

ATTRIBUTE	DESCRIPTION
signature	The signature whose free monad this is. TYPE: EffectSignature
depth	The depth bound for the operation-tree carrier. TYPE: int

pure

pure(A: SetObject) -> Morphism

pure : A → Free^d(A) is the leaf injection.

Source code in src/quivers/monadic/algebraic.py

def pure(self, A: SetObject) -> Morphism:
    """``pure : A → Free^d(A)`` is the leaf injection."""
    carrier = self._carrier(A)
    if carrier == A:
        return id_morph(A)
    data = torch.full((A.size, carrier.size), PRODUCT_FUZZY.zero)
    for a in range(A.size):
        data[a, a] = PRODUCT_FUZZY.unit
    # Reshape to (*A.shape, *carrier.shape).
    data = data.reshape(*A.shape, *carrier.shape)
    return observed(A, carrier, data)

fmap

fmap(A: SetObject, B: SetObject, f: Morphism) -> Morphism

Lift f : A → B to Free^d(f) : Free^d(A) → Free^d(B).

Acts as f on the leaf summand and recursively as Free^{d-1}(f) on each operation's continuation slots. Each operation node's continuation function-space is re-encoded through the recursive fmap's deterministic image.

Source code in src/quivers/monadic/algebraic.py

def fmap(self, A: SetObject, B: SetObject, f: Morphism) -> Morphism:
    """Lift ``f : A → B`` to ``Free^d(f) : Free^d(A) → Free^d(B)``.

    Acts as ``f`` on the leaf summand and recursively as
    ``Free^{d-1}(f)`` on each operation's continuation slots.
    Each operation node's continuation function-space is re-encoded
    through the recursive ``fmap``'s deterministic image.
    """
    carrier_A = self._carrier(A)
    carrier_B = self._carrier(B)
    if carrier_A == A and carrier_B == B:
        return f
    sub_A_size = _carrier_size(self.signature, A.size, self.depth - 1)
    sub_B_size = _carrier_size(self.signature, B.size, self.depth - 1)
    sub_recur = FreeMonad(signature=self.signature, depth=self.depth - 1).fmap(
        A, B, f
    )
    sub_t = sub_recur.tensor.reshape(sub_A_size, sub_B_size)
    data = torch.full((carrier_A.size, carrier_B.size), PRODUCT_FUZZY.zero)
    # Leaf branch: route through f.
    f_t = f.tensor.reshape(A.size, B.size)
    for a in range(A.size):
        for b in range(B.size):
            v = float(f_t[a, b].item())
            if v > 0:
                data[a, b] = v
    # Op branches.
    offset_A = A.size
    offset_B = B.size
    for op in self.signature.operations:
        cont_A_card = sub_A_size**op.result.size
        cont_B_card = sub_B_size**op.result.size
        block_A = op.parameter.size * cont_A_card
        block_B = op.parameter.size * cont_B_card
        for p_idx in range(op.parameter.size):
            for cont_A_flat in range(cont_A_card):
                outs_A = _decode_continuation(
                    cont_A_flat, sub_A_size, op.result.size
                )
                # Build per-slot distributions over sub_B.
                per_slot: list[list[tuple[int, float]]] = []
                for s in outs_A:
                    nonzero = [
                        (j, float(sub_t[s, j].item()))
                        for j in range(sub_B_size)
                        if sub_t[s, j].item() > 0
                    ]
                    per_slot.append(nonzero)
                if any(not slot for slot in per_slot):
                    continue
                src = offset_A + p_idx * cont_A_card + cont_A_flat
                stack: list[tuple[int, float, list[int]]] = [(0, 1.0, [])]
                while stack:
                    slot_idx, w, choices = stack.pop()
                    if slot_idx == op.result.size:
                        cont_B_flat = _encode_continuation(
                            tuple(choices), sub_B_size
                        )
                        tgt = offset_B + p_idx * cont_B_card + cont_B_flat
                        cur = data[src, tgt].item()
                        data[src, tgt] = max(cur, w)
                        continue
                    for j, jw in per_slot[slot_idx]:
                        stack.append((slot_idx + 1, w * jw, choices + [j]))
        offset_A += block_A
        offset_B += block_B
    return observed(carrier_A, carrier_B, data)

join

join(A: SetObject) -> Morphism

join : Free^d(Free^d(A)) → Free^d(A) via tree splicing.

