diff --git a/pytato/analysis/__init__.py b/pytato/analysis/__init__.py
index 9bf351a5c..a1c2ac62a 100644
--- a/pytato/analysis/__init__.py
+++ b/pytato/analysis/__init__.py
@@ -28,9 +28,10 @@
 
 from typing import TYPE_CHECKING, Any
 
+from typing_extensions import Self
+
 from loopy.tools import LoopyKeyBuilder
 from pymbolic.mapper.optimize import optimize_mapper
-from pytools import memoize_method
 
 from pytato.array import (
     Array,
@@ -76,7 +77,7 @@
 
 # {{{ NUserCollector
 
-class NUserCollector(Mapper[None, []]):
+class NUserCollector(Mapper[None, None, []]):
     """
     A :class:`pytato.transform.CachedWalkMapper` that records the number of
     times an array expression is a direct dependency of other nodes.
@@ -317,7 +318,7 @@ def is_einsum_similar_to_subscript(expr: Einsum, subscripts: str) -> bool:
 
 # {{{ DirectPredecessorsGetter
 
-class DirectPredecessorsGetter(Mapper[frozenset[ArrayOrNames], []]):
+class DirectPredecessorsGetter(Mapper[frozenset[ArrayOrNames], None, []]):
     """
     Mapper to get the
     `direct predecessors
@@ -413,9 +414,14 @@ class NodeCountMapper(CachedWalkMapper[[]]):
        Dictionary mapping node types to number of nodes of that type.
     """
 
-    def __init__(self, count_duplicates: bool = False) -> None:
+    def __init__(
+            self,
+            count_duplicates: bool = False,
+            _visited_functions: set[Any] | None = None,
+            ) -> None:
+        super().__init__(_visited_functions=_visited_functions)
+
         from collections import defaultdict
-        super().__init__()
         self.expr_type_counts: dict[type[Any], int] = defaultdict(int)
         self.count_duplicates = count_duplicates
 
@@ -423,6 +429,16 @@ def get_cache_key(self, expr: ArrayOrNames) -> int | ArrayOrNames:
         # Returns unique nodes only if count_duplicates is False
         return id(expr) if self.count_duplicates else expr
 
+    def get_function_definition_cache_key(
+            self, expr: FunctionDefinition) -> int | FunctionDefinition:
+        # Returns unique nodes only if count_duplicates is False
+        return id(expr) if self.count_duplicates else expr
+
+    def clone_for_callee(self, function: FunctionDefinition) -> Self:
+        return type(self)(
+            count_duplicates=self.count_duplicates,
+            _visited_functions=self._visited_functions)
+
     def post_visit(self, expr: Any) -> None:
         if not isinstance(expr, DictOfNamedArrays):
             self.expr_type_counts[type(expr)] += 1
@@ -488,15 +504,20 @@ class NodeMultiplicityMapper(CachedWalkMapper[[]]):
 
     .. autoattribute:: expr_multiplicity_counts
     """
-    def __init__(self) -> None:
+    def __init__(self, _visited_functions: set[Any] | None = None) -> None:
+        super().__init__(_visited_functions=_visited_functions)
+
         from collections import defaultdict
-        super().__init__()
         self.expr_multiplicity_counts: dict[Array, int] = defaultdict(int)
 
     def get_cache_key(self, expr: ArrayOrNames) -> int:
         # Returns each node, including nodes that are duplicates
         return id(expr)
 
+    def get_function_definition_cache_key(self, expr: FunctionDefinition) -> int:
+        # Returns each node, including nodes that are duplicates
+        return id(expr)
+
     def post_visit(self, expr: Any) -> None:
         if not isinstance(expr, DictOfNamedArrays):
             self.expr_multiplicity_counts[expr] += 1
@@ -530,14 +551,16 @@ class CallSiteCountMapper(CachedWalkMapper[[]]):
        The number of nodes.
     """
 
-    def __init__(self) -> None:
-        super().__init__()
+    def __init__(self, _visited_functions: set[Any] | None = None) -> None:
+        super().__init__(_visited_functions=_visited_functions)
         self.count = 0
 
     def get_cache_key(self, expr: ArrayOrNames) -> int:
         return id(expr)
 
-    @memoize_method
+    def get_function_definition_cache_key(self, expr: FunctionDefinition) -> int:
+        return id(expr)
+
     def map_function_definition(self, expr: FunctionDefinition) -> None:
         if not self.visit(expr):
             return
diff --git a/pytato/codegen.py b/pytato/codegen.py
index d08445517..85ac4052d 100644
--- a/pytato/codegen.py
+++ b/pytato/codegen.py
@@ -68,9 +68,9 @@
 
 
 if TYPE_CHECKING:
-    from collections.abc import Mapping
+    from collections.abc import Hashable, Mapping
 
-    from pytato.function import NamedCallResult
+    from pytato.function import FunctionDefinition, NamedCallResult
     from pytato.target import Target
 
 
@@ -136,10 +136,13 @@ class CodeGenPreprocessor(ToIndexLambdaMixin, CopyMapper):  # type: ignore[misc]
     ======================================  =====================================
     """
 
-    def __init__(self, target: Target,
-                 kernels_seen: dict[str, lp.LoopKernel] | None = None
-                 ) -> None:
-        super().__init__()
+    def __init__(
+            self,
+            target: Target,
+            kernels_seen: dict[str, lp.LoopKernel] | None = None,
+            _function_cache: dict[Hashable, FunctionDefinition] | None = None
+            ) -> None:
+        super().__init__(_function_cache=_function_cache)
         self.bound_arguments: dict[str, DataInterface] = {}
         self.var_name_gen: UniqueNameGenerator = UniqueNameGenerator()
         self.target = target
@@ -266,13 +269,16 @@ def normalize_outputs(
 
