Common Library¶

This page summarizes library functions available in the Cairo common library. The common library is written in Cairo, and its code can be found here. It provides a level of abstraction for common and useful components that can be imported for use in any Cairo program.

The libraries available are listed below, organized alphabetically. The functions within each library are outlined under the relevant library heading.

alloc.
bitwise.
cairo_builtins.
default_dict.
dict.
dict_access.
find_element.
set.

Some descriptions state that the library function “requires the implicit argument <argument>”. See Implicit arguments for more information on this topic.

`alloc`¶

This section refers to the common library’s alloc module.

`alloc()`¶

Returns a pointer to a newly allocated memory segment. This is useful when defining dynamically allocated arrays. As more elements are added, more memory will be allocated.

from starkware.cairo.common.alloc import alloc

// Allocate a memory segment.
let (array_ptr: felt*) = alloc();

// Allocate a memory segment for an array of structs.
let (local struct_array_ptr: MyStruct*) = alloc();

`cairo_builtins`¶

This section refers to the common library’s cairo_builtins module.

`BitwiseBuiltin`¶

A struct specifying the bitwise builtin memory structure. This struct is used by functions from the common library that use the bitwise builtin. For example, the bitwise_xor() function accepts an implicit argument of type BitwiseBuiltin*, which is used internally to track the next available builtin instance. See the function here.

The struct has the following members of type felt:

x, the first oprand.
y, the second operand.
x_and_y, the result of bitwise AND operation on x and y.
x_xor_y, the result of bitwise XOR operation on x and y.
x_or_y, the result of bitwise OR operation on x and y.

A pointer to the bitwise builtin, bitwise_ptr, has the type BitwiseBuiltin*.

`HashBuiltin`¶

A struct specifying the hash builtin memory structure. This struct is used by functions from the common library that use a hash builtin, such as the pedersen builtin. For example, the hash2() function accepts an implicit argument of type HashBuiltin*, which is used internally to track the next available builtin instance. See the function here.

The struct has the following members of type felt:

x, the first input being hashed.
y, the second input being hashed.
result, the hash of x and y.

A pointer to the pedersen builtin, pedersen_ptr, has the type HashBuiltin*.

`SignatureBuiltin`¶

A struct specifying the signature builtin memory structure. This struct is used by functions from the common library that use a signature builtin, such as the ecdsa builtin. For example, the verify_ecdsa_signature() function accepts an implicit argument of type SignatureBulitin*, which is used internally to track the next available builtin instance. See the function here.

The struct has the following members of type felt:

pub_key, an ECDSA public key.
message, a message signed by the pub_key.

A pointer to the ecdsa builtin, ecdsa_ptr, has the type SignatureBuiltin*.

`bitwise`¶

This section refers to the common library’s bitwise module.

`bitwise_and()`¶

Returns the result of the bitwise AND operation of two elements. Requires an implicit argument, bitwise_ptr of type BitwiseBuiltin*.

The function accepts the explicit arguments:

x of type felt, the first operand.
y of type felt, the second operand.

The function returns the value:

x_and_y of type felt, the result of the bitwise AND operation a & b.

The example below shows the operation on binary inputs 1100 and 1010 results in 1000:

from starkware.cairo.common.bitwise import bitwise_and

let (result) = bitwise_and(12, 10);  // Binary (1100, 1010).
assert result = 8;  // Binary 1000.

`bitwise_xor()`¶

Returns the result of the bitwise XOR operation on two elements. Requires an implicit argument, bitwise_ptr of type BitwiseBuiltin*.

The function accepts the explicit arguments:

x of type felt, the first operand.
y of type felt, the second operand.

The function returns the value:

x_xor_y of type felt, the result of the bitwise XOR operation a ^ b.

The example below shows the operation on binary inputs 1100 and 1010 results in 0110:

from starkware.cairo.common.bitwise import bitwise_xor

let (result) = bitwise_xor(12, 10);  // Binary (1100, 1010).
assert result = 6;  // Binary 0110.

`bitwise_or()`¶

Returns the result of the bitwise OR operation on two elements. Requires an implicit argument, bitwise_ptr of type BitwiseBuiltin*.

The function accepts the explicit arguments:

x of type felt, the first operand.
y of type felt, the second operand.

The function returns the value:

x_or_y of type felt, the result of the bitwise OR operation a | b.

