Review articles
# Proving Program Termination

The program termination problem, also known as the uniform halting problem, can be defined as follows:

Using only a finite amount of time, determine whether a given program will always finish running or could execute forever.

This problem rose to prominence before the invention of the
modern computer, in the era of Hilbert's
*Entscheidungsproblem:*^{a} the
challenge to formalize all of mathematics and use algorithmic
means to determine the validity of all statements. In hopes of
either solving Hilbert's challenge, or showing it impossible,
logicians began to search for possible instances of undecidable
problems. Turing's proof^{38} of
termination's undecidability is the most famous of those
findings.^{b}

The termination problem is structured as an infinite set of queries: to solve the problem we would need to invent a method capable of accurately answering either "terminates" or "doesn't terminate" when given any program drawn from this set. Turing's result tells us that any tool that attempts to solve this problem will fail to return a correct answer on at least one of the inputs. No number of extra processors nor terabytes of storage nor new sophisticated algorithms will lead to the development of a true oracle for program termination.

Unfortunately, many have drawn too strong of a conclusion about the prospects of automatic program termination proving and falsely believe we are always unable to prove termination, rather than more benign consequence that we are unable to always prove termination. Phrases like "but that's like the termination problem" are often used to end discussions that might otherwise have led to viable partial solutions for real but undecidable problems. While we cannot ignore termination's undecidability, if we develop a slightly modified problem statement we can build useful tools. In our new problem statement we will still require that a termination proving tool always return answers that are correct, but we will not necessarily require an answer. If the termination prover cannot prove or disprove termination, it should return "unknown."

Using only a finite amount of time, determine whether a given program will always finish running or could execute forever, or return the answer "unknown."

This problem can clearly be solved, as we could simply always
return "unknown." The challenge is to solve this problem while
keeping the occurrences of the answer "unknown" to within a
tolerable threshold, in the same way that we hope Web browsers
will *usually* succeed to download Web pages, although we
know they will sometimes fail. Note that the principled use of
unknown in tools attempting to solve undecidable or intractable
problems is increasingly common in computer science; for example,
in program analysis, type systems, and networking.

In recent years, powerful new termination tools have emerged
that return "unknown" infrequently enough that they are useful in
practice.^{35} These termination tools
can automatically prove or disprove termination of many famous
complex examples such as Ackermann's function or McCarthy's 91
function as well as moderately sized industrial examples such as
Windows device drivers. Furthermore, entire families of
industrially useful termination-like properties—called
*liveness properties*—such as "Every call to
`lock`

is eventually followed by a call to
`unlock`

" are now automatically provable using
termination proving
techniques.^{12,29}
With every month, we now see more powerful applications of
automatic termination proving. As an example, recent work has
demonstrated the utility of automatic termination proving to the
problem of showing concurrent algorithms to be
non-blocking.^{20} With further research
and development, we will see more powerful and more scalable
tools.

We could also witness a shift in the power of software, as techniques from termination proving could lead to tools for other problems of equal difficulty. Whereas in the past a software developer hoping to build practical tools for solving something related to termination might have been frightened off by a colleague's retort "but that's like the termination problem," perhaps in the future the developer will instead adapt techniques from within modern termination provers in order to develop a partial solution to the problem of interest.

The purpose of this article is to familiarize the reader with the recent advances in program termination proving, and to catalog the underlying techniques for those interested in adapting the techniques to other domains. We also discuss current work and possible avenues for future investigation. Concepts and strategies will be introduced informally, with citations to original papers for those interested in more detail. Several sidebars are included for readers with backgrounds in mathematical logic.

Thirteen years after publishing his original undecidability
result, Turing proposed the now classic method of proving program
termination.^{39} His solution divides
the problem into two parts:

*Termination argument search:* Find a potential
termination argument in the form of a function that maps every
program state to a value in a mathematical structure called a
well-order. We will not define well-orders here, the reader can
assume for now that we are using the natural numbers (a.k.a. the
positive integers).

