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blink_tree.tla
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------------------------ MODULE blink_tree ------------------------
EXTENDS SequencesExt, Naturals, FiniteSets, TLC, action, StateDB
CONSTANTS KEY_INIT
CONSTANTS KEY_OPER
CONSTANTS KEY_MAX
CONSTANTS PAGE_NUM
CONSTANTS FAN_OUT_NUM
CONSTANTS SESSION_NUM
CONSTANTS NULL
CONSTANTS ENABLE_ACTION
VARIABLES
v_root_id,
v_tree_level,
v_root_seq,
v_page,
v_latch,
v_stack,
v_depth,
v_command,
v_operation,
__action__
variables == <<
v_root_id,
v_tree_level,
v_root_seq,
v_page,
v_latch,
v_stack,
v_depth,
v_command,
v_operation,
__action__
>>
vars_view == <<
v_root_seq,
v_root_id,
v_tree_level,
v_page,
v_latch,
v_stack,
v_depth,
v_command,
v_operation
>>
AI == "AccessIntent"
ND == "NodeDelete"
RL == "ReadLock"
WL == "WriteLock"
PM == "ParentModification"
PAGE_ID == 1..PAGE_NUM
SESSION == 1..SESSION_NUM
S_IDLE == "IDLE"
S_SEARCH_NON_LEAF == "SEARCH_NON_LEAF"
S_SEARCH_LEAF == "SEARCH_LEAF"
S_READ_WRITE == "READ_WRITE"
S_COMMAND == "COMMAND"
S_SCAN_LEAF == "SCAN_LEAF"
S_DONE == "DONE"
C_INSERT_KEY_VALUE == "INSERT_KEY_VALUE"
C_SPLIT_PAGE == "SPLIT_PAGE"
C_SEARCH_KEY == "SEARCH_KEY"
C_CONSOLIDATE_PAGE == "CONSOLIDATE_PAGE"
C_DELETE_PAGE == "DELETE_PAGE"
C_DELETE_SLOT == "DELETE_SLOT"
C_UPDATE_SLOT == "UPDATE_SLOT"
C_INSERT_SLOT_GUT == "INSERT_SLOT_GUT"
C_UPDATE_SLOT_GUT == "UPDATE_SLOT_GUT"
C_DELETE_SLOT_GUT == "DELETE_SLOT_GUT"
C_LATCH_ACQUIRE == "LATCH_ACQUIRE"
C_LATCH_RELEASE == "LATCH_RELEASE"
C_INSERT_SLOT == "INSERT_SLOT"
C_SEARCH_KEY_LEAF == "SEARCH_KEY_LEAF"
C_VISIT_PARENT == "VISIT_PARENT"
C_CHECK_ROOT == "CHECK_ROOT"
C_UPDATE_TREE_LEVEL == "UPDATE_TREE_LEVEL"
C_SEARCH_PARENT == "SEARCH_PARENT"
OP_SEARCH_KEY == "SEARCH_KEY"
OP_INSERT_KEY == "INSERT_KEY"
OP_UPDATE_KEY == "UPDATE_KEY"
OP_DELETE_KEY == "DELETE_KEY"
OP_WRITE == {OP_INSERT_KEY, OP_UPDATE_KEY, OP_DELETE_KEY}
D_SETUP == "SETUP"
D_LATCH_ACQUIRE == "LATCH_ACQUIRE"
D_LATCH_RELEASE == "LATCH_RELEASE"
D_LATCH_ACQUIRED == "LATCH_ACQUIRED"
D_SEARCH_RESULT == "SEARCH_RESULT"
D_DELETE_RESULT == "DELETE_RESULT"
D_INSERT_RESULT == "INSERT_RESULT"
D_UPDATE_RESULT == "UPDATE_RESULT"
D_SPLIT_PAGE == "SPLIT_PAGE"
D_SEARCH_TOP_DOWN == "SEARCH_TOP_DOWN"
\* Return the minimum value from a set, or undefined if the set is empty.
Min(s) == CHOOSE x \in s : \A y \in s : x <= y
\* Return the maximum value from a set, or undefined if the set is empty.