The example below shows the operation on binary inputs 1100 and 1010 results in 1110:

from starkware.cairo.common.bitwise import bitwise_or

let (result) = bitwise_or(12, 10);  // Binary (1100, 1010).
assert result = 14;  // Binary 1110.

`bitwise_operations()`¶

Returns the result of the bitwise AND, XOR and OR operations on two elements. Requires an implicit argument, bitwise_ptr of type BitwiseBuiltin*.

The function accepts the explicit arguments:

x of type felt, the first operand.
y of type felt, the second operand.

The function returns the values:

x_and_y of type felt, the result of the bitwise AND operation a & b.
x_xor_y of type felt, the result of the bitwise XOR operation a ^ b.
x_or_y of type felt, the result of the bitwise OR operation a | b.

The example below shows the operation on binary inputs 1100 and 1010 results in 1000, 0110 and 1110:

from starkware.cairo.common.bitwise import bitwise_operations

// Binary (1100, 1010).
let (and_, xor_, or_) = bitwise_operations(12, 10);
assert and_ = 8;  // Binary 1000.
assert xor_ = 6;  // Binary 0110.
assert or_ = 14;  // Binary 1110.

`default_dict`¶

This section refers to the common library’s default_dict module.

`default_dict_new()`¶

Returns a new dictionary where all keys are initialized with a given default value. One can interact with the dictionary using the dict_read(), dict_write() operations discussed in the dict module. Note that in order to enforce the consistency of subsequent dictionary accesses with the default values, one must eventually call default_dict_finalize() (which in turn calls dict_squash(), as discussed in the corresponding section). Otherwise, this is only enforced by hints which can be bypassed by a malicious prover.

The function expects the explicit argument:

default_value, the default value.

The function returns:

res, of type DictAccess*, a pointer to the new dictionary.

`default_dict_finalize()`¶

Squashes the dictionary and verifies consistency with respect to the default value. A squashed dictionary is one whose intermediate updates have been summarized and each key appears exactly once with its most recent value. For more details see dict_squash() from the dict module.

The function expects three explicit arguments:

dict_accesses_start, a pointer to the initial dictionary (first operation).
dict_accesses_end, a pointer to the end of the dictionary (last operation).
default_value, the expected initial value of each key.

The function returns the values:

squashed_dict_start, a pointer to the start of the squashed dictionary.
squashed_dict_end, a pointer to the end of the squashed dictionary.

Note that one must eventually call default_dict_finalize() to verify both the internal consistency of the DictAccess entries forming the dictionary and of the consistency with the default value.

In the example below we create and finalize a default dictionary, and explain what may happen if default_dict_finalize() is not called.

Example¶

%builtins range_check

from starkware.cairo.common.default_dict import (
    default_dict_new,
    default_dict_finalize,
)
from starkware.cairo.common.dict import dict_write, dict_update

func main{range_check_ptr}() -> () {
    alloc_locals;
    let (local my_dict_start) = default_dict_new(
        default_value=7
    );
    let my_dict = my_dict_start;
    dict_write{dict_ptr=my_dict}(key=0, new_value=8);
    // The following is an inconsistent update, the entry with
    // key 1 still contains the default value 7.
    // This will fail while using the library's hints
    // but can be made to pass by a malicious prover.

    // For a honest prover, this will fail in the library's hints,
    // but a malicious prover can make the following dict_update
    // pass. However, if it does, the code will necessarily fail
    // at default_dict_finalize.
    // dict_update{dict_ptr=my_dict}(key=1, prev_value=8, new_value=9);

    // Finalize fails for the malicious prover with extra update.
    let (
        finalized_dict_start, finalized_dict_end
    ) = default_dict_finalize(my_dict_start, my_dict, 7);
    return ();
}

`dict`¶

This section refers to the common library’s dict module for working with user defined dictionaries, abstracting away Cairo’s simulation of dictionaries as an array of read/write logs.

`dict_new()`¶

Returns a new dictionary. The function does not require any arguments. A new dictionary is initially populated by using a hint with the expression initial_dict. The dictionary associated with that expression will be found by the __dict_manager.