*Termination argument checking:* Proves the termination
argument to be valid for the program under consideration by
proving that result of the function decreases for every possible
program transition. That is, if *f* is the termination
argument and the program can transition from some state *s*
to state *s*', then *f(s) > f(s')*.

(Readers with a background in logic may be interested in the formal explanation contained in the sidebar here.)

A well-order can be thought of as a terminating
program—in the example of the natural numbers, the program
is one that counts from some initial value in the natural numbers
down to 0. Thus, no matter which initial value is chosen the
program will still terminate. Given this connection between
well-orders and terminating programs, in essence Turing is
proposing that we search for a map from the program we are
interested in proving terminating into a program known to
terminate such that all steps in the first program have analogous
steps in the second program. This map to a well-order is usually
called a *progress measure* or a *ranking function* in
the literature. Until recently, all known methods of proving
termination were in essence minor variations on the original
technique.

The problem with Turing's method is that finding a single, or monolithic, ranking function for the whole program is typically difficult, even for simple programs. In fact, we are often forced to use ranking functions into well-orders that are much more complex than the natural numbers. Luckily, once a suitable ranking function has been found, checking validity is in practice fairly easy.

The key trend that has led toward current progress in
termination proving has been the move away from the search for a
single ranking function and toward a search for a set of ranking
functions. We think of the set as a choice of ranking functions
and talk about a disjunctive termination argument. This
terminology refers to the proof rule of disjunctively
well-founded transition invariants.^{31}
The recent approaches for proving termination for general
programs^{3,4,9,12,14,26,32}
are based on this proof rule. The proof rule itself is based on
Ramsey's theorem,^{34} and it has been
developed in the effort to give a logical foundation to the
termination analysis based on size-change
graphs.^{24}

The principle it expresses appears implicitly in previously
developed termination algorithms for rewrite systems, logic, and
functional programs, see
refs^{10,15,17,24}.

The advantage to the new style of termination argument is that it is usually easier to find, because it can be expressed in small, mutually independent pieces. Each piece can be found separately or incrementally using various known methods for the discovery of monolithic termination arguments. As a trade-off, when using a disjunctive termination argument, a more difficult validity condition must be checked. This difficulty can be mitigated thanks to recent advances in assertion checking tools (as discussed in a later section).

*Example using a monolithic termination argument.*
Consider the example code fragment in Figure 1.
In this code the collection of user-provided input is performed
via the function *input*(). We will assume the user always
enters a new value when prompted. Furthermore, we will assume for
now that variables range over possibly negative integers with
arbitrary precision (that is, mathematical integers as opposed to
32-bit words, 64-bit words, and so on). Before reading further,
please answer the question: "Does this program terminate, no
matter what values the user gives via the `input()`

function?" The answer is given
below.^{c}

Using Turing's traditional method we can define a ranking
function from program variables to the natural numbers. One
ranking function that will work is `2x + y`

, though
there are many others. Here we are using the formula ```
2x +
y
```

as shorthand for a function that takes a program
configuration as its input and returns the natural number
computed by looking up the value of `x`

in the memory,
multiplying that by 2 and then adding in *y*'s
value—thus `2x + y`

represents a mapping from
program configurations to natural numbers. This ranking function
meets the constraints required to prove termination: the
valuation of `2x + y`

when executing at line 9 in the
program will be strictly one less than its valuation during the
same loop iteration at line 4. Furthermore, we know the function
always produces natural numbers (thus it is a map into a
well-order), as `2x + y`

is greater than 0 at lines 4
through 9.

Automatically proving the validity of a monolithic termination
argument like `2x + y`

is usually easy using tools
that check verification conditions (for example,
Slam^{2}). However, as mentioned
previously, the actual search for a valid argument is famously
tricky. As an example, consider the case in Figure
2, where we have replaced the command "y := y + 1;" in
Figure 1 with "y := `input ()`

;". In
this case no function into the natural numbers exists that
suffices to prove termination; instead we must resort to a
*lexicographic* ranking function (a ranking function into
ordinals, a more advanced well-order than the naturals).

*Example using a disjunctive termination argument.*
Following the trend toward the use of disjunctive termination
arguments, we could also prove the termination of Figure
1 by defining an argument as the unordered finite
collection of measures `x`

and `y`

. The
termination argument in this case should be read as:

`x`

goes down by at least 1 and is larger than
0.

*or*

`y`

goes down by at least 1 and is larger than
0

We have constructed this termination argument with two ranking
functions: `x`

and `y`

. The use of
"*or*" is key: the termination argument is modular because
it is easy to enlarge using additional measures via additional
uses of "*or*." As an example, we could enlarge the
termination argument by adding "*or* ```
2w –
y
```