 @optimize_mapper(drop_args=True, drop_kwargs=True, inline_get_cache_key=True)
 class NamesValidityChecker(CachedWalkMapper[[]]):
-    def __init__(self) -> None:
+    def __init__(self, _visited_functions: set[Any] | None = None) -> None:
         self.name_to_input: dict[str, InputArgumentBase] = {}
-        super().__init__()
+        super().__init__(_visited_functions=_visited_functions)
 
     def get_cache_key(self, expr: ArrayOrNames) -> int:
         return id(expr)
 
+    def get_function_definition_cache_key(self, expr: FunctionDefinition) -> int:
+        return id(expr)
+
     def post_visit(self, expr: Any) -> None:
         if isinstance(expr, Placeholder | SizeParam | DataWrapper):
             if expr.name is not None:
diff --git a/pytato/distributed/partition.py b/pytato/distributed/partition.py
index 4a0eb5897..8e5940a06 100644
--- a/pytato/distributed/partition.py
+++ b/pytato/distributed/partition.py
@@ -288,8 +288,9 @@ def __init__(self,
                  recvd_ary_to_name: Mapping[Array, str],
                  sptpo_ary_to_name: Mapping[Array, str],
                  name_to_output: Mapping[str, Array],
+                 _function_cache: dict[Hashable, FunctionDefinition] | None = None,
                  ) -> None:
-        super().__init__()
+        super().__init__(_function_cache=_function_cache)
 
         self.recvd_ary_to_name = recvd_ary_to_name
         self.sptpo_ary_to_name = sptpo_ary_to_name
@@ -303,7 +304,7 @@ def clone_for_callee(
             self, function: FunctionDefinition) -> _DistributedInputReplacer:
         # Function definitions aren't allowed to contain receives,
         # stored arrays promoted to part outputs, or part outputs
-        return type(self)({}, {}, {})
+        return type(self)({}, {}, {}, _function_cache=self._function_cache)
 
     def map_placeholder(self, expr: Placeholder) -> Placeholder:
         self.user_input_names.add(expr.name)
@@ -456,7 +457,7 @@ def _recv_to_comm_id(
 
 
 class _LocalSendRecvDepGatherer(
-        CombineMapper[frozenset[CommunicationOpIdentifier]]):
+        CombineMapper[frozenset[CommunicationOpIdentifier], None]):
     def __init__(self, local_rank: int) -> None:
         super().__init__()
         self.local_comm_ids_to_needed_comm_ids: \
diff --git a/pytato/distributed/verify.py b/pytato/distributed/verify.py
index 5e8aa526d..bde9e2277 100644
--- a/pytato/distributed/verify.py
+++ b/pytato/distributed/verify.py
@@ -144,8 +144,8 @@ class MissingRecvError(DistributedPartitionVerificationError):
 
 @optimize_mapper(drop_args=True, drop_kwargs=True, inline_get_cache_key=True)
 class _SeenNodesWalkMapper(CachedWalkMapper[[]]):
-    def __init__(self) -> None:
-        super().__init__()
+    def __init__(self, _visited_functions: set[Any] | None = None) -> None:
+        super().__init__(_visited_functions=_visited_functions)
         self.seen_nodes: set[ArrayOrNames] = set()
 
     def get_cache_key(self, expr: ArrayOrNames) -> int:
diff --git a/pytato/equality.py b/pytato/equality.py
index 1ef2a88ed..6eed44b48 100644
--- a/pytato/equality.py
+++ b/pytato/equality.py
@@ -27,8 +27,6 @@
 
 from typing import TYPE_CHECKING, Any
 
-from pytools import memoize_method
-
 from pytato.array import (
     AbstractResultWithNamedArrays,
     AdvancedIndexInContiguousAxes,
@@ -85,26 +83,31 @@ class EqualityComparer:
           more on this.
     """
     def __init__(self) -> None:
+        # Uses the same cache for both arrays and functions
         self._cache: dict[tuple[int, int], bool] = {}
 
-    def rec(self, expr1: ArrayOrNames, expr2: Any) -> bool:
+    def rec(self, expr1: ArrayOrNames | FunctionDefinition, expr2: Any) -> bool:
         cache_key = id(expr1), id(expr2)
         try:
             return self._cache[cache_key]
         except KeyError:
-
-            method: Callable[[Array | AbstractResultWithNamedArrays, Any],
-                             bool]
-
-            try:
-                method = getattr(self, expr1._mapper_method)
-            except AttributeError:
-                if isinstance(expr1, Array):
-                    result = self.handle_unsupported_array(expr1, expr2)
+            if expr1 is expr2:
+                result = True
+            elif isinstance(expr1, ArrayOrNames):
+                method: Callable[[ArrayOrNames, Any], bool]
+                try:
+                    method = getattr(self, expr1._mapper_method)
+                except AttributeError:
+                    if isinstance(expr1, Array):
+                        result = self.handle_unsupported_array(expr1, expr2)
+                    else:
+                        result = self.map_foreign(expr1, expr2)
                 else:
-                    result = self.map_foreign(expr1, expr2)
+                    result = method(expr1, expr2)
+            elif isinstance(expr1, FunctionDefinition):
+                result = self.map_function_definition(expr1, expr2)
             else:
-                result = (expr1 is expr2) or method(expr1, expr2)
+                result = self.map_foreign(expr1, expr2)
 
             self._cache[cache_key] = result
             return result
@@ -296,7 +299,6 @@ def map_distributed_recv(self, expr1: DistributedRecv, expr2: Any) -> bool:
                 and expr1.tags == expr2.tags
                 )
 