Note that Cairo has no way to enforce that subsequent read/writes are consistent with the initial_dict hint (this is only enforced at the Python level). Technically, the return value is a pointer to an empty DictAccess array. Soundness with respect to the initial values can only be achieved with explicit initialization of the dictionary or by using the default_dict_new() from the default_dict module instead, which is not based on user-defined hints (and is thus also available in StarkNet, unlike dict_new()). One must eventually call dict_squash() when using the dictionary.

The function returns the argument:

res, of type DictAccess*, a pointer to the new dictionary.

from starkware.cairo.common.dict import dict_new
from starkware.cairo.common.dict_access import DictAccess

alloc_locals;
%{
    initial_dict = {
        5: 8,
        12: 35,
        33: 198
    }
%}
let (local my_dict: DictAccess*) = dict_new();
// 'my_dict' is now associated with the hint's 'initial_dict'.

`dict_read()`¶

Returns the value of a specified dictionary key. Must be passed an implicit argument, dict_ptr, of type DictAccess*, representing the pointer to the end of the dictionary. Only available for dictionaries created via dict_new() or default_dict_new().

Note that the consistency of the returned value from dict_read() is only verified at the hint level (technically, dict_read() appends one DictAccess instruction to the dictionary). To make sure that a malicious prover won’t be able to return a different value one must eventually call dict_squash().

The function accepts the explicit argument:

key, of type felt, the requested key.

The function returns the argument:

value, of type felt, the value assigned to key.

The example below shows, for an existing dictionary whose end pointer is my_dict, how the value associated with the key 12 can be read. Note how the pointer my_dict is passed as an implicit argument.

from starkware.cairo.common.dict import dict_read

// my_dict has key:val pairs {5: 8, 12: 35, 33: 198}.
let (local val: felt) = dict_read{dict_ptr=my_dict}(key=12);
assert val = 35;

`dict_write()`¶

Overrides the current value of an existing key. In order to work with a dynamic dictionary one can initialize it with default_dict_new() rather than dict_new() (in which case all keys are assumed to be present, initially with the default value). Must be passed a pointer to the end of the dictionary, dict_ptr, of type DictAccess*, as an implicit argument. No values are returned. Only available for dictionaries created via dict_new() or default_dict_new().

The function accepts the explicit arguments:

key, of type felt, the key to override.
new_value, of type felt, the value to be assigned to key.

The example below shows how, for an existing dictionary whose pointer is my_dict, the value associated with the key 12 can be changed from 35 to 34. Note how the pointer my_dict is passed as an implicit argument.

from starkware.cairo.common.dict import dict_write

// my_dict has key:val pairs {5: 8, 12: 35, 33: 198}.

// The value associated with key=12 is changed.
dict_write{dict_ptr=my_dict}(key=12, new_value=34);

let key_12_val = dict_read{dict_ptr=my_dict}(key=12);
// dict_squash() must be called here (omitted for brevity).
assert key_12_val = 34;

`dict_update()`¶

Updates the value of a given key in a dictionary. dict_ptr, of type DictAccess*, representing a pointer to the end of the dictionary, must be passed as an implicit argument to this function. Only available for dictionaries created via dict_new() or default_dict_new(). No values are returned.

The function expects three explicit arguments of type felt:

key, the key to update.
prev_value, the current value assigned to key.
new_value, the value to be assigned to key.

It is possible to get prev_value from __dict_manager using the hint:

%{ ids.new_value = __dict_manager.get_dict(ids.dict_ptr)[ids.key] %}

The example demonstrates how to update the value of a specified key for a dictionary whose end pointer is referenced by dict_end.

%builtins range_check

from starkware.cairo.common.dict import (
    dict_new,
    dict_write,
    dict_update,
    dict_squash,
)

func main{range_check_ptr}() -> () {
    %{ initial_dict = {0: 0} %}
    let (dict_start) = dict_new();
    let dict_end = dict_start;
    dict_write{dict_ptr=dict_end}(key=0, new_value=1);
    dict_update{dict_ptr=dict_end}(
        key=0, prev_value=1, new_value=2
    );
    return ();
}

One can think of dict_update() as a conditional write. Passing prev_value ensures that an override will only occur in case the current value equals prev_value. Note that this is only verified at the hint level and consistency relies on eventual squashing.