goes down by at least 1 and is greater than 1,000."
Furthermore, as we will discuss later, independently finding
these pieces of the termination argument is easier in practice
than finding a single monolithic ranking function.

The expert reader will notice the relationship between our disjunctive termination argument and complex lexicographic ranking functions. The advantage here is that we do not need to find an order on the pieces of the argument, thus making the pieces of the argument independent from one another.

The difficulty with disjunctive termination arguments in
comparison to monolithic ones is that they are more difficult to
prove valid: for the benefit of modularity we pay the price in
the fact that the termination arguments must consider the
transitions in all possible loop unrollings and not just single
passes through a loop. That is to say: the disjunctive
termination argument must hold not only between the states before
and after any single iteration of the loop, but before and after
*any number* of iterations of the loop (one iteration, two
iterations, three iterations, and so on). This is a much more
difficult condition to automatically prove. In the case of
Figure 1 we can prove the more complex condition
using techniques described later.

Note that this same termination argument now works for the
tricky program in Figure 2, where we replaced
"`y := y + 1`

;" with "`y := input()`

;." On
every possible unrolling of the loop we will still see that
either `x`

or `y`

has gone down and is
larger than 0.

To see why we cannot use the same validity check for
disjunctive termination arguments as we do for monolithic ones,
consider the slightly modified example in Figure
3. For every single iteration of the loop it is true that
either `x`

goes down by at least one and
`x`

is greater than 0 or y goes down by at least one
and `y`

is greater than 0. Yet, the program does not
guarantee termination. As an example input sequence that triggers
non-termination, consider 5, 5, followed by 1, 0, 1, 0, 1, 0,
.... If we consider *all possible unrollings* of the
loop, however, we will see that after two iterations it is
possible (in the case that the user supplied the inputs 1 and 0
during the two loop iterations) that neither `x`

nor
`y`

went down, and thus the disjunctive termination
argument is not valid for the program in Figure
3.

While validity checking for disjunctive termination arguments
is more difficult than checking for monolithic arguments, we can
adapt the problem statement such that recently developed tools
for proving the validity of assertions in programs (such as
Slam^{2}).

An assertion statement can be put in a program to check if a
condition is true. For example, `assert (y ≥ 1);`

checks that `y ≥ 1`

after executing the command. We
can use an assertion checking tool to formally investigate at
compile time whether the conditions passed to assertion
statements always evaluate to true. For example, most assertion
checking tools will be able to prove the `assert`

statement at line 3 in Figure 4 never fails. Note
that compile-time assertion checking is itself an undecidable
problem, although it is technically in an easier class of
difficulty than termination.^{d}

The reason that assertion checking is so important to
termination is the validity of disjunctive termination arguments
can be encoded as an assertion statement, where the statement
fails only in the case that the termination argument is not
valid. Once we are given an argument of the form
*T*_{1} or *T*_{2} or ... or
*T*_{n}, to check validity we simply want to
prove the following statement:

Each time an execution passes through one state and then
through another one, *T*_{1} or *T*_{2}
or ... or *T*_{n} holds between these
two states. That is, there does not exist a pair of states, one
being reachable from the other, possibly via the unrolling of a
loop, such that neither *T*_{1} nor
*T*_{2} nor ... nor *T*_{n}
holds between this pair of states.

This statement can be verified a program transformation where
we introduce new variables into the program to record the state
before the unrolling of the loop and then use an assertion
statement to check the termination argument always holds between
the current state and the recorded state. If the assertion
checker can prove the assert cannot fail, it has proved the
validity of the termination argument. We can use encoding tricks
to force the assertion checker to consider *all* possible
unrollings.