Max(s) == CHOOSE x \in s : \A y \in s : x >= y
RECURSIVE __SeqRemoveIndex(_, _, _)
__SeqRemoveIndex(_seq_in, _seq_out, index) ==
IF Cardinality(index) = 0 THEN
_seq_out
ELSE
LET min_index == Min(index)
IN _seq_out \o <<_seq_in[min_index]>>
_SeqSelectAllInSeq(seq, Test(_)) ==
(*************************************************************************)
(* Selects the index of the first element such that Test(seq[i]) is true *)
(* Equals 0 if Test(seq[i]) is FALSE for all elements. *)
(*************************************************************************)
LET I == { i \in 1..Len(seq) : Test(seq[i]) }
IN I
_SeqRemoveMatch(_seq, _test(_)) ==
LET index == _SeqSelectAllInSeq(_seq, _test)
IN __SeqRemoveIndex(_seq,<<>>, index)
_HalfCeiling(_n) ==
LET v == _n \div 2
IN
IF _n % 2 = 0 THEN
v
ELSE
v + 1
_StateBLink(_id) ==
[
task_id |-> _id,
root_id |-> v_root_id,
page |-> v_page,
stack |-> v_stack[_id],
latch |-> v_latch,
operation |-> v_operation[_id]
]
_ActionSetup ==
ActionsFromHandle(
_StateBLink,
SESSION,
ActionSetup,
D_SETUP
)
_KeyLess(key1, key2) ==
CASE key1 = KEY_MAX -> (
FALSE
)
[] key2 = KEY_MAX -> (
TRUE
)
[] OTHER -> (
key1 < key2
)
_KeyGreat(key1, key2) ==
_KeyLess(key2, key1)
_KeyLessEqual(key1, key2) ==
~_KeyGreat(key1, key2)
_KeyEqual(key1, key2) ==
/\ ~_KeyLess(key1, key2)
/\ ~_KeyLess(key2, key1)
_ChoosePageId(_page_id) ==
CHOOSE _p \in _page_id: TRUE
_HighKey(
_page
) ==
IF Len(_page.slot) = 0 THEN
NULL
ELSE
_page.slot[Len(_page.slot)].key
RECURSIVE _ConstructSlotNonLeaf(_, _)
_ConstructSlotNonLeaf(_slot_array, _page_children) ==
LET _t == PrintT(<<_slot_array, _page_children>>)
IN
IF Cardinality(_page_children) = 0 THEN
_slot_array
ELSE
LET page ==
CHOOSE s1 \in _page_children:
\A s2 \in _page_children \ {s1}:
_KeyLess(_HighKey(s1), _HighKey(s2))
slot == [key |-> _HighKey(page), page_id |-> page.page_id]
IN _ConstructSlotNonLeaf(_slot_array \o <<slot>>, _page_children \ {page})
RECURSIVE _ConstructSlotLeaf(_, _)
_ConstructSlotLeaf(_slot_array, _slot_set) ==
IF Cardinality(_slot_set) = 0 THEN
_slot_array
ELSE
LET slot ==
CHOOSE s1 \in _slot_set:
\A s2 \in _slot_set \ {s1}:
_KeyLess(s1.key, s2.key)
IN _ConstructSlotLeaf(_slot_array \o <<slot>>, _slot_set \ {slot})
_SelectLevel(_page_set, _level) ==
{p \in _page_set: p.level = _level}
RECURSIVE _ConstructTreeNonLeaf(_, _, _, _, _, _)
_ConstructTreeNonLeaf(_page_set_l1, _page_set, _page_ids, _current_page_id, _level, _fan_out) ==
LET page_id ==
IF _current_page_id = NULL THEN
CHOOSE _p \in _page_ids: TRUE
ELSE
_current_page_id
IN CASE Cardinality(_page_set) = 0 -> (
_page_set_l1
)
[] Cardinality(_page_set) < _fan_out -> (
LET pages == _page_set
slot == _ConstructSlotNonLeaf(<<>>, pages)
page == [
page_id |-> page_id,
level |-> _level,
right_id |-> NULL,
slot |-> slot
]
IN _page_set_l1 \cup {page}
)
[] OTHER -> (
LET pages == CHOOSE pages \in SUBSET(_page_set):
/\ Cardinality(pages) = _fan_out
/\ \A p1 \in pages:
\A p2 \in _page_set \ pages:
_KeyLess(_HighKey(p1), _HighKey(p2))
slot == _ConstructSlotNonLeaf(<<>>, pages)
right_page_id == IF Cardinality(_page_set) = _fan_out THEN
\* no page left, no right siblings
NULL
ELSE
CHOOSE id \in _page_ids : id /= page_id
page == [
page_id |-> page_id,
level |-> _level,
right_id |-> right_page_id,
slot |-> slot
]
IN _ConstructTreeNonLeaf(
_page_set_l1 \cup {page},
_page_set \ pages,
