The Tortoise and the Hare in TLA+

Problem: Determine if a linked list contains a cycle, using O(1) space.

Robert Floyd invented an algorithm for solving this problem in the late 60s, which is called “The Tortoise and the Hare”[1].

(This is supposedly a popular question to ask in technical interviews. I’m not a fan of expecting interviewees to re-invent the algorithms of famous computer scientists on the spot).

As an exercise, I implemented the algorithm in TLA+, using PlusCal.

The algorithm itself is very simple: the hardest part was deciding how to model linked lists in TLA+. We want to model the set of all linked lists, to express that algorithm should work for any element of the set. However, the model checker can only work with finite sets. The typical approach is to do something like “the set of all linked lists which contain up to N nodes”, and then run the checker against different values of N.

What I ended up doing was generating N nodes, giving each node a successor

(a next element, which could be the terminal value NIL), and then selecting

the start of the list from the set of nodes:

CONSTANTS N

ASSUME N \in Nat

Nodes == 1..N

NIL == CHOOSE NIL : NIL \notin Nodes

start \in Nodes
succ \in [Nodes -> Nodes \union {NIL}]

(Note: I originally defined NIL as a constant, but this is incorrect, see the comment by Ron Pressler for more details).
This is not the most efficient way from a model checker point of view, because the model checker will generate nodes that are irrelevant because they aren’t in the list. However, it does generate all possible linked lists up to length N.

Superficially, this doesn’t look like the pointer-and-structure linked list you’d see in C, but it behaves the same way at a high level. It’s possible to model a memory address space and pointers in TLA+, but I chose not to do so.

In addition, the nodes of a linked list typically have an associated value, but Floyd’s algorithm doesn’t use this value, so I didn’t model it.

Here’s my implementation of the algorithm:

EXTENDS Naturals

CONSTANTS N

ASSUME N \in Nat

Nodes == 1..N

NIL == CHOOSE NIL : NIL \notin Nodes

(*
--fair algorithm TortoiseAndHare

variables
    start \in Nodes,
    succ \in [Nodes -> Nodes \union {NIL}],
    cycle, tortoise, hare, done;
begin
h0: tortoise := start;
    hare := start;
    done := FALSE;
h1: while ~done do
        h2: tortoise := succ[tortoise];
            hare := LET hare1 == succ[hare] IN
                    IF hare1 \in DOMAIN succ THEN succ[hare1] ELSE NIL;
        h3: if tortoise = NIL \/ hare = NIL then
                cycle := FALSE;
                done := TRUE;
            elsif tortoise = hare then
                cycle := TRUE;
                done := TRUE;
            end if;
    end while;

end algorithm
*)

I wanted to use the model checker to verify the the implementation was correct:

PartialCorrectness == pc="Done" => (cycle <=> HasCycle(start))

(See the Correctness wikipedia page for why this is called “partial correctness”).

To check correctness, I needed to implement my HasCycle operator (without resorting to Floyd’s algorithm). I used the transitive closure of the successor function for this, which is called TC here. If the transitive closure contains the pair (node, NIL), then the list that starts with node does not contain a cycle:

HasCycle(node) == LET R == {<<s, t>> \in Nodes \X (Nodes \union {NIL}): succ[s] = t }
                  IN <<node, NIL>> \notin TC(R)

The above definition is “cheating” a bit, since we’ve defined a cycle by what it isn’t. It also wouldn’t work if we allowed infinite lists, since those don’t have cycles nor do they have NIL nodes.

Here’s a better definition: a linked list that starts with node contains a cycle if there exists some node n that is reachable from node and from itself.

HasCycle2(node) ==
  LET R == {<<s, t>> \in Nodes \X (Nodes \union {NIL}): succ[s] = t }
  IN \E n \in Nodes : /\ <<node, n>> \in TC(R) 
                      /\ <<n, n>> \in TC(R)

To implement the transitive closure in TLA+, I used an existing implementation

from the TLA+ repository itself:

TC(R) ==
  LET Support(X) == {r[1] : r \in X} \cup {r[2] : r \in X}
      S == Support(R)
      RECURSIVE TCR(_)
      TCR(T) == IF T = {} 
                  THEN R
                  ELSE LET r == CHOOSE s \in T : TRUE
                           RR == TCR(T \ {r})
                       IN  RR \cup {<<s, t>> \in S \X S : 
                                      <<s, r>> \in RR /\ <<r, t>> \in RR}
  IN  TCR(S)

The full model is the lorin/tla-tortoise-hare repo on GitHub.