Figure 5 offers such an example, where we have
used the termination argument "`x`

goes down by at
least one and `x`

is greater than 0" using the
encoding given in Cook et al.^{14} The
new code (introduced as a part of the encoding) is given in red,
whereas the original program from Figure 1 is in
black. We make use of an extra call to `input()`

to
decide when the unrolling begins. The new variables oldx and oldy
are used for recording a state. Note that the assertion checker
must consider all values possibly returned by
`input()`

during its proof, thus the proof of
termination is valid for any starting position. This has the
effect of considering any possible unrolling of the loop. After
some state has been recorded, from this point out the termination
argument is checked using the recorded state and the current
state. In this case the assertion can fail, meaning that the
termination argument is not valid.

If we were to attempt to check this condition in a naïve way (for example, by simply executing the program) we would never find a proof for all but the most trivial of cases. Thus, assertion checkers must be cleverly designed to find proofs about all possible executions without actually executing all of the paths. A plethora of recently developed techniques now make this possible. Many recent assertion checkers are designed to produce a path to a bug in the case that the assertion statement cannot be proved. For example, a path leading to the assertion failure is 1 → 2 → 3 → 4 → 5 → 7 → 8 → 9 → 10 → 11 → 12 → 16 → 17 → 4 → 5 → 6. This path can be broken into parts, each representing different phases of the execution: the prefix-path 1 → 2 → 3 → 4 is the path from the program initial state to the recorded state in the failing pair of states. The second part of the path 4 → 5 →...5 → 6 represents how we reached the current state from the recorded one. That is: this is the unrolling found that demonstrates that the assertion statement can fail. What we know is that the termination argument does not currently cover the case where this path is repeated forever.

See Figure 6 for a version using the same encoding, but with the valid termination argument:

`x`

goes down by at least 1 and is larger than
0

*or*

`y`

goes down by at least 1 and is larger than
0.

This assertion cannot fail. The fact that it cannot fail can be proved by a number of assertion verification tools.

We have examined how we can check a termination argument's validity via a translation to a program with an assertion statement. We now discuss known methods for finding monolithic termination arguments.

*Rank function synthesis.* In some cases simple ranking
functions can be automatically found. We call a ranking function
simple if it can be defined by a linear arithmetic expression
(for example, `–3x = –2y + 100`

). The most
popular approach for finding this class of ranking function uses
a result from Farkas^{16} together with
tools for solving linear constraint systems. (See Colón
and Sipma^{11} or Polelski and
Rybalchecko^{30} for examples of tools
using Farkas' lemma.) Many other approaches for finding ranking
functions for different classes of programs have been proposed
(see
refs^{1,6–8,19,37}).
Tools for the synthesis of ranking functions are sometimes
applied directly to programs, but more frequently they are used
(on small and simplified program fragments) internally within
termination proving tools for suggesting the single ranking
functions that appear in a disjunctive termination argument.

*Termination analysis.* Numerous approaches have been
developed for finding disjunctive termination arguments in
which—in effect—the validity condition for
disjunctive termination arguments is almost guaranteed to hold by
construction. In some cases—for example, Berdine et
al.^{3}—to prove termination we
need only check that the argument indeed represents a set of
measures. In other cases, such as Lee et
al.^{24} or Manolios and
Vroon,^{26} the tool makes a one-time
guess as to the termination argument and then checks it using
techniques drawn from abstract interpretation.

Consider the modified program in Figure 7. The
termination strategy described in Berdine et
al.^{3} and Podelski and
Rybalchenko^{32} essentially builds a
program like this and then applies a custom program analysis to
find the following candidate termination argument:

for the program at line 4—meaning we could pass this
complex expression to the assertion at line 4 in Figure
7 and know that the assertion cannot fail. We know this
statement is true of any unrolling of the loop in the original
Figure 1. What remains is to prove that each
piece of the candidate argument represents a measure that
decreases—here we can use rank function synthesis tools to
prove that `oldx > x + 1`