_page_ids \ {page_id, right_page_id},
right_page_id, _level, _fan_out)
)
RECURSIVE _ConstructTreeLeaf(_, _, _, _, _)
_ConstructTreeLeaf(page_set, _keys, _page_ids, _current_page_id, _fan_out) ==
LET page_id == IF _current_page_id = NULL THEN
CHOOSE _p \in _page_ids: TRUE
ELSE
_current_page_id
IN
IF Cardinality(_keys) < _fan_out THEN
LET
slot == _ConstructSlotLeaf(<<>>, [key : _keys, value : {NULL}]) \o <<[key |-> KEY_MAX, value |-> NULL]>>
page == [
level |-> 0,
page_id |-> page_id,
right_id |-> NULL,
slot |-> slot
]
IN page_set \cup {page}
ELSE
LET key_page ==
IF Cardinality(_keys) < _fan_out THEN
_keys
ELSE
CHOOSE key_page \in SUBSET(_keys):
/\ Cardinality(key_page) = _fan_out
/\ \A k1 \in key_page:
\A k2 \in _keys \ key_page:
_KeyLess(k1, k2)
right_page_id == CHOOSE id \in _page_ids : id /= page_id
slot == _ConstructSlotLeaf(<<>>, [key : key_page, value : {NULL}])
page == [
page_id |-> page_id,
level |-> 0,
right_id |-> right_page_id,
slot |-> slot
]
IN _ConstructTreeLeaf(page_set \cup {page}, _keys \ key_page, _page_ids \ {page_id, right_page_id}, right_page_id, _fan_out)
_PageIdSet(_page_set) ==
{ p.page_id : p \in _page_set}
RECURSIVE _ConstructTreeGut(_, _, _, _, _)
_ConstructTreeGut(_page, _keys, _page_ids, _level, _fan_out) ==
IF _level = 0 THEN
LET pages == _ConstructTreeLeaf(_page, _keys, _page_ids, NULL, _fan_out)
IN _ConstructTreeGut(pages \cup _page, _keys, _page_ids \ _PageIdSet(pages \cup _page), 1, _fan_out)
ELSE
LET pages_l0 == _SelectLevel(_page, _level - 1)
IN IF Cardinality(pages_l0) = 1 THEN
_page
ELSE
LET pages_l1 == _ConstructTreeNonLeaf({}, pages_l0, _page_ids, NULL, _level, _fan_out)
IN _ConstructTreeGut(pages_l1 \cup _page, _keys, _page_ids \ _PageIdSet(pages_l1 \cup _page), _level + 1, _fan_out)
_ConstructTree(_keys, _page_ids, _fan_out) ==
LET pages == _ConstructTreeGut({}, _keys, _page_ids, 0, _fan_out)
root_page ==
CHOOSE p0 \in pages:
\A p1 \in pages \ {p0}:
p0.level > p1.level
root_id == root_page.page_id
level == root_page.level + 1
IN [root_id |-> root_id, page |-> pages, level |-> level]
_TestConstructTreeLeaf ==
LET pages == _ConstructTreeLeaf({}, 1..4, {"p1", "p2", "p3"}, NULL, 2)
IN /\ pages =
{
[slot |-> <<[key |-> KEY_MAX, value |-> NULL]>>, page_id |-> "p3", level |-> 0, right_id |-> NULL],
[slot |-> <<[key |-> 1, value |-> NULL], [key |-> 2, value |-> NULL]>>, page_id |-> "p1", level |-> 0, right_id |-> "p2"],
[slot |-> <<[key |-> 3, value |-> NULL], [key |-> 4, value |-> NULL]>>, page_id |-> "p2", level |-> 0, right_id |-> "p3"]
}
/\ LET non_leaf_pages == _ConstructTreeNonLeaf({}, pages, {"p4", "p5", "p6", "p7"}, NULL, 1, 2)
IN non_leaf_pages = {
[slot |-> <<[key |-> KEY_MAX, page_id |-> "p3"]>>, page_id |-> "p5", level |-> 1, right_id |-> NULL],
[slot |-> <<[key |-> 2, page_id |-> "p1"],
[key |-> 4, page_id |-> "p2"]>>, page_id |-> "p4", level |-> 1, right_id |-> "p5"]
}
_TestConstructTree ==
LET tree == _ConstructTree(1..4, {"p1", "p2", "p3", "p4", "p5", "p6", "p7"}, 2)
IN /\ tree.root_id = "p6"
/\ tree.level = 3
/\ tree.page = {
[
page_id |-> "p1", level |-> 0, right_id |-> "p2",
slot |-> <<[key |-> 1, value |-> NULL], [key |-> 2, value |-> NULL]>>
],
[
page_id |-> "p2", level |-> 0, right_id |-> "p3",
slot |-> <<[key |-> 3, value |-> NULL], [key |-> 4, value |-> NULL]>>
],
[
page_id |-> "p3", level |-> 0, right_id |-> NULL,
