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EWD840.tla
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EWD840.tla
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------------------------------- MODULE EWD840 -------------------------------
(***************************************************************************)
(* TLA+ specification of an algorithm for distributed termination *)
(* detection on a ring, due to Dijkstra, published as EWD 840: *)
(* Derivation of a termination detection algorithm for distributed *)
(* computations (with W.H.J.Feijen and A.J.M. van Gasteren). *)
(***************************************************************************)
EXTENDS Naturals
CONSTANT N
ASSUME NAssumption == N \in Nat \ {0}
VARIABLES active, color, tpos, tcolor
Node == 0 .. N-1
Color == {"white", "black"}
TypeOK ==
/\ active \in [Node -> BOOLEAN] \* status of nodes (active or passive)
/\ color \in [Node -> Color] \* color of nodes
/\ tpos \in Node \* token position
/\ tcolor \in Color \* token color
(***************************************************************************)
(* Initially the token is black. The other variables may take any *)
(* "type-correct" values. *)
(***************************************************************************)
Init ==
/\ active \in [Node -> BOOLEAN]
/\ color \in [Node -> Color]
/\ tpos \in Node
/\ tcolor = "black"
(***************************************************************************)
(* Node 0 may initiate a probe when it has the token and when either it is *)
(* black or the token is black. It passes a white token to node N-1 and *)
(* paints itself white. *)
(***************************************************************************)
InitiateProbe ==
/\ tpos = 0
/\ tcolor = "black" \/ color[0] = "black"
/\ tpos' = N-1
/\ tcolor' = "white"
/\ active' = active
/\ color' = [color EXCEPT ![0] = "white"]
(***************************************************************************)
(* A node i different from 0 that possesses the token may pass it to node *)
(* i-1 under the following circumstances: *)
(* - node i is inactive or *)
(* - node i is colored black or *)
(* - the token is black. *)
(* Note that the last two conditions will result in an inconclusive round, *)
(* since the token will be black. The token will be stained if node i is *)
(* black, otherwise its color is unchanged. Node i will be made white. *)
(***************************************************************************)
PassToken(i) ==
/\ tpos = i
/\ ~ active[i] \/ color[i] = "black" \/ tcolor = "black"
/\ tpos' = i-1
/\ tcolor' = IF color[i] = "black" THEN "black" ELSE tcolor
/\ active' = active
/\ color' = [color EXCEPT ![i] = "white"]
(***************************************************************************)
(* token passing actions controlled by the termination detection algorithm *)
(***************************************************************************)
System == InitiateProbe \/ \E i \in Node \ {0} : PassToken(i)
(***************************************************************************)
(* An active node i may activate another node j by sending it a message. *)
(* If j>i (hence activation goes against the direction of the token being *)
(* passed), then node i becomes black. *)
(***************************************************************************)
SendMsg(i) ==
/\ active[i]
/\ \E j \in Node \ {i} :
/\ active' = [active EXCEPT ![j] = TRUE]
/\ color' = [color EXCEPT ![i] = IF j>i THEN "black" ELSE @]
/\ UNCHANGED <<tpos, tcolor>>
(***************************************************************************)
(* Any active node may become inactive at any moment. *)
(***************************************************************************)
Deactivate(i) ==
/\ active[i]
/\ active' = [active EXCEPT ![i] = FALSE]
/\ UNCHANGED <<color, tpos, tcolor>>
(***************************************************************************)
(* actions performed by the underlying algorithm *)
(***************************************************************************)
Environment == \E i \in Node : SendMsg(i) \/ Deactivate(i)
(***************************************************************************)
(* next-state relation: disjunction of above actions *)
(***************************************************************************)
Next == System \/ Environment
vars == <<active, color, tpos, tcolor>>
Spec == Init /\ [][Next]_vars /\ WF_vars(System)
-----------------------------------------------------------------------------
(***************************************************************************)
(* Non-properties, useful for validating the specification with TLC. *)
(***************************************************************************)
TokenAlwaysBlack == tcolor = "black"
NeverChangeColor == [][ UNCHANGED color ]_vars
(***************************************************************************)
(* Main safety property: if there is a white token at node 0 then every *)
(* node is inactive. *)
(***************************************************************************)
terminationDetected ==
/\ tpos = 0 /\ tcolor = "white"
/\ color[0] = "white" /\ ~ active[0]
TerminationDetection ==
terminationDetected => \A i \in Node : ~ active[i]
(***************************************************************************)
(* Liveness property: termination is eventually detected. *)
(***************************************************************************)
Liveness ==
(\A i \in Node : ~ active[i]) ~> terminationDetected
(***************************************************************************)
(* The following property asserts that when every process always *)
(* eventually terminates then eventually termination will be detected. *)
(* It does not hold since processes can wake up each other. *)
(***************************************************************************)
FalseLiveness ==
(\A i \in Node : []<> ~ active[i]) ~> terminationDetected
(***************************************************************************)
(* The following property says that eventually all nodes will terminate *)
(* assuming that from some point onwards no messages are sent. It is *)
(* not supposed to hold: any node may indefinitely perform local *)
(* computations. However, this property is verified if the fairness *)
(* condition WF_vars(Next) is used instead of only WF_vars(System) that *)
(* requires fairness of the actions controlled by termination detection. *)
(***************************************************************************)
SpecWFNext == Init /\ [][Next]_vars /\ WF_vars(Next)
AllNodesTerminateIfNoMessages ==
<>[][\A i \in Node : ~ SendMsg(i)]_vars => <>(\A i \in Node : ~ active[i])
(***************************************************************************)
(* Dijkstra's inductive invariant *)
(***************************************************************************)
Inv ==
\/ P0:: \A i \in Node : tpos < i => ~ active[i]
\/ P1:: \E j \in 0 .. tpos : color[j] = "black"
\/ P2:: tcolor = "black"
(***************************************************************************)
(* Use the following specification to let TLC check that the predicate *)
(* TypeOK /\ Inv is inductive for EWD 840: verify that it is an *)
(* (ordinary) invariant of a specification obtained by replacing the *)
(* initial condition by that conjunction. *)
(***************************************************************************)
CheckInductiveSpec == TypeOK /\ Inv /\ [][Next]_vars
(***************************************************************************)
(* The algorithm implements the high-level specification of termination *)
(* detection in a ring with synchronous communication between nodes. *)
(* Note that the parameters of the module SyncTerminationDetection are *)
(* instantiated by the symbols of the same name in the present module. *)
(***************************************************************************)
TD == INSTANCE SyncTerminationDetection
THEOREM Spec => TD!Spec
=============================================================================
\* Modification History
\* Last modified Thu Jan 21 16:09:40 CET 2021 by merz
\* Created Mon Sep 09 11:33:10 CEST 2013 by merz