and ```
oldx >
0
```

. . . represents the measure based on `x`

. If
each piece between the ors in fact represents a measure (with the
exception of copied `≠ 1`

which comes from the
encoding) then we have proved termination.

One difficulty with this style of termination proving is that, in the case that the program doesn't terminate, the tools can only report "unknown," as the techniques used inside the abstract interpretation tools have lost so much detail that it is impossible to find a non-terminating execution from the failed proof and then prove it non-terminating. The advantage when compared to other known techniques is it is much faster.

*Finding arguments by refinement.* Another method for
discovering a termination argument is to follow the approach of
Cook et al.^{14} or Chawdhary et
al.^{9} and search for counterexamples to
(possibly invalid) termination arguments and then refine them
based on new ranking functions found via the counterexamples.

In recent years, powerful new termination tools have emerged that return "unknown" infrequently enough that they are useful in practice.

Recall Figure 5, which encoded the invalid termination argument for the program in Figure 1, and the path leading to the failure of the assertion: is 1 → 2 → 3 → 4 → 5 → 7 → 8 → 9 → 10 → 11 → 12 → 16 → 17 → 4 → 5 → 6. Recall this path represents two phases of the program's execution: the path to the loop, and some unrolling of the loop such that the termination condition doesn't hold. In this case the path 4 → 5 → . . . 6 represents how we reached the second failing state from the first. This is a counterexample to the validity of the termination argument, meaning that the current termination argument does not take this path and others like it into account.

If the path can be repeated forever during the program's
execution then we have found a real counterexample. Known
approaches (for example, Gupta et
al.^{21}) can be used to try and prove
this path can be repeated forever. In this case, however, we know
that the path cannot be repeated forever, as `y`

is
decremented on each iteration through the path and also
constrained via a conditional statement to be positive. Thus this
path is a spurious counterexample to termination and can be ruled
out via a refinement to the termination argument. Again, using
rank function synthesis tools we can automatically find a ranking
function that demonstrates the spuriousness of this path. In this
case a rank function synthesis tool will find `y`

,
meaning that the reason this path cannot be repeated forever is
that "`y`

always goes down by at least one and is
larger than 0." We can then refine the current termination
argument used in Figure 5:

`x`

goes down by at least 1 and is larger than 0
with the larger termination argument:

`x`

goes down by at least 1 and is larger than
0

*or*

`y`

goes down by at least 1 and is larger than
0

We can then check the validity of this termination argument using a tool such as IMPACT on the program in Figure 6. IMPACT can prove this assertion never fails, thus proving the termination of the program in Figure 1.

With fresh advances in methods for proving the termination of sequential programs that operate over mathematical numbers, we are now in the position to begin proving termination of more complex programs, such as those with dynamically allocated data structures, or multithreading. Furthermore, these new advances open up new potential for proving properties beyond termination, and finding conditions that would guarantee termination. We now discuss these avenues of future research and development in some detail.

*Dynamically allocated heap.* Consider the C loop in
Figure 8, which walks down a list and removes
links with data elements equaling 5. Does this loop guarantee
termination? What termination argument should we use?

The problem here is that there are no arithmetic variables in
the program from which we can begin to construct an
argument—instead we would want to express the termination
argument over the lengths of paths to `NULL`

via the
next field. Furthermore, the programmer has obviously intended
for this loop to be used on acyclic singly linked lists, but how
do we know that the lists pointed to by head will always be
acyclic? The common solution to these problems is to use *shape
analysis* tools (which are designed to automatically discover
the shapes of data-structures) and then to create new auxiliary
variables in the program that track the sizes of those data
structures, thus allowing for arithmetic ranking functions to be
more easily expressed (examples include
refs^{4,5,25}).
The difficultly with this approach is that we are now dependent
on the accuracy and scalability of current shape analysis
tools—to date the best known shape analysis
tool^{40} supports only lists and trees
(cyclic and acyclic, singly and doubly linked) and scales only to
relatively simple programs of size less than 30,000 LOC.
Furthermore, the auxiliary variables introduced by methods such
as Magill et al.^{25} sometimes do not
track enough information in order to prove termination (for
example, imagine a case with lists of lists in which the sizes of
the nested lists are important). In order to improve the state of