slot |-> <<[key |-> KEY_MAX, value |-> NULL]>>
],
[
page_id |-> "p4", level |-> 1, right_id |-> "p5",
slot |-> <<[key |-> 2, page_id |-> "p1"], [key |-> 4, page_id |-> "p2"]>>
],
[
page_id |-> "p5", level |-> 1, right_id |-> NULL,
slot |-> <<[key |-> KEY_MAX, page_id |-> "p3"]>>
],
[
page_id |-> "p6", level |-> 2, right_id |-> NULL,
slot |-> <<[key |-> 4, page_id |-> "p4"], [key |-> KEY_MAX, page_id |-> "p5"]>>
]
}
Init ==
LET tree == _ConstructTree(KEY_INIT, PAGE_ID, FAN_OUT_NUM)
IN
/\ v_root_id = tree.root_id
/\ v_tree_level = tree.level
/\ v_page = [ id \in {page.page_id: page \in tree.page} |-> CHOOSE page \in tree.page: page.page_id = id]
/\ v_operation = [s \in SESSION |-> [operation |-> NULL]]
/\ v_depth = [s \in SESSION |-> 0]
/\ v_latch = [s \in PAGE_ID |-> {}]
/\ v_stack = [s \in SESSION |-> <<>>]
/\ v_command = [s \in SESSION |-> <<>>]
/\ InitActionT(__action__, _ActionSetup, _ActionSetup)
/\ v_root_seq = 0
(*
Latch Requested
AI ND RL WL PM
E AccessIntent Y N Y Y Y
X NodeDelete N/A N/A Y Y Y
I ReadLock Y Y Y N Y
S WriteLock Y Y N N Y
T ParentModification Y Y Y Y N
*)
_First(
_command_seq
) ==
_command_seq[1]
_Last(
_command_seq
) ==
_command_seq[Len(_command_seq)]
_FirstCommandType(
_command_seq
) ==
_command_seq[1].command_type
_LastCommandType(
_command_seq
) ==
_command_seq[Len(_command_seq)].command_type
_PopLast(
_command_seq
) ==
IF Len(_command_seq) <= 1 THEN
<<>>
ELSE
SubSeq(_command_seq, 1, Len(_command_seq) - 1)
_PopFirst(
_command_seq
) ==
IF Len(_command_seq) <= 1 THEN
<<>>
ELSE
SubSeq(_command_seq, 2, Len(_command_seq))
_CommandExcluded(_cmd_seq, _cmd_type_set, _page_id_set) ==
[
i \in DOMAIN _cmd_seq |->
/\ _cmd_seq[i].command_type \in _cmd_type_set
/\ _cmd_seq[i].page_id \in _page_id_set
]
\*
\* since all references to it have previously been removed from the tree,
\* AcesssIntent, NodeDelete latch requesting cannot happend when existing ,
\* But for the root id is an exception
\*
_LatchNA(
_latch_existing,
_latch_request
) ==
/\ _latch_existing = ND
/\ _latch_request \in {AI, ND}
_LatchCapatible(
_latch_existing,
_latch_request
) ==
CASE _latch_existing = AI -> (
_latch_request \in {AI, RL, WL, PM}
)
[] _latch_existing = ND -> (
IF _latch_request \in {AI, ND} THEN
\* since all references to it have previously been removed from the tree
"AcesssIntent, NodeDelete latch requesting cannot happend when existing NodeDelete"
ELSE
_latch_request \in {RL, WL, PM}
)
[] _latch_existing = RL -> (
_latch_request \in {AI, ND, RL, PM}
)
[] _latch_existing = WL -> (
_latch_request \in {AI, ND, PM}
)
[] _latch_existing = PM -> (
_latch_request \in {AI, ND, RL, WL}
)
[] OTHER -> (
"unknown latch type"
)
_PopLastLevel(_stack) ==
IF Len(_stack) = 0 THEN
<<>>
ELSE
LET last_depth == _stack[Len(_stack)].depth
index == CHOOSE i \in 1..Len(_stack):
/\ \A j \in 1..i - 1:
_stack[j].depth < last_depth
/\ \A k \in i .. Len(_stack):
_stack[k].depth = last_depth
IN SubSeq(_stack, 1, index - 1)
_ParentLevelPageId(_stack) ==
IF Len(_stack) <= 1 THEN
NULL
ELSE
LET s == _PopLastLevel(_stack)
IN IF Len(s) = 0 THEN
NULL
ELSE
\* choose the first index
LET _index == {
_i \in 1..Len(s):
_stack[Len(s)].depth = _stack[_i].depth
}
index == Min(_index)
IN _stack[index].page_id
YES == "YES"
NO == "NO"
NA == "NA"
_LatchCanAcquire(_s, _existing_latch, mode) ==
IF \E l \in _existing_latch:
/\ l.task_id /= _s