`dict_squash()`¶

Squashes a dictionary represented by an array of read/write logs. A squashed dictionary is one whose intermediate updates have been summarized and each key appears exactly once with its most recent value. This is the only function in this module that asserts the consistency of accesses to the dictionary represented by the DictAccess array. A program that uses dict operations without invoking dict_squash() can run successfully even if it contains inconsistent dictionary operations (see example below).

The function uses the range_check builtin and thus requires range_check_ptr as an implicit argument.

The function expects two explicit arguments of type DictAccess*:

dict_accesses_start, a pointer to the start of the dictionary (first operation).
dict_accesses_end, a pointer to the end of the dictionary (last operation).

The function returns two values of type DictAccess*:

squashed_dict_start, a pointer to the start of the squashed dictionary.
squashed_dict_end, a pointer to the end of the squashed dictionary.

The only function that uses dict_accesses_start is dict_squash(). All other dictionary operations append to the array of DictAccess instances.

%builtins range_check

from starkware.cairo.common.dict import (
    dict_new,
    dict_write,
    dict_update,
    dict_squash,
)

func main{range_check_ptr}() -> () {
    %{ initial_dict = {0: 0} %}
    let (dict_start) = dict_new();
    let dict_end = dict_start;
    dict_write{dict_ptr=dict_end}(0, 1);
    dict_update{dict_ptr=dict_end}(0, 1, 2);
    let (squashed_dict_start, squashed_dict_end) = dict_squash{
        range_check_ptr=range_check_ptr
    }(dict_start, dict_end);
    // The following is an inconsistent update, 'prev_value'
    // is now '2'. This will fail while using the library's hints
    // but can be made to pass by a malicious prover.
    dict_update{dict_ptr=squashed_dict_end}(
        key=0, prev_value=3, new_value=2
    );
    // Squash fails. Even a malicious prover can't pass
    // verification for a failed dict_squash operation.
    let (squashed_dict_start, squashed_dict_end) = dict_squash{
        range_check_ptr=range_check_ptr
    }(squashed_dict_start, squashed_dict_end);
    return ();
}

`dict_access`¶

This section refers to the common library’s dict_access module.

`DictAccess`¶

A struct specifying the DictAccess memory structure. Cairo simulates dictionaries by an array of read-modify-write instructions, which are logged by the DictAccess struct. The consistency of such an array can be verified by applying squash_dict().

For libraries that abstract Cairo’s representation of dictionaries and allow a more standard dictionary interface than what will be shown here, see the dict and default_dict modules in the common library.

The struct has the following members of type felt:

key, the key of a key-value pair.
prev_value, the previous value iof a key-value pair.
new_value, the current value of a key-value pair.

In the example below, a dictionary is created by adding DictAccess structs to an array and manually incrementing a pointer to the end of the array.

%builtins range_check

from starkware.cairo.common.dict import dict_squash
from starkware.cairo.common.squash_dict import squash_dict
from starkware.cairo.common.alloc import alloc
from starkware.cairo.common.dict_access import DictAccess

func main{range_check_ptr}() -> () {
    alloc_locals;
    let (dict_start: DictAccess*) = alloc();
    assert dict_start[0] = DictAccess(
        key=0, prev_value=100, new_value=100
    );
    assert dict_start[1] = DictAccess(
        key=1, prev_value=200, new_value=200
    );

    let dict_end = dict_start + 2 * DictAccess.SIZE;
    // (dict_start, dict_end) now represents the dictionary
    // {0: 100, 1: 200}.

    // Now pass the dictionary to a function for inspection.
    check_key_ratio{dict_ptr=dict_end}(a=0, b=1);

    // Squash the dictionary from an array of 4 DictAccess
    // structs to an array of 2, with a single DictAccess
    // entry per key.
    // Fails if the prover changed 'value_a' and 'value_b'.
    let (local squashed_dict_start: DictAccess*) = alloc();
    let (squashed_dict_end) = squash_dict{
        range_check_ptr=range_check_ptr
    }(dict_start, dict_end, squashed_dict_start);
    return ();
}

check_key_ratio() checks that the value of key b is double the value of key a. This will only be enforced if we eventually call squash_dict().