Splices outer trees: an outer-leaf carrying an inner tree produces that inner tree directly; an outer-operation node op(p, k) produces op(p, k') where each continuation slot is the recursive join of the corresponding outer subtree. Truncation at depth d is enforced by recursive descent: outer ops at depth d produce inner ops whose sub-trees come from depth-d-1 recursion.

Source code in src/quivers/monadic/algebraic.py

def join(self, A: SetObject) -> Morphism:
    """``join : Free^d(Free^d(A)) → Free^d(A)`` via tree splicing.

    Splices outer trees: an outer-leaf carrying an inner tree
    produces that inner tree directly; an outer-operation node
    ``op(p, k)`` produces ``op(p, k')`` where each continuation
    slot is the recursive join of the corresponding outer
    subtree. Truncation at depth ``d`` is enforced by recursive
    descent: outer ops at depth ``d`` produce inner ops whose
    sub-trees come from depth-``d-1`` recursion.
    """
    carrier_A = self._carrier(A)
    carrier_FA = _tree_carrier(self.signature, carrier_A, self.depth)
    FA_size = carrier_FA.size
    A_carrier_size = carrier_A.size
    data = torch.full((FA_size, A_carrier_size), PRODUCT_FUZZY.zero)

    def splice(outer_flat: int, depth_remaining: int) -> dict[int, float]:
        """Compute join image of an outer index at the given depth."""
        if depth_remaining == 0 or not self.signature.operations:
            # The outer tree is a single leaf in carrier_A; identity.
            return {outer_flat: 1.0}
        # Outer-leaf summand: indices [0, A_carrier_size).
        if outer_flat < A_carrier_size:
            return {outer_flat: 1.0}
        # Outer-op summand.
        sub_outer_size = _carrier_size(
            self.signature, A_carrier_size, depth_remaining - 1
        )
        rest = outer_flat - A_carrier_size
        offset = 0
        for i, op in enumerate(self.signature.operations):
            cont_card = sub_outer_size**op.result.size
            block = op.parameter.size * cont_card
            if rest < block:
                p_idx = rest // cont_card
                cont_flat = rest % cont_card
                cont_outs = _decode_continuation(
                    cont_flat, sub_outer_size, op.result.size
                )
                # Recursively join each slot.
                per_slot: list[dict[int, float]] = [
                    splice(co, depth_remaining - 1) for co in cont_outs
                ]
                # Map each slot's image (in Free^{depth_remaining - 1}(A))
                # back into the inner-carrier's op-summand cont
                # function-space. The inner carrier is
                # Free^{self.depth}(A); its sub-carrier at depth-1
                # corresponds to Free^{self.depth - 1}(A). But here
                # the recursive `splice` produces values in
                # `_tree_carrier(sig, A, depth_remaining - 1)` —
                # which has size sub_inner_size.
                sub_inner_size = _carrier_size(
                    self.signature, A.size, depth_remaining - 1
                )
                # The image's inner-cont function-space cardinality
                # is sub_inner_size ** op.result.size.
                # Build the offset for this op in carrier_A's
                # op-summand block (at depth = self.depth).
                inner_offset_at_depth = _carrier_op_offset(
                    self.signature, A.size, self.depth, i
                )
                inner_cont_card = sub_inner_size**op.result.size
                out: dict[int, float] = {}
                stack: list[tuple[int, float, list[int]]] = [(0, 1.0, [])]
                while stack:
                    slot_idx, w, choices = stack.pop()
                    if slot_idx == op.result.size:
                        inner_cont_flat = _encode_continuation(
                            tuple(choices), sub_inner_size
                        )
                        target_idx = (
                            inner_offset_at_depth
                            + p_idx * inner_cont_card
                            + inner_cont_flat
                        )
                        if target_idx < A_carrier_size:
                            out[target_idx] = max(out.get(target_idx, 0.0), w)
                        continue
                    for choice, choice_w in per_slot[slot_idx].items():
                        if choice_w > 0 and choice < sub_inner_size:
                            stack.append(
                                (slot_idx + 1, w * choice_w, choices + [choice])
                            )
                return out
            rest -= block
            offset += block
        return {}

    for outer_flat in range(FA_size):
        dist = splice(outer_flat, self.depth)
        for inner_idx, w in dist.items():
            data[outer_flat, inner_idx] = w
    return observed(carrier_FA, carrier_A, data)

bind

bind(A: SetObject, B: SetObject, k: Morphism) -> Morphism

bind(m, k) = join_B ∘ Free^d(k)(m).