/\ _LatchNA(l.mode, mode)
THEN
NA
ELSE
IF \A l \in _existing_latch:
\/ l.task_id = _s
\/ _LatchCapatible(l.mode, mode)
THEN
YES
ELSE
NO
_LatchAcquireCommand(_page_id, _mode) ==
[command_type |-> C_LATCH_ACQUIRE, page_id |-> _page_id, latch_mode |-> _mode]
_LatchReleaseCommand(_page_id, _mode) ==
[command_type |-> C_LATCH_RELEASE, page_id |-> _page_id, latch_mode |-> _mode]
_CheckRootCommand(_page_id, _seq) ==
[command_type |-> C_CHECK_ROOT, page_id |-> _page_id, sequence |-> _seq]
_UpdateTreeLevelCommand(_page_id, _level) ==
[command_type |-> C_UPDATE_TREE_LEVEL, page_id |-> _page_id, tree_level |-> _level]
\* search the index I, where:
\* the first index _slot[i] >= _k
__SearchKey(_slot_seq, _key, _choose_max_if_more_than_one) ==
CASE Len(_slot_seq) = 0 -> (
0
)
[] OTHER -> (
LET set ==
{
idx \in 1..Len(_slot_seq):
/\ ( \A i \in 1..idx - 1:
_KeyLess(_slot_seq[i].key, _key)
)
/\ (\/ (/\ _slot_seq[idx].key = _key
/\ ( ~\E i \in idx + 1 .. Len(_slot_seq):
_KeyLess(_slot_seq[i].key, _key)
)
)
\/ (/\ _KeyGreat(_slot_seq[idx].key, _key)
/\ ( ~\E i \in idx + 1 .. Len(_slot_seq):
~_KeyGreat(_slot_seq[i].key, _key)
)
)
)
}
IN CASE Cardinality(set) = 1 ->
CHOOSE index \in set: TRUE
[] Cardinality(set) = 0 ->
0
[] OTHER -> {
IF _choose_max_if_more_than_one THEN
Max(set)
ELSE
Min(set)
}
)
_SearchKey(_slot_seq, _key) ==
__SearchKey(_slot_seq, _key, TRUE)
_SearchSlot(_slot_seq, _slot) ==
IF Len(_slot_seq) = 0 THEN
0
ELSE
IF ~ \E i \in 1..Len(_slot_seq):
_slot_seq[i] = _slot
THEN
0
ELSE
CHOOSE i \in 1..Len(_slot_seq): _slot_seq[i] = _slot
_TestSearchKey ==
LET slot == <<[key |-> 3], [key |-> 4], [key |-> 4], [key |-> 6], [key |-> 8]>>
slot2 == <<>>
slot3 == <<[key |-> 10]>>
IN /\ 2 = _SearchKey(slot, 4)
/\ 4 = _SearchKey(slot, 5)
/\ 5 = _SearchKey(slot, 8)
/\ 0 = _SearchKey(slot, 10)
/\ 1 = _SearchKey(slot, 1)
/\ 1 = _SearchKey(slot3, 10)
/\ 1 = _SearchKey(slot3, 1)
\* search the next in the same depth
_PageRightLink(_page, _key) ==
CASE Len(_page.slot) = 0 -> ( \*empty page, search right sibling
_page.right_id
)
[] _KeyLess(_HighKey(_page), _key) -> ( \*high key < searched key
_page.right_id
)
[] OTHER -> (
NULL
)
_NonLeafRightOrChild(_page, _key) ==
CASE _PageRightLink(_page, _key) /= NULL -> (
_PageRightLink(_page, _key)
)
[] OTHER -> (
LET index == _SearchKey(_page.slot, _key)
IN IF index = 0 \/ index > Len(_page.slot) THEN
NULL
ELSE
_page.slot[index].page_id
)
_TestSearchPageNonLeaf ==
/\ LET page ==
[
level |-> 1,
right_id |-> "right",
slot |-> <<
[key |-> 2, page_id |-> "p1"],
[key |-> 3, page_id |-> "p3"],
[key |-> 6, page_id |-> "p6"],
[key |-> 8, page_id |-> "p8"]
>>
]
IN /\ _NonLeafRightOrChild(page, 3) = "p3"
/\ _NonLeafRightOrChild(page, 4) = "p6"
/\ _NonLeafRightOrChild(page, 7) = "p8"
/\ _NonLeafRightOrChild(page, 8) = "p8"
/\ _NonLeafRightOrChild(page, 2) = "p1"
/\ _NonLeafRightOrChild(page, 1) = "p1"
/\ _NonLeafRightOrChild(page, 11) = "right"
/\ LET page ==
[
level |-> 1,
right_id |-> "right",
slot |-> <<[key |-> 3, page_id |-> "p1"], [key |-> 6, page_id |-> "p2"], [key |-> 8, page_id |-> "p3"]>>
]
IN _NonLeafRightOrChild(page, 1) = "p1"
/\ LET page ==
[
level |-> 1,
right_id |-> NULL,
slot |-> <<[key |-> KEY_MAX, page_id |-> "p1"]>>
]
IN _NonLeafRightOrChild(page, 1) = "p1"
_PrevAccessed(_stack) ==
IF Len(_stack) = 0 THEN
NULL
ELSE
LET last_index == Len(_stack)
stack_top == _stack[last_index]