func check_key_ratio{dict_ptr: DictAccess*}(a: felt, b: felt) {
    alloc_locals;
    // Adds more DictAccess entries to the existing array.
    // Values match previous entries and will be squashed.
    local value_a;
    local value_b;
    %{
        ids.value_a = 100  # Malicious prover may change it.
        ids.value_b = 200
    %}
    assert value_a * 2 = value_b;
    // Simulate dictionary read by appending a 'DictAccess'
    // instruction with 'prev_value=new_value=current_value'.
    assert dict_ptr[0] = DictAccess(
        key=a, prev_value=value_a, new_value=value_a
    );
    assert dict_ptr[1] = DictAccess(
        key=b, prev_value=value_b, new_value=value_b
    );
    let dict_end = dict_ptr + 2 * DictAccess.SIZE;
    // A call to squash_dict() will ensure that the prover
    // used values that are consistent with the input dictionary.
    return ();
}

`find_element`¶

This section refers to the common library’s find_element module.

`find_element()`¶

Returns the pointer to an element in an array whose key matches a specified key. The function requires the implicit argument range_check_ptr. Note that if the array contains multiple elements with the requested key, the function may return a pointer to any of them.

The function requires four explicit arguments:

array_ptr, a pointer to an array.
elm_size, the size (in memory cells) of each element in the array.
n_elms, the number of elements in the array.
key, the key to look for (the key is assumed to be the first member of each element in the array).

The function returns:

elm_ptr, the pointer to an element whose first memory cell is key (namely, [elm_ptr]=key).

The function has the ability to receive the index of that element via a hint, which may save proving time. If key is not found then a ValueError exception will be raised while processing the library’s hint. Note that a malicious prover can’t cause find_element() to succeed by changing the hint, as the Cairo program will fail when the key is not present in the array.

%builtins range_check
from starkware.cairo.common.find_element import find_element
from starkware.cairo.common.alloc import alloc

struct MyStruct {
    a: felt,
    b: felt,
}

func main{range_check_ptr}() -> () {
    // Create an array with MyStruct elements (1,2), (3,4), (5,6).
    alloc_locals;
    let (local array_ptr: MyStruct*) = alloc();
    assert array_ptr[0] = MyStruct(a=1, b=2);
    assert array_ptr[1] = MyStruct(a=3, b=4);
    assert array_ptr[2] = MyStruct(a=5, b=6);

    // Find any element with key '5'.
    let (element_ptr: MyStruct*) = find_element(
        array_ptr=array_ptr,
        elm_size=MyStruct.SIZE,
        n_elms=3,
        key=5,
    );
    // A pointer to the element with index 2 is returned.
    assert element_ptr.a = 5;
    assert element_ptr.b = 6;

    // Pass a known index in a hint to save proving time.
    %{ __find_element_index = 2 %}
    let (element_ptr: MyStruct*) = find_element(
        array_ptr=array_ptr,
        elm_size=MyStruct.SIZE,
        n_elms=3,
        key=5,
    );
    assert element_ptr.a = 5;
    assert element_ptr.b = 6;
    return ();
}

`search_sorted_lower()`¶

Returns the pointer to the first element in the array whose first field is at least key. The array elements must be sorted by the first field in ascending order. If no such item exists, it returns a pointer to the end of the array (after the last item). The function requires the implicit argument range_check_ptr.

The function accepts the arguments:

array_ptr, a pointer to a sorted array.
elm_size, the size (in memory cells) of each element in the array.
n_elms, the number of elements in the array.
key, the key lower bound (the key is assumed to be the first member of each element in the array).

The function returns:

elm_ptr, the pointer to the first element whose key is greater or equal to the lower bound.

Continuing with the example above, with lower bound 2, the middle element is returned.

from starkware.cairo.common.find_element import (
    search_sorted_lower,
)

let (smallest_ptr: MyStruct*) = search_sorted_lower(
    array_ptr=array_ptr, elm_size=2, n_elms=3, key=2
);
assert smallest_ptr.a = 3;
assert smallest_ptr.b = 4;

`search_sorted()`¶

Returns both the pointer to the first element in the array whose key matches a specified key, and an indicator for the success of the search. The array elements must be sorted by the first field in ascending order. If no such item exists, returns an undefined pointer, and success=0. The function requires the implicit argument range_check_ptr.

The function accepts the arguments:

array_ptr, the pointer to a sorted array.
elm_size, the size (in memory cells) of each element in the array.
n_elms, the number of elements in the array.
key, the key to look for (the key is assumed to be the first member of each element in the array).