-    @memoize_method
     def map_function_definition(self, expr1: FunctionDefinition, expr2: Any
                                 ) -> bool:
         return (expr1.__class__ is expr2.__class__
@@ -310,7 +312,7 @@ def map_function_definition(self, expr1: FunctionDefinition, expr2: Any
 
     def map_call(self, expr1: Call, expr2: Any) -> bool:
         return (expr1.__class__ is expr2.__class__
-                and self.map_function_definition(expr1.function, expr2.function)
+                and self.rec(expr1.function, expr2.function)
                 and frozenset(expr1.bindings) == frozenset(expr2.bindings)
                 and all(self.rec(bnd,
                                  expr2.bindings[name])
diff --git a/pytato/stringifier.py b/pytato/stringifier.py
index f2e7f066b..952b8f8e2 100644
--- a/pytato/stringifier.py
+++ b/pytato/stringifier.py
@@ -31,8 +31,6 @@
 import numpy as np
 from immutabledict import immutabledict
 
-from pytools import memoize_method
-
 from pytato.array import (
     Array,
     Axis,
@@ -58,7 +56,7 @@
 
 # {{{ Reprifier
 
-class Reprifier(Mapper[str, [int]]):
+class Reprifier(Mapper[str, str, [int]]):
     """
     Stringifies :mod:`pytato`-types to closely resemble CPython's implementation
     of :func:`repr` for its builtin datatypes.
@@ -71,6 +69,7 @@ def __init__(self,
         self.truncation_depth = truncation_depth
         self.truncation_string = truncation_string
 
+        # Uses the same cache for both arrays and functions
         self._cache: dict[tuple[int, int], str] = {}
 
     def rec(self, expr: Any, depth: int) -> str:
@@ -82,6 +81,15 @@ def rec(self, expr: Any, depth: int) -> str:
             self._cache[cache_key] = result
             return result
 
+    def rec_function_definition(self, expr: FunctionDefinition, depth: int) -> str:
+        cache_key = (id(expr), depth)
+        try:
+            return self._cache[cache_key]
+        except KeyError:
+            result = super().rec_function_definition(expr, depth)
+            self._cache[cache_key] = result
+            return result
+
     def __call__(self, expr: Any, depth: int = 0) -> str:
         return self.rec(expr, depth)
 
@@ -171,7 +179,6 @@ def _get_field_val(field: str) -> str:
                         for field in dataclasses.fields(type(expr)))
                 + ")")
 
-    @memoize_method
     def map_function_definition(self, expr: FunctionDefinition, depth: int) -> str:
         if depth > self.truncation_depth:
             return self.truncation_string
@@ -194,7 +201,7 @@ def map_call(self, expr: Call, depth: int) -> str:
 
         def _get_field_val(field: str) -> str:
             if field == "function":
-                return self.map_function_definition(expr.function, depth+1)
+                return self.rec_function_definition(expr.function, depth+1)
             else:
                 return self.rec(getattr(expr, field), depth+1)
 
diff --git a/pytato/target/loopy/codegen.py b/pytato/target/loopy/codegen.py
index 528de5d0e..84249c3b8 100644
--- a/pytato/target/loopy/codegen.py
+++ b/pytato/target/loopy/codegen.py
@@ -384,7 +384,7 @@ def update_t_unit(self, t_unit: lp.TranslationUnit) -> None:
 
 # {{{ codegen mapper
 
-class CodeGenMapper(Mapper[ImplementedResult, [CodeGenState]]):
+class CodeGenMapper(Mapper[ImplementedResult, None, [CodeGenState]]):
     """A mapper for generating code for nodes in the computation graph.
     """
     exprgen_mapper: InlinedExpressionGenMapper
diff --git a/pytato/target/python/numpy_like.py b/pytato/target/python/numpy_like.py
index 0c9bd413c..2561f57f7 100644
--- a/pytato/target/python/numpy_like.py
+++ b/pytato/target/python/numpy_like.py
@@ -62,6 +62,7 @@
     SizeParam,
     Stack,
 )
+from pytato.function import FunctionDefinition
 from pytato.raising import BinaryOpType, C99CallOp
 from pytato.reductions import (
     AllReductionOperation,
@@ -171,7 +172,7 @@ def _is_slice_trivial(slice_: NormalizedSlice,
 }
 
 
-class NumpyCodegenMapper(CachedMapper[str, []]):
+class NumpyCodegenMapper(CachedMapper[str, FunctionDefinition, []]):
     """
     .. note::
 
diff --git a/pytato/transform/__init__.py b/pytato/transform/__init__.py
index 393583037..b4be9aceb 100644
--- a/pytato/transform/__init__.py
+++ b/pytato/transform/__init__.py
@@ -43,7 +43,6 @@
 from typing_extensions import Self
 
 from pymbolic.mapper.optimize import optimize_mapper
-from pytools import memoize_method
 
 from pytato.array import (
     AbstractResultWithNamedArrays,
@@ -86,9 +85,6 @@
 ArrayOrNames: TypeAlias = Array | AbstractResultWithNamedArrays
 MappedT = TypeVar("MappedT",
                   Array, AbstractResultWithNamedArrays, ArrayOrNames)
-TransformMapperResultT = TypeVar("TransformMapperResultT",  # used in TransformMapper
-                            Array, AbstractResultWithNamedArrays, ArrayOrNames)
-CachedMapperT = TypeVar("CachedMapperT")  # used in CachedMapper
 IndexOrShapeExpr = TypeVar("IndexOrShapeExpr")
 R = frozenset[Array]
 
@@ -142,13 +138,15 @@
 
     A type variable representing the input type of a :class:`Mapper`.
 