IN stack_top
_PrevAccessedPageId(_stack) ==
_PrevAccessed(_stack).page_id
_PrevAccessedDepth(_stack) ==
_PrevAccessed(_stack).depth
_LatchAdd(
_latch,
_task_id,
_latch_mode,
_page_id) ==
[_latch EXCEPT
![_page_id] = _latch[_page_id] \cup {[mode |-> _latch_mode, task_id |-> _task_id]}
]
_LatchRemove(
_latch,
_task_id,
_latch_mode,
_page_id) ==
[_latch EXCEPT
![_page_id] = _latch[_page_id] \ {[mode |-> _latch_mode, task_id |-> _task_id]}
]
_LatchHold(
_latch,
_session,
_latch_mode,
_page_id) ==
\E s \in _latch[_page_id]:
/\ s.task_id = _session
/\ s.mode = _latch_mode
__CommandSearchPageLeafForOperation(_v_command, _v_operation, _s, _key, _page_id) ==
CASE _v_operation[_s].operation = OP_SEARCH_KEY -> (
[_v_command EXCEPT ![_s] =
<<
[command_type |-> C_SEARCH_KEY_LEAF, page_id |-> _page_id, key |-> _key],
_LatchReleaseCommand(_page_id, RL)
>> \o _PopFirst(_v_command[_s])]
)
[] _v_operation[_s].operation = OP_INSERT_KEY -> (
[_v_command EXCEPT ![_s] =
<<
[command_type |-> C_INSERT_SLOT, page_id |-> _page_id, slot |-> [key |-> _key, value |-> _s]],
_LatchReleaseCommand(_page_id, WL)
>> \o _PopFirst(_v_command[_s])]
)
[] _v_operation[_s].operation = OP_UPDATE_KEY -> (
[_v_command EXCEPT ![_s] =
<<
[command_type |-> C_UPDATE_SLOT, page_id |-> _page_id, slot |-> [key |-> _key, value |-> _s],
slot_new |-> [key |-> _key, value |-> _s]],
_LatchReleaseCommand(_page_id, WL)
>> \o _PopFirst(_v_command[_s])]
)
[] _v_operation[_s].operation = OP_DELETE_KEY -> (
[_v_command EXCEPT ![_s] =
<<
[command_type |-> C_DELETE_SLOT, page_id |-> _page_id, slot |-> [key |-> _key, value |-> _s]],
_LatchReleaseCommand(_page_id, WL)
>> \o _PopFirst(_v_command[_s])]
)
[] OTHER -> (
<<"error, unknown operation type", _v_operation[_s].operation>>
)
_ActionLatchAcquire(_s, _page_id, _latch_mode) ==
Action(ActionInternal, MessageLocal(_s, D_LATCH_ACQUIRE, [task_id |-> _s, page_id |-> _page_id, latch_mode |-> _latch_mode]))
_ActionLatchAcquired(_s, _page_id, _latch_mode) ==
Action(ActionInternal, MessageLocal(_s, D_LATCH_ACQUIRED, [task_id |-> _s, page_id |-> _page_id, latch_mode |-> _latch_mode]))
_ActionLatchRelease(_s, _page_id, _latch_mode) ==
Action(ActionInternal, MessageLocal(_s, D_LATCH_RELEASE, [task_id |-> _s, page_id |-> _page_id, latch_mode |-> _latch_mode]))
_ActionSearchResult(_s, _page_id, _index, _success) ==
Action(ActionOutput, MessageLocal(_s, D_SEARCH_RESULT, [task_id |-> _s, page_id |-> _page_id, index |-> _index, success |-> _success]))
_ActionDeleteResult(_s, _page_id, _index, _success) ==
Action(ActionOutput, MessageLocal(_s, D_DELETE_RESULT, [task_id |-> _s, page_id |-> _page_id, index |-> _index, success |-> _success]))
_ActionUpdateResult(_s, _page_id, _index, _success) ==
Action(ActionOutput, MessageLocal(_s, D_UPDATE_RESULT, [task_id |-> _s, page_id |-> _page_id, index |-> _index, success |-> _success]))
_ActionInsertResult(_s, _page_id, _index, _success) ==
Action(ActionOutput, MessageLocal(_s, D_INSERT_RESULT, [task_id |-> _s, page_id |-> _page_id, index |-> _index, success |-> _success]))
_ActionSearchPage(_s, _page_id) ==
Action(ActionInternal, MessageLocal(_s, D_SEARCH_TOP_DOWN, [task_id |-> _s, page_id |-> _page_id]))
__UpdatePageId(_command_seq, _page_id) ==
[i \in DOMAIN _command_seq |->
IF /\ "no_parent" \in DOMAIN _command_seq[i]
/\ _command_seq[i].no_parent
/\ _command_seq[i].page_id = NULL