Source code in src/quivers/monadic/algebraic.py

def bind(self, A: SetObject, B: SetObject, k: Morphism) -> Morphism:
    """``bind(m, k) = join_B ∘ Free^d(k)(m)``."""
    return self.fmap(A, self._carrier(B), k) >> self.join(B)

apply

apply(A: SetObject, B: SetObject) -> Morphism

apply : Free^d([A → B]) ⊗ Free^d(A) → Free^d(B).

Derived from lift_a2 and the function-space evaluator.

Source code in src/quivers/monadic/algebraic.py

def apply(self, A: SetObject, B: SetObject) -> Morphism:
    """``apply : Free^d([A → B]) ⊗ Free^d(A) → Free^d(B)``.

    Derived from `lift_a2` and the function-space evaluator.
    """
    from quivers.monadic.instances import _evaluation_morphism, _function_space

    fn_space = _function_space(A, B)
    return self.lift_a2(fn_space, A, B, _evaluation_morphism(A, B))

lift_a2

lift_a2(A: SetObject, B: SetObject, C: SetObject, f: Morphism) -> Morphism

lift_a2(f) : Free^d(A) ⊗ Free^d(B) → Free^d(C).

Recursive definition (the standard free-monad applicative):

• liftA2(f)(leaf(a), leaf(b)) = leaf(f(a, b)).
• liftA2(f)(op(p, k), n) = op(p, λr. liftA2(f)(k(r), n)).
• liftA2(f)(leaf(a), op(p, k)) = op(p, λr. liftA2(f)(leaf(a), k(r))).

The recursion descends one depth at a time on the side carrying the operation, while the other side stays at its current depth. Sub-results are produced in :math:\mathrm{Free}^{\max(d_a, d_b) - 1}(C) and re-encoded into the parent's op-summand continuation slots through the appropriate sub-carrier function-space cardinality. Truncation is enforced uniformly when the output depth would exceed the bound d: such sub-results are dropped, preserving the bounded-depth contract.

Source code in src/quivers/monadic/algebraic.py

def lift_a2(
    self, A: SetObject, B: SetObject, C: SetObject, f: Morphism
) -> Morphism:
    """``lift_a2(f) : Free^d(A) ⊗ Free^d(B) → Free^d(C)``.

    Recursive definition (the standard free-monad applicative):

    - ``liftA2(f)(leaf(a), leaf(b)) = leaf(f(a, b))``.
    - ``liftA2(f)(op(p, k), n) = op(p, λr. liftA2(f)(k(r), n))``.
    - ``liftA2(f)(leaf(a), op(p, k)) = op(p, λr. liftA2(f)(leaf(a), k(r)))``.

    The recursion descends one depth at a time on the side carrying
    the operation, while the other side stays at its current
    depth. Sub-results are produced in
    :math:`\\mathrm{Free}^{\\max(d_a, d_b) - 1}(C)` and re-encoded
    into the parent's op-summand continuation slots through the
    appropriate sub-carrier function-space cardinality. Truncation
    is enforced uniformly when the output depth would exceed the
    bound ``d``: such sub-results are dropped, preserving the
    bounded-depth contract.
    """
    carrier_A = self._carrier(A)
    carrier_B = self._carrier(B)
    carrier_C = self._carrier(C)
    f_t = f.tensor.reshape(A.size, B.size, C.size)
    data = torch.full(
        (carrier_A.size, carrier_B.size, carrier_C.size),
        PRODUCT_FUZZY.zero,
    )
    sig = self.signature

    def recur(
        da: int, db: int, dc: int, a_flat: int, b_flat: int
    ) -> dict[int, float]:
        """Distribution over flat ``Free^dc(C)`` indices.