-.. class:: CombineT
+.. class:: ResultT
 
-    A type variable representing the type of a :class:`CombineMapper`.
+    A type variable representing the result type of a :class:`Mapper` when mapping
+    a :class:`pytato.Array` or :class:`pytato.AbstractResultWithNamedArrays`.
 
-.. class:: ResultT
+.. class:: FunctionResultT
 
-    A type variable representing the result type of a :class:`Mapper`.
+    A type variable representing the result type of a :class:`Mapper` when mapping
+    a :class:`pytato.function.FunctionDefinition`.
 
 .. class:: Scalar
 
@@ -165,10 +163,11 @@ class UnsupportedArrayError(ValueError):
 # {{{ mapper base class
 
 ResultT = TypeVar("ResultT")
+FunctionResultT = TypeVar("FunctionResultT")
 P = ParamSpec("P")
 
 
-class Mapper(Generic[ResultT, P]):
+class Mapper(Generic[ResultT, FunctionResultT, P]):
     """A class that when called with a :class:`pytato.Array` recursively
     iterates over the DAG, calling the *_mapper_method* of each node. Users of
     this class are expected to override the methods of this class or create a
@@ -194,7 +193,7 @@ def handle_unsupported_array(self, expr: MappedT,
         raise UnsupportedArrayError(
                 f"{type(self).__name__} cannot handle expressions of type {type(expr)}")
 
-    def map_foreign(self, expr: Any, *args: P.args, **kwargs: P.kwargs) -> ResultT:
+    def map_foreign(self, expr: Any, *args: P.args, **kwargs: P.kwargs) -> Any:
         """Mapper method that is invoked for an object of class for which a
         mapper method does not exist in this mapper.
         """
@@ -203,7 +202,7 @@ def map_foreign(self, expr: Any, *args: P.args, **kwargs: P.kwargs) -> ResultT:
 
     def rec(self, expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         """Call the mapper method of *expr* and return the result."""
-        method: Callable[..., Array] | None
+        method: Callable[..., Any] | None
 
         try:
             method = getattr(self, expr._mapper_method)
@@ -218,11 +217,25 @@ def rec(self, expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs) -> ResultT:
                 else:
                     return self.handle_unsupported_array(expr, *args, **kwargs)
             else:
-                return self.map_foreign(expr, *args, **kwargs)
+                return cast("ResultT", self.map_foreign(expr, *args, **kwargs))
 
         assert method is not None
         return cast("ResultT", method(expr, *args, **kwargs))
 
+    def rec_function_definition(
+            self, expr: FunctionDefinition, *args: P.args, **kwargs: P.kwargs
+            ) -> FunctionResultT:
+        """Call the mapper method of *expr* and return the result."""
+        method: Callable[..., Any] | None
+
+        try:
+            method = self.map_function_definition  # type: ignore[attr-defined]
+        except AttributeError:
+            return cast("FunctionResultT", self.map_foreign(expr, *args, **kwargs))
+
+        assert method is not None
+        return cast("FunctionResultT", method(expr, *args, **kwargs))
+
     def __call__(self,
                  expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         """Handle the mapping of *expr*."""
@@ -233,23 +246,42 @@ def __call__(self,
 
 # {{{ CachedMapper
 
-class CachedMapper(Mapper[ResultT, P]):
+class CachedMapper(Mapper[ResultT, FunctionResultT, P]):
     """Mapper class that maps each node in the DAG exactly once. This loses some
     information compared to :class:`Mapper` as a node is visited only from
     one of its predecessors.
 
     .. automethod:: get_cache_key
+    .. automethod:: get_function_definition_cache_key
+    .. automethod:: clone_for_callee
     """
 
-    def __init__(self) -> None:
+    def __init__(
+            self,
+            # Arrays are cached separately for each call stack frame, but
+            # functions are cached globally
+            _function_cache: dict[Hashable, FunctionResultT] | None = None
+            ) -> None:
         super().__init__()
         self._cache: dict[Hashable, ResultT] = {}
 
+        if _function_cache is not None:
+            function_cache = _function_cache
+        else:
+            function_cache = {}
+
+        self._function_cache: dict[Hashable, FunctionResultT] = function_cache
+
     def get_cache_key(
                 self, expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs
             ) -> Hashable:
         return (expr, *args, tuple(sorted(kwargs.items())))
 
+    def get_function_definition_cache_key(
+                self, expr: FunctionDefinition, *args: P.args, **kwargs: P.kwargs
+            ) -> Hashable:
+        return (expr, *args, tuple(sorted(kwargs.items())))
+
     def rec(self, expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs) -> ResultT:
         key = self.get_cache_key(expr, *args, **kwargs)
         try:
@@ -259,12 +291,31 @@ def rec(self, expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs) -> ResultT:
             self._cache[key] = result
             return result
 