the art for termination proving of programs using data
structures, we must develop better methods of finding arguments
over data structure shapes, and we must also improve the accuracy
and scalability of existing shape analysis tools.

*Bit vectors.* In the examples used until now we have
considered only variables that range over mathematical numbers.
The reality is that most programs use variables that range over
fixed-width numbers, such as 32-bit integers or 64-bit
floatingpoint numbers, with the possibility of overflow or
underflow. If a program uses only fixed-width numbers and does
not use dynamically allocated memory, then termination proving is
decidable (though still not easy). In this case we simply need to
look for a repeated state, as the program will diverge if and
only if there exists some state that is repeated during
execution. Furthermore, we cannot ignore the fixed-width
semantics, as overflow and underflow can cause non-termination in
programs that would otherwise terminate, an example is included
in Figure 9. Another complication when
considering this style of program is that of bit-level
operations, such as left- or right-shift.

*Binary executables.* Until now we have discussed proving
termination of programs at their source level, perhaps in C or
Java. The difficulty with this strategy is the compilers that
then take these source programs and convert them into executable
artifacts can introduce termination bugs that do not exist in the
original source program. Several potential strategies could help
mitigate this problem: We might try to prove termination of the
executable binaries instead of the source level programs, or we
might try to equip the compiler with the ability to prove that
the resulting binary program preserves termination, perhaps by
first proving the termination of the source program and then
finding a map from the binary to the source-level program and
proving that the composition with the source-level termination
argument forms a valid termination argument for the binary-level
program.

*Non-linear systems.* Current termination provers largely
ignore nonlinear arithmetic. When non-linear updates to variables
do occur (for example `x := y * z;`

), current
termination provers typically treat them as if they were the
instruction x := input();. This modification is
sound—meaning when the termination prover returns the
answer "terminating," we know the proof is valid. Unfortunately,
this method is not precise: the treatment of these commands can
lead to the result "unknown" for programs that actually
terminate. Termination provers are also typically unable to find
or check non-linear termination arguments
(`x`

, for example) when they are required.
Some preliminary efforts in this direction have been
made,^{2}^{1,6} but these
techniques are weak. To improve the current power of termination
provers, further developments in non-linear reasoning are
required.

*Concurrency.* Concurrency adds an extra layer of
difficulty when attempting to prove program termination. The
problem here is that we must consider all possible interactions
between concurrently executing threads. This is especially true
for modern fine-grained concurrent algorithms, in which threads
interact in subtle ways through dynamically allocated data
structures. Rather than attempting to explicitly consider all
possible interleavings of the threads (which does not scale to
large programs) the usual method for proving concurrent programs
correct is based on *rely-guarantee* or
*assume-guarantee* style of reasoning, which considers every
thread in isolation under assumptions on its environment and thus
avoids reasoning about thread interactions directly. Much of the
power of a rely-guarantee proof system (such as
Jones^{22} and Misra and
Chandy^{28}) comes from the cyclic proof
rules, where we can assume a property of the second thread while
proving property of the first thread, and then assume the
recently proved property of the first thread when proving the
assumed property of the second thread. This strategy can be
extended to liveness properties using induction over time, for
example, Gotsman et al.^{20} and
McMillan.^{27}

As an example, consider the two code fragments in
Figure 10. Imagine that we are executing these
two fragments concurrently. To prove the termination of the left
thread we must prove that it does not get stuck waiting for the
call to **lock**. To prove this we can assume the other thread
will always eventually release the lock—but to prove this
of the code on the right we must assume the analogous property of
the thread on the left, and so on. In this case we can certainly
just consider all possible interleavings of the threads, thus
turning the concurrent program into a sequential model
representing its executions, but this approach does not scale
well to larger programs. The challenge is to develop automatic
methods of finding non-circular rely-guarantee termination
arguments. Recent steps^{20} have
developed heuristics that work for non-blocking algorithms, but
more general techniques are still required.

With fresh advances in methods for proving the termination of sequential programs that operate over mathematical numbers, we are now in the position to begin proving termination of more complex programs.