The function returns:

elm_ptr, the pointer to the first element whose first member is key, namely [elm_ptr] = key.
success, a felt which equals 1 if the key was found and 0 otherwise.

Continuing with the same example, since the array is sorted, searching for the key 5 leads to the last element.

from starkware.cairo.common.find_element import search_sorted

let (first_ptr: MyStruct*, success_val) = search_sorted(
    array_ptr=array_ptr, elm_size=2, n_elms=3, key=5
);
assert success_val = 1;
assert first_ptr.a = 5;
assert first_ptr.b = 6;
// There is no element with key=2.
let (first_ptr: MyStruct*, success_val) = search_sorted(
    array_ptr=array_ptr, elm_size=2, n_elms=3, key=2
);
assert success_val = 0;

`set`¶

This section refers to the common library’s set module.

`set_add()`¶

This function either appends an element to a given array or asserts that it exists. An honest prover should not append the element if it is already present, but this is not verified. The function requires the implicit arguments set_end_ptr (the pointer to the end of the list) and range_check_ptr.

The function expects three explicit arguments:

set_ptr, the pointer to the start of the list.
elm_size, the size of each list element.
elm_ptr, a pointer to the element being added.

%builtins range_check

from starkware.cairo.common.alloc import alloc
from starkware.cairo.common.set import set_add

struct MyStruct {
    a: felt,
    b: felt,
}

func main{range_check_ptr}() {
    alloc_locals;

    // An array containing two structs.
    let (local my_list: MyStruct*) = alloc();
    assert my_list[0] = MyStruct(a=1, b=3);
    assert my_list[1] = MyStruct(a=5, b=7);

    // Suppose that we want to add the element
    // MyStruct(a=1, b=3), but only if it is not already
    // present (for the purpose of the example the contents of
    // the array are known, but this doesn't have to be the
    // case).
    let list_end: felt* = &my_list[2];
    let (new_elm: MyStruct*) = alloc();
    assert new_elm[0] = MyStruct(a=2, b=3);

    set_add{set_end_ptr=list_end}(
        set_ptr=my_list, elm_size=MyStruct.SIZE, elm_ptr=new_elm
    );
    return ();
}

Common Library¶

`alloc`¶

`alloc()`¶

`cairo_builtins`¶

`BitwiseBuiltin`¶

`HashBuiltin`¶

`SignatureBuiltin`¶

`bitwise`¶

`bitwise_and()`¶

`bitwise_xor()`¶

`bitwise_or()`¶

`bitwise_operations()`¶

`default_dict`¶

`default_dict_new()`¶

`default_dict_finalize()`¶

Example¶

`dict`¶

`dict_new()`¶

`dict_read()`¶

`dict_write()`¶

`dict_update()`¶

`dict_squash()`¶

`dict_access`¶

`DictAccess`¶

`find_element`¶

`find_element()`¶

`search_sorted_lower()`¶

`search_sorted()`¶

`set`¶

`set_add()`¶

Cairo

Navigation

Related Topics

Common Library¶

alloc¶

alloc()¶

cairo_builtins¶

BitwiseBuiltin¶

HashBuiltin¶

SignatureBuiltin¶

bitwise¶

bitwise_and()¶

bitwise_xor()¶

bitwise_or()¶

bitwise_operations()¶

default_dict¶

default_dict_new()¶

default_dict_finalize()¶

Example¶

dict¶

dict_new()¶

dict_read()¶

dict_write()¶

dict_update()¶

dict_squash()¶

dict_access¶

DictAccess¶

find_element¶

find_element()¶

search_sorted_lower()¶

search_sorted()¶

set¶

set_add()¶

`alloc`¶

`alloc()`¶

`cairo_builtins`¶

`BitwiseBuiltin`¶

`HashBuiltin`¶

`SignatureBuiltin`¶

`bitwise`¶

`bitwise_and()`¶

`bitwise_xor()`¶

`bitwise_or()`¶

`bitwise_operations()`¶

`default_dict`¶

`default_dict_new()`¶

`default_dict_finalize()`¶

`dict`¶

`dict_new()`¶

`dict_read()`¶

`dict_write()`¶

`dict_update()`¶

`dict_squash()`¶

`dict_access`¶

`DictAccess`¶

`find_element`¶

`find_element()`¶

`search_sorted_lower()`¶

`search_sorted()`¶

`set`¶

`set_add()`¶