+    def rec_function_definition(
+                self, expr: FunctionDefinition, *args: P.args, **kwargs: P.kwargs
+            ) -> FunctionResultT:
+        key = self.get_function_definition_cache_key(expr, *args, **kwargs)
+        try:
+            return self._function_cache[key]
+        except KeyError:
+            result = super().rec_function_definition(expr, *args, **kwargs)
+            self._function_cache[key] = result
+            return result
+
+    def clone_for_callee(
+            self, function: FunctionDefinition) -> Self:
+        """
+        Called to clone *self* before starting traversal of a
+        :class:`pytato.function.FunctionDefinition`.
+        """
+        return type(self)(_function_cache=self._function_cache)
+
 # }}}
 
 
 # {{{ TransformMapper
 
-class TransformMapper(CachedMapper[ArrayOrNames, []]):
+class TransformMapper(CachedMapper[ArrayOrNames, FunctionDefinition, []]):
     """Base class for mappers that transform :class:`pytato.array.Array`\\ s into
     other :class:`pytato.array.Array`\\ s.
 
@@ -272,28 +323,20 @@ class TransformMapper(CachedMapper[ArrayOrNames, []]):
     arrays (e.g., calling :meth:`~CachedMapper.get_cache_key` on them). Does not
     implement default mapper methods; for that, see :class:`CopyMapper`.
 
-    .. automethod:: clone_for_callee
     """
     def rec_ary(self, expr: Array) -> Array:
         res = self.rec(expr)
         assert isinstance(res, Array)
         return res
 
-    def clone_for_callee(self, function: FunctionDefinition) -> Self:
-        """
-        Called to clone *self* before starting traversal of a
-        :class:`pytato.function.FunctionDefinition`.
-        """
-        return type(self)()
-
 # }}}
 
 
 # {{{ TransformMapperWithExtraArgs
 
 class TransformMapperWithExtraArgs(
-            CachedMapper[ArrayOrNames, P],
-            Mapper[ArrayOrNames, P]
+            CachedMapper[ArrayOrNames, FunctionDefinition, P],
+            Mapper[ArrayOrNames, FunctionDefinition, P]
         ):
     """
     Similar to :class:`TransformMapper`, but each mapper method takes extra
@@ -301,21 +344,12 @@ class TransformMapperWithExtraArgs(
 
     The logic in :class:`TransformMapper` purposely does not take the extra
     arguments to keep the cost of its each call frame low.
-
-    .. automethod:: clone_for_callee
     """
     def rec_ary(self, expr: Array, *args: P.args, **kwargs: P.kwargs) -> Array:
         res = self.rec(expr, *args, **kwargs)
         assert isinstance(res, Array)
         return res
 
-    def clone_for_callee(self, function: FunctionDefinition) -> Self:
-        """
-        Called to clone *self* before starting traversal of a
-        :class:`pytato.function.FunctionDefinition`.
-        """
-        return type(self)()
-
 # }}}
 
 
@@ -492,7 +526,6 @@ def map_distributed_recv(self, expr: DistributedRecv) -> Array:
                dtype=expr.dtype, tags=expr.tags, axes=expr.axes,
                non_equality_tags=expr.non_equality_tags)
 
-    @memoize_method
     def map_function_definition(self,
                                 expr: FunctionDefinition) -> FunctionDefinition:
         # spawn a new mapper to avoid unsound cache hits, since the namespace of the
@@ -503,7 +536,7 @@ def map_function_definition(self,
         return dataclasses.replace(expr, returns=immutabledict(new_returns))
 
     def map_call(self, expr: Call) -> AbstractResultWithNamedArrays:
-        return Call(self.map_function_definition(expr.function),
+        return Call(self.rec_function_definition(expr.function),
                     immutabledict({name: self.rec(bnd)
                          for name, bnd in expr.bindings.items()}),
                     tags=expr.tags,
@@ -717,7 +750,7 @@ def map_function_definition(
 
     def map_call(self, expr: Call,
                  *args: P.args, **kwargs: P.kwargs) -> AbstractResultWithNamedArrays:
-        return Call(self.map_function_definition(expr.function, *args, **kwargs),
+        return Call(self.rec_function_definition(expr.function, *args, **kwargs),
                     immutabledict({name: self.rec(bnd, *args, **kwargs)
                          for name, bnd in expr.bindings.items()}),
                     tags=expr.tags,
@@ -734,7 +767,7 @@ def map_named_call_result(self, expr: NamedCallResult,
 
 # {{{ CombineMapper
 
-class CombineMapper(Mapper[ResultT, []]):
+class CombineMapper(Mapper[ResultT, FunctionResultT, []]):
     """
     Abstract mapper that recursively combines the results of user nodes
     of a given expression.
@@ -744,6 +777,9 @@ class CombineMapper(Mapper[ResultT, []]):
     def __init__(self) -> None:
         super().__init__()
         self.cache: dict[ArrayOrNames, ResultT] = {}
+        # Don't need to pass function cache as argument here, because unlike
+        # CachedMapper we're not creating a new mapper for each call
+        self.function_cache: dict[FunctionDefinition, FunctionResultT] = {}
 
     def rec_idx_or_size_tuple(self, situp: tuple[IndexOrShapeExpr, ...]
                               ) -> tuple[ResultT, ...]:
@@ -756,6 +792,14 @@ def rec(self, expr: ArrayOrNames) -> ResultT:
         self.cache[expr] = result
         return result
 