        ``da`` is the depth at which ``a_flat`` lives in
        ``Free^da(A)``; ``db`` likewise for ``b_flat``; ``dc`` is
        the target output depth (``dc = max(da, db)`` keeps the
        recursion well-typed against the bounded-depth target).
        """
        a_kind, a_op, a_pay, a_cont = _decompose_carrier_index(
            sig, A.size, da, a_flat
        )
        b_kind, b_op, b_pay, b_cont = _decompose_carrier_index(
            sig, B.size, db, b_flat
        )
        if a_kind == "leaf" and b_kind == "leaf":
            # leaf(a) ★ leaf(b) = leaf(f(a, b))
            out: dict[int, float] = {}
            for c in range(C.size):
                v = float(f_t[a_pay, b_pay, c].item())
                if v > 0:
                    out[c] = v
            return out
        if a_kind == "op":
            # op(p_a, k_a) ★ n = op(p_a, λr. recur(k_a(r), n))
            if dc < 1:
                return {}
            op = sig.operations[a_op]
            sub_a_size = _carrier_size(sig, A.size, da - 1)
            cont_outs = _decode_continuation(a_cont, sub_a_size, op.result.size)
            per_slot: list[dict[int, float]] = [
                recur(da - 1, db, dc - 1, k_r, b_flat) for k_r in cont_outs
            ]
            if any(not slot for slot in per_slot):
                return {}
            sub_c_size = _carrier_size(sig, C.size, dc - 1)
            offset_c = _carrier_op_offset(sig, C.size, dc, a_op)
            cont_card_c = sub_c_size**op.result.size
            out_dist: dict[int, float] = {}
            stack: list[tuple[int, float, list[int]]] = [(0, 1.0, [])]
            while stack:
                slot_idx, w, choices = stack.pop()
                if slot_idx == op.result.size:
                    cont_c_flat = _encode_continuation(tuple(choices), sub_c_size)
                    c_idx = offset_c + a_pay * cont_card_c + cont_c_flat
                    out_dist[c_idx] = max(out_dist.get(c_idx, 0.0), w)
                    continue
                for cc, cw in per_slot[slot_idx].items():
                    if cc < sub_c_size:
                        stack.append((slot_idx + 1, w * cw, choices + [cc]))
            return out_dist
        # a is leaf, b is op
        if dc < 1:
            return {}
        op = sig.operations[b_op]
        sub_b_size = _carrier_size(sig, B.size, db - 1)
        cont_outs = _decode_continuation(b_cont, sub_b_size, op.result.size)
        per_slot = [recur(da, db - 1, dc - 1, a_flat, k_r) for k_r in cont_outs]
        if any(not slot for slot in per_slot):
            return {}
        sub_c_size = _carrier_size(sig, C.size, dc - 1)
        offset_c = _carrier_op_offset(sig, C.size, dc, b_op)
        cont_card_c = sub_c_size**op.result.size
        out_dist = {}
        stack = [(0, 1.0, [])]
        while stack:
            slot_idx, w, choices = stack.pop()
            if slot_idx == op.result.size:
                cont_c_flat = _encode_continuation(tuple(choices), sub_c_size)
                c_idx = offset_c + b_pay * cont_card_c + cont_c_flat
                out_dist[c_idx] = max(out_dist.get(c_idx, 0.0), w)
                continue
            for cc, cw in per_slot[slot_idx].items():
                if cc < sub_c_size:
                    stack.append((slot_idx + 1, w * cw, choices + [cc]))
        return out_dist

    for a_flat in range(carrier_A.size):
        for b_flat in range(carrier_B.size):
            dist = recur(self.depth, self.depth, self.depth, a_flat, b_flat)
            for c_idx, w in dist.items():
                if c_idx < carrier_C.size:
                    data[a_flat, b_flat, c_idx] = w
    source = ProductSet(components=(carrier_A, carrier_B))
    data = data.reshape(*source.shape, *carrier_C.shape)
    return observed(source, carrier_C, data)