/\ _command_seq[i].command_type \in {C_UPDATE_SLOT, C_DELETE_SLOT, C_INSERT_SLOT}
THEN
[_command_seq[i] EXCEPT !.page_id = _page_id]
ELSE
_command_seq[i]
]
__RemoveNullPageCommand(_command_seq) ==
_SeqRemoveMatch(_command_seq,
LAMBDA e :
/\ "no_parent" \in DOMAIN e
/\ e.no_parent
/\ e.page_id = NULL
/\ e.command_type \in {C_UPDATE_SLOT, C_DELETE_SLOT, C_INSERT_SLOT}
)
_LevelDown(_v_depth, _v_stack, _s, _page_id, _is_down) ==
/\ IF _is_down THEN
_v_depth' = [_v_depth EXCEPT ![_s] = _v_depth[_s] + 1]
ELSE
/\ UNCHANGED <<_v_depth>>
/\ _v_stack' = [_v_stack EXCEPT ![_s] = _v_stack[_s] \o << [depth |-> _v_depth[_s], page_id |-> _page_id]>>]
SearchPageTopDown(_s) ==
/\ Len(v_command[_s]) > 0
/\ _FirstCommandType(v_command[_s]) = C_SEARCH_KEY
/\ LET cmd == _First(v_command[_s])
page_id == cmd.page_id
key == cmd.key
mode == IF /\ cmd.current_level = 0 \* the leaf page
/\ v_operation[_s].operation \in OP_WRITE
THEN WL ELSE RL
current_level == cmd.current_level
IN IF ~ _LatchHold( v_latch, _s, mode, page_id) THEN (
/\ LET a == _ActionLatchAcquire(_s, page_id, mode)
IN SetAction(__action__, _ActionSetup, a, ENABLE_ACTION)
/\ v_command' = [v_command EXCEPT ![_s] =
<<_LatchAcquireCommand(page_id, mode)>> \o
<<_LatchReleaseCommand(page_id, AI)>> \o
<<cmd>> \o
<<_LatchReleaseCommand(page_id, mode)>> \o
_PopFirst(v_command[_s])]
/\ UNCHANGED <<v_stack, v_depth>>
) ELSE (
LET page == v_page[page_id]
IN IF page.level /= 0 THEN \* this is not a leaf node
LET next_page_id == _NonLeafRightOrChild(page, key)
is_sibling == next_page_id /= NULL /\ next_page_id = page.right_id
inc_depth == IF is_sibling THEN 0 ELSE 1
current_depth == v_depth[_s] + inc_depth
IN
IF next_page_id /= NULL THEN
(LET next_mode ==
IF /\ ~is_sibling
/\ page.level = 1
/\ v_operation[_s].operation \in OP_WRITE \* the child is leaf page
THEN WL ELSE RL
IN
/\ v_command' = [v_command EXCEPT ![_s] =
<<
_LatchAcquireCommand(next_page_id, AI),
_LatchReleaseCommand(page_id, mode),
_LatchAcquireCommand(next_page_id, next_mode),
_LatchReleaseCommand(next_page_id, AI),
[command_type |-> C_SEARCH_KEY, key |-> key, page_id |-> next_page_id, current_level |-> current_level - inc_depth],
_LatchReleaseCommand(next_page_id, next_mode)
>> \o _PopFirst(v_command[_s])]
/\ _LevelDown(v_depth, v_stack, _s, page_id, ~is_sibling)
/\ LET a == _ActionSearchPage(_s, page_id)
IN SetAction(__action__, _ActionSetup, a, ENABLE_ACTION)
)
ELSE
/\ PrintT(<<page, key>>)
/\ "SearchPageTopDown, error"
ELSE
/\ _LevelDown(v_depth, v_stack, _s, page_id, TRUE)
/\ v_command' = __CommandSearchPageLeafForOperation(v_command, v_operation, _s, key, page_id)
/\ UNCHANGED << __action__>>
)
/\ UNCHANGED <<v_latch, v_operation, v_page, v_root_id, v_tree_level, v_root_seq>>
SearchParent(_s) ==
/\ Len(v_command[_s]) > 0
/\ _FirstCommandType(v_command[_s]) = C_SEARCH_PARENT
/\ LET cmd == _First(v_command[_s])
page_id == cmd.page_id
slot == cmd.slot
key == cmd.slot.key
mode == IF cmd.level + 1 = cmd.current_level
THEN WL ELSE RL
IN
/\ IF ~ _LatchHold( v_latch, _s, mode, page_id) THEN (
/\ LET a == _ActionLatchAcquire(_s, page_id, mode)
IN SetAction(__action__, _ActionSetup, a, ENABLE_ACTION)
/\ v_command' = [v_command EXCEPT ![_s] =
<<_LatchAcquireCommand(page_id, mode)>> \o
<<_LatchReleaseCommand(page_id, AI)>> \o
<<cmd>> \o
<<_LatchReleaseCommand(page_id, mode)>> \o