+    def rec_function_definition(
+            self, expr: FunctionDefinition) -> FunctionResultT:
+        if expr in self.function_cache:
+            return self.function_cache[expr]
+        result: FunctionResultT = super().rec_function_definition(expr)
+        self.function_cache[expr] = result
+        return result
+
     def __call__(self, expr: ArrayOrNames) -> ResultT:
         return self.rec(expr)
 
@@ -839,8 +883,7 @@ def map_distributed_send_ref_holder(
     def map_distributed_recv(self, expr: DistributedRecv) -> ResultT:
         return self.combine(*self.rec_idx_or_size_tuple(expr.shape))
 
-    @memoize_method
-    def map_function_definition(self, expr: FunctionDefinition) -> ResultT:
+    def map_function_definition(self, expr: FunctionDefinition) -> FunctionResultT:
         raise NotImplementedError("Combining results from a callee expression"
                                   " is context-dependent. Derived classes"
                                   " must override map_function_definition.")
@@ -858,7 +901,7 @@ def map_named_call_result(self, expr: NamedCallResult) -> ResultT:
 
 # {{{ DependencyMapper
 
-class DependencyMapper(CombineMapper[R]):
+class DependencyMapper(CombineMapper[R, R]):
     """
     Maps a :class:`pytato.array.Array` to a :class:`frozenset` of
     :class:`pytato.array.Array`'s it depends on.
@@ -920,14 +963,13 @@ def map_distributed_send_ref_holder(
     def map_distributed_recv(self, expr: DistributedRecv) -> R:
         return self.combine(frozenset([expr]), super().map_distributed_recv(expr))
 
-    @memoize_method
     def map_function_definition(self, expr: FunctionDefinition) -> R:
         # do not include arrays from the function's body as it would involve
         # putting arrays from different namespaces into the same collection.
         return frozenset()
 
     def map_call(self, expr: Call) -> R:
-        return self.combine(self.map_function_definition(expr.function),
+        return self.combine(self.rec_function_definition(expr.function),
                             *[self.rec(bnd) for bnd in expr.bindings.values()])
 
     def map_named_call_result(self, expr: NamedCallResult) -> R:
@@ -958,7 +1000,8 @@ def combine(self, *args: frozenset[Array]) -> frozenset[Array]:
 
 # {{{ InputGatherer
 
-class InputGatherer(CombineMapper[frozenset[InputArgumentBase]]):
+class InputGatherer(
+        CombineMapper[frozenset[InputArgumentBase], frozenset[InputArgumentBase]]):
     """
     Mapper to combine all instances of :class:`pytato.array.InputArgumentBase` that
     an array expression depends on.
@@ -977,7 +1020,6 @@ def map_data_wrapper(self, expr: DataWrapper) -> frozenset[InputArgumentBase]:
     def map_size_param(self, expr: SizeParam) -> frozenset[SizeParam]:
         return frozenset([expr])
 
-    @memoize_method
     def map_function_definition(self, expr: FunctionDefinition
                                 ) -> frozenset[InputArgumentBase]:
         # get rid of placeholders local to the function.
@@ -999,7 +1041,7 @@ def map_function_definition(self, expr: FunctionDefinition
         return frozenset(result)
 
     def map_call(self, expr: Call) -> frozenset[InputArgumentBase]:
-        return self.combine(self.map_function_definition(expr.function),
+        return self.combine(self.rec_function_definition(expr.function),
             *[
                 self.rec(bnd)
                 for name, bnd in sorted(expr.bindings.items())])
@@ -1009,7 +1051,8 @@ def map_call(self, expr: Call) -> frozenset[InputArgumentBase]:
 
 # {{{ SizeParamGatherer
 
-class SizeParamGatherer(CombineMapper[frozenset[SizeParam]]):
+class SizeParamGatherer(
+        CombineMapper[frozenset[SizeParam], frozenset[SizeParam]]):
     """
     Mapper to combine all instances of :class:`pytato.array.SizeParam` that
     an array expression depends on.
@@ -1022,14 +1065,13 @@ def combine(self, *args: frozenset[SizeParam]
     def map_size_param(self, expr: SizeParam) -> frozenset[SizeParam]:
         return frozenset([expr])
 
-    @memoize_method
     def map_function_definition(self, expr: FunctionDefinition
                                 ) -> frozenset[SizeParam]:
         return self.combine(*[self.rec(ret)
                               for ret in expr.returns.values()])
 
     def map_call(self, expr: Call) -> frozenset[SizeParam]:
-        return self.combine(self.map_function_definition(expr.function),
+        return self.combine(self.rec_function_definition(expr.function),
             *[
                 self.rec(bnd)
                 for name, bnd in sorted(expr.bindings.items())])
@@ -1039,7 +1081,7 @@ def map_call(self, expr: Call) -> frozenset[SizeParam]:
 
 # {{{ WalkMapper
 
-class WalkMapper(Mapper[None, P]):
+class WalkMapper(Mapper[None, None, P]):
     """
     A mapper that walks over all the arrays in a :class:`pytato.Array`.
 
@@ -1228,7 +1270,7 @@ def map_call(self, expr: Call, *args: P.args, **kwargs: P.kwargs) -> None:
         if not self.visit(expr, *args, **kwargs):
             return
 
-        self.map_function_definition(expr.function, *args, **kwargs)
+        self.rec_function_definition(expr.function, *args, **kwargs)
         for bnd in expr.bindings.values():
             self.rec(bnd, *args, **kwargs)
 
@@ -1255,24 +1297,46 @@ class CachedWalkMapper(WalkMapper[P]):
     one of its predecessors.
     """
 
-    def __init__(self) -> None:
+    def __init__(
+            self,
+            _visited_functions: set[Any] | None = None) -> None:
         super().__init__()
-        self._visited_nodes: set[Any] = set()
+        self._visited_arrays_or_names: set[Any] = set()
+
+        if _visited_functions is not None:
+            visited_functions = _visited_functions
+        else:
+            visited_functions = set()
+
+        self._visited_functions: set[Any] = visited_functions
 
     def get_cache_key(self, expr: ArrayOrNames, *args: Any, **kwargs: Any) -> Any:
         raise NotImplementedError
 
+    def get_function_definition_cache_key(
+            self, expr: FunctionDefinition, *args: Any, **kwargs: Any) -> Any:
+        raise NotImplementedError
+
     def rec(self, expr: ArrayOrNames, *args: Any, **kwargs: Any
             ) -> None:
         cache_key = self.get_cache_key(expr, *args, **kwargs)
-        if cache_key in self._visited_nodes:
+        if cache_key in self._visited_arrays_or_names:
             return
 
         super().rec(expr, *args, **kwargs)
-        self._visited_nodes.add(cache_key)
+        self._visited_arrays_or_names.add(cache_key)
+
+    def rec_function_definition(self, expr: FunctionDefinition,
+                                *args: Any, **kwargs: Any) -> None:
+        cache_key = self.get_function_definition_cache_key(expr, *args, **kwargs)
+        if cache_key in self._visited_functions:
+            return
+
+        super().rec_function_definition(expr, *args, **kwargs)
+        self._visited_functions.add(cache_key)
 
     def clone_for_callee(self, function: FunctionDefinition) -> Self:
-        return type(self)()
+        return type(self)(_visited_functions=self._visited_functions)
 
 # }}}
 
@@ -1291,8 +1355,10 @@ class TopoSortMapper(CachedWalkMapper[[]]):
         :class:`~pytato.function.FunctionDefinition`.
     """
 
-    def __init__(self) -> None:
-        super().__init__()
+    def __init__(
+            self,
+            _visited_functions: set[Any] | None = None) -> None:
+        super().__init__(_visited_functions=_visited_functions)
         self.topological_order: list[Array] = []
 
     def get_cache_key(self, expr: ArrayOrNames) -> int:
@@ -1301,7 +1367,6 @@ def get_cache_key(self, expr: ArrayOrNames) -> int:
     def post_visit(self, expr: Any) -> None:
         self.topological_order.append(expr)
 
-    @memoize_method
     def map_function_definition(self, expr: FunctionDefinition) -> None:
         # do nothing as it includes arrays from a different namespace.
         return
@@ -1317,13 +1382,17 @@ class CachedMapAndCopyMapper(CopyMapper):
     traversals are memoized i.e. each node is mapped via *map_fn* exactly once.
     """
 
-    def __init__(self, map_fn: Callable[[ArrayOrNames], ArrayOrNames]) -> None:
-        super().__init__()
+    def __init__(
+            self,
+            map_fn: Callable[[ArrayOrNames], ArrayOrNames],
+            _function_cache: dict[Hashable, FunctionDefinition] | None = None
+            ) -> None:
+        super().__init__(_function_cache=_function_cache)
         self.map_fn: Callable[[ArrayOrNames], ArrayOrNames] = map_fn
 
     def clone_for_callee(
             self, function: FunctionDefinition) -> Self:
-        return type(self)(self.map_fn)
+        return type(self)(self.map_fn, _function_cache=self._function_cache)
 
     def rec(self, expr: ArrayOrNames) -> ArrayOrNames:
         if expr in self._cache:
@@ -1371,7 +1440,7 @@ def _materialize_if_mpms(expr: Array,
         return MPMSMaterializerAccumulator(materialized_predecessors, expr)
 
 
-class MPMSMaterializer(Mapper[MPMSMaterializerAccumulator, []]):
+class MPMSMaterializer(Mapper[MPMSMaterializerAccumulator, None, []]):
     """
     See :func:`materialize_with_mpms` for an explanation.
 
@@ -1648,7 +1717,7 @@ def materialize_with_mpms(expr: DictOfNamedArrays) -> DictOfNamedArrays:
 
 # {{{ UsersCollector
 
-class UsersCollector(CachedMapper[None, []]):
+class UsersCollector(CachedMapper[None, None, []]):
     """
     Maps a graph to a dictionary representation mapping a node to its users,
     i.e. all the nodes using its value.
@@ -1772,7 +1841,6 @@ def map_distributed_send_ref_holder(
     def map_distributed_recv(self, expr: DistributedRecv) -> None:
         self.rec_idx_or_size_tuple(expr, expr.shape)
 
-    @memoize_method
     def map_function_definition(self, expr: FunctionDefinition) -> None:
         raise AssertionError("Control shouldn't reach at this point."
                              " Instantiate another UsersCollector to"
diff --git a/pytato/transform/calls.py b/pytato/transform/calls.py
index bc3d69909..ea08a6ac6 100644
--- a/pytato/transform/calls.py
+++ b/pytato/transform/calls.py
@@ -58,6 +58,7 @@ class PlaceholderSubstitutor(CopyMapper):
     """
 
     def __init__(self, substitutions: Mapping[str, Array]) -> None:
+        # Ignoring function cache, not needed
         super().__init__()
         self.substitutions = substitutions
 
diff --git a/pytato/transform/lower_to_index_lambda.py b/pytato/transform/lower_to_index_lambda.py
index e952e4e9e..ec00e9f45 100644
--- a/pytato/transform/lower_to_index_lambda.py
+++ b/pytato/transform/lower_to_index_lambda.py
@@ -637,7 +637,7 @@ def map_axis_permutation(self, expr: AxisPermutation) -> IndexLambda:
                            non_equality_tags=expr.non_equality_tags)
 