_PopFirst(v_command[_s])]
/\ UNCHANGED <<v_stack, v_depth>>
) ELSE (
LET page == v_page[page_id]
IN IF cmd.level + 1 < cmd.current_level THEN \* not this level, search next level
LET next_page_id == _NonLeafRightOrChild(page, key)
is_sibling == next_page_id /= NULL /\ next_page_id = page.right_id
inc_depth == IF is_sibling THEN 0 ELSE 1
current_depth == v_depth[_s] + inc_depth
IN
IF next_page_id /= NULL THEN
(LET next_mode ==
IF /\ ~is_sibling
/\ page.level = cmd.level + 1
/\ v_operation[_s].operation \in OP_WRITE
THEN WL ELSE RL
IN
/\ v_command' = [v_command EXCEPT ![_s] =
<<
_LatchAcquireCommand(next_page_id, AI),
_LatchReleaseCommand(page_id, mode),
_LatchAcquireCommand(next_page_id, next_mode),
_LatchReleaseCommand(next_page_id, AI),
[
command_type |-> C_SEARCH_PARENT, slot |-> slot, level |-> cmd.level,
current_level |-> cmd.current_level - inc_depth, page_id |-> next_page_id
],
_LatchReleaseCommand(next_page_id, next_mode)
>> \o _PopFirst(v_command[_s])]
/\ _LevelDown(v_depth, v_stack, _s, page_id, ~is_sibling )
/\ LET a == _ActionSearchPage(_s, page_id)
IN SetAction(__action__, _ActionSetup, a, ENABLE_ACTION)
)
ELSE
/\ PrintT(<<page, key>>)
/\ "SearchPageTopDown, error"
ELSE
/\ v_command' = [v_command EXCEPT ![_s] = __UpdatePageId(_PopFirst(v_command[_s]), page_id)] \* update parent id
/\ _LevelDown(v_depth, v_stack, _s, page_id, TRUE)
/\ UNCHANGED << __action__>>
)
/\ UNCHANGED <<v_latch, v_operation, v_page, v_root_id, v_tree_level, v_root_seq>>
_InsertIntoPage(
_page,
_index,
_slot
) ==
LET s0 == _page.slot
s == IF _index >= 1 THEN
LET s1 == SubSeq(s0, 1, _index - 1)
s2 == SubSeq(s0, _index, Len(s0))
IN s1 \o <<_slot>> \o s2
ELSE
<<_slot>> \o s0
IN [_page EXCEPT !.slot = s]
\* delte _page.slot[_index], _index = 0 for invalid index, _page would be return
_DeleteFromPage(
_page,
_index
) ==
LET s0 == _page.slot
s == IF _index >= 1 THEN
LET s1 == SubSeq(s0, 1, _index - 1)
s2 == SubSeq(s0, _index + 1, Len(s0))
IN s1 \o s2
ELSE
s0
IN [_page EXCEPT !.slot = s]
_UpdatePage(
_page,
_index,
_slot
) ==
LET s0 == _page.slot
s == IF _index >= 1 THEN
LET s1 == SubSeq(s0, 1, _index - 1)
s2 == SubSeq(s0, _index + 1, Len(s0))
IN s1 \o <<_slot>> \o s2
ELSE
<<_slot>> \o s0
IN [_page EXCEPT !.slot = s]
_TestInsertIntoPage ==
/\ LET _page == [slot |-> <<[key |-> 3, page_id |-> 3], [key |-> 6, page_id |-> 6], [key |-> 9, page_id |-> 9]>>]
IN /\ _InsertIntoPage(_page, 1, [key |-> 1, page_id |-> 1]).slot =
<<[key |-> 1, page_id |-> 1], [key |-> 3, page_id |-> 3], [key |-> 6, page_id |-> 6], [key |-> 9, page_id |-> 9]>>
/\ _InsertIntoPage(_page, 2, [key |-> 4, page_id |-> 4]).slot =
<<[key |-> 3, page_id |-> 3], [key |-> 4, page_id |-> 4], [key |-> 6, page_id |-> 6], [key |-> 9, page_id |-> 9]>>
/\ LET _page == [slot |-> <<>>]
IN _InsertIntoPage(_page, 0, [key |-> 1, page_id |-> 1]).slot = <<[key |-> 1, page_id |-> 1]>>
_TestDeleteFromPage ==
/\ LET _page == [slot |-> <<[key |-> 3, page_id |-> 3], [key |-> 6, page_id |-> 6], [key |-> 9, page_id |-> 9]>>]
IN /\ _DeleteFromPage(_page, 1).slot =
<<[key |-> 6, page_id |-> 6], [key |-> 9, page_id |-> 9]>>
/\ _DeleteFromPage(_page, 2).slot =
<<[key |-> 3, page_id |-> 3], [key |-> 9, page_id |-> 9]>>
SearchKeyLeaf(_s) ==
/\ Len(v_command[_s]) > 0
/\ _FirstCommandType(v_command[_s]) = C_SEARCH_KEY_LEAF
/\ LET cmd == _First(v_command[_s])
page_id == cmd.page_id