 
-class ToIndexLambdaMapper(Mapper[Array, []], ToIndexLambdaMixin):
+class ToIndexLambdaMapper(Mapper[Array, None, []], ToIndexLambdaMixin):
 
     def handle_unsupported_array(self, expr: Any) -> Any:
         raise CannotBeLoweredToIndexLambda(type(expr))
diff --git a/pytato/transform/metadata.py b/pytato/transform/metadata.py
index 8cd520d94..4154c5962 100644
--- a/pytato/transform/metadata.py
+++ b/pytato/transform/metadata.py
@@ -90,9 +90,9 @@
 
 
 if TYPE_CHECKING:
-    from collections.abc import Collection, Mapping
+    from collections.abc import Collection, Hashable, Mapping
 
-    from pytato.function import NamedCallResult
+    from pytato.function import FunctionDefinition, NamedCallResult
     from pytato.loopy import LoopyCall
 
 
@@ -101,7 +101,7 @@
 
 # {{{ AxesTagsEquationCollector
 
-class AxesTagsEquationCollector(Mapper[None, []]):
+class AxesTagsEquationCollector(Mapper[None, None, []]):
     r"""
     Records equations arising from operand/output axes equivalence for an array
     operation. This mapper implements a default set of propagation rules,
@@ -595,8 +595,9 @@ class AxisTagAttacher(CopyMapper):
     """
     def __init__(self,
                  axis_to_tags: Mapping[tuple[Array, int], Collection[Tag]],
-                 tag_corresponding_redn_descr: bool):
-        super().__init__()
+                 tag_corresponding_redn_descr: bool,
+                 _function_cache: dict[Hashable, FunctionDefinition] | None = None):
+        super().__init__(_function_cache=_function_cache)
         self.axis_to_tags: Mapping[tuple[Array, int], Collection[Tag]] = axis_to_tags
         self.tag_corresponding_redn_descr: bool = tag_corresponding_redn_descr
 
diff --git a/pytato/utils.py b/pytato/utils.py
index aedd0dc05..78e5463b3 100644
--- a/pytato/utils.py
+++ b/pytato/utils.py
@@ -269,7 +269,7 @@ def cast_to_result_type(
 
 # {{{ dim_to_index_lambda_components
 
-class ShapeExpressionMapper(CachedMapper[ScalarExpression, []]):
+class ShapeExpressionMapper(CachedMapper[ScalarExpression, None, []]):
     """
     Mapper that takes a shape component and returns it as a scalar expression.
     """
@@ -372,7 +372,7 @@ def are_shapes_equal(shape1: ShapeType, shape2: ShapeType) -> bool:
 
 # {{{ ShapeToISLExpressionMapper
 
-class ShapeToISLExpressionMapper(CachedMapper[isl.Aff, []]):
+class ShapeToISLExpressionMapper(CachedMapper[isl.Aff, None, []]):
     """
     Mapper that takes a shape component and returns it as :class:`isl.Aff`.
     """
diff --git a/pytato/visualization/dot.py b/pytato/visualization/dot.py
index 7e685aba9..3d4512079 100644
--- a/pytato/visualization/dot.py
+++ b/pytato/visualization/dot.py
@@ -178,7 +178,7 @@ def stringify_shape(shape: ShapeType) -> str:
     return "(" + ", ".join(components) + ")"
 
 
-class ArrayToDotNodeInfoMapper(CachedMapper[ArrayOrNames, []]):
+class ArrayToDotNodeInfoMapper(CachedMapper[None, None, []]):
     def __init__(self) -> None:
         super().__init__()
         self.node_to_dot: dict[ArrayOrNames, _DotNodeInfo] = {}
diff --git a/pytato/visualization/fancy_placeholder_data_flow.py b/pytato/visualization/fancy_placeholder_data_flow.py
index 3d06309fd..ace1c32b4 100644
--- a/pytato/visualization/fancy_placeholder_data_flow.py
+++ b/pytato/visualization/fancy_placeholder_data_flow.py
@@ -23,6 +23,7 @@
     Placeholder,
     Stack,
 )
+from pytato.function import FunctionDefinition
 from pytato.transform import CachedMapper
 
 
@@ -100,7 +101,7 @@ def _get_dot_node_from_predecessors(node_id: str,
         return NoShowNode(), frozenset()
 
 
-class FancyDotWriter(CachedMapper[_FancyDotWriterNode, []]):
+class FancyDotWriter(CachedMapper[_FancyDotWriterNode, FunctionDefinition, []]):
     def __init__(self) -> None:
         super().__init__()
         self.vng = UniqueNameGenerator()
diff --git a/test/testlib.py b/test/testlib.py
index 36857197b..59d4f9f50 100644
--- a/test/testlib.py
+++ b/test/testlib.py
@@ -32,7 +32,7 @@
 
 # {{{ tools for comparison to numpy
 
-class NumpyBasedEvaluator(Mapper[Any, []]):
+class NumpyBasedEvaluator(Mapper[Any, None, []]):
     """
     Mapper to return the result according to an eager evaluation array package
     *np*.