*Advanced programming features.* The industrial adoption
of high-level programming features such as virtual functions,
inheritance, higher-order functions, or closures make the task of
proving industrial programs more of a challenge. With few
exceptions (such as Giesl et al.^{18}),
this area has not been well studied.

Untyped or dynamically typed programs also contribute difficulty when proving termination, as current approaches are based on statically discovering data-structure invariants and finding arithmetic measures in order to prove termination. Data in untyped programs is often encoded in strings, using pattern matching to marshal data in and out of strings. Termination proving tools for JavaScript would be especially welcome, given the havoc that nonterminating JavaScript causes daily for Web browsers.

*Finding preconditions that guarantee termination.* In
the case that a program does not guarantee termination from all
initial configurations, we may want to automatically discover the
conditions under which the program does guarantee termination.
That is, when calling some function provided by a library: what
are the conditions under which the code is guaranteed to return
with a result? The challenge in this area is to find the right
precondition: the empty precondition is correct but useless,
whereas the weakest precondition for even very simple programs
can often be expressed only in complex domains not supported by
today's tools. Furthermore, they should be computed quickly (the
weakest precondition expressible in the target logic may be too
expensive to compute). Recent work has shown some preliminary
progress in this
direction.^{13,33}

*Liveness.* We have alluded to the connection between
liveness properties and the program termination problem.
Formally, liveness properties expressed in temporal logics can be
converted into questions of *fair
termination*—termination proving were certain
non-terminating executions are deemed unfair via given fairness
constraints, and thus ignored. Current tools, in fact, either
perform this reduction, or simply require the user to express
liveness constraints directly as the set of fairness
constraints.^{12,29}
Neither approach is optimal: the reduction from liveness to
fairness is inefficient in the size of the conversion, and
fairness constraints are difficult for humans to understand when
used directly. An avenue for future work would be to directly
prove liveness properties, perhaps as an adaption of existing
termination proving techniques.

*Dynamic analysis and crash dumps for liveness bugs.* In
this article we have focused only on *static,* or
*compile-time,* proof techniques rather than techniques for
diagnosing divergence during execution. Some effort has been
placed into the area of automatically detecting deadlock during
execution time. With new developments in the area of program
termination proving we might find that automatic methods of
discovering *livelock* could also now be possible. Temporary
modifications to scheduling, or other techniques, might also be
employed to help programs not diverge even in cases where they do
not guarantee termination or other liveness properties. Some
preliminary work has begun to emerge in this area (see Jula et
al.^{23}) but more work is needed.

*Scalability, performance, and precision.* Scalability to
large and complex programs is currently a problem for modern
termination provers—current techniques are known, at best,
to scale to simple systems code of 30,000 lines of code. Another
problem we face is one of precision. Some small programs
currently cannot be proved terminating with existing tools.
Turing's undecidability result, of course, states that this will
always be true, but this does preclude us from improving
precision for various classes of programs and concrete examples.
The most famous example is that of the Collatz' problem, which
amounts to proving the termination or non-termination of the
program in Figure 11. Currently no proof of this
program's termination behavior is known.

This article has surveyed recent advances in program termination proving techniques for sequential programs, and pointed toward ongoing work and potential areas for future development. The hope of many tool builders in this area is that the current and future termination proving techniques will become generally available for developers wishing to directly prove termination or liveness. We also hope that termination-related applications—such as detecting livelock at runtime or Wang's tiling problem—will also benefit from these advances.

The authors would like to thank Lucas Bourdeaux, Abigail See, Tim Harris, Ralf Herbrich, Peter O'Hearn, and Hongseok Yang for their reading of early drafts of this article and suggestions for improvement.

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a. In English: "decision problem."

b. There is a minor controversy as to
whether or not Turing proved the undecidability
in^{38}. Technically he did not, but
termination's undecidability is an easy consequence of the result
that is proved. A simple proof can be found in
Strachey.^{36}

d. Checking validity of an assertion statement is an undecidable but co-recursively enumerable problem, whereas termination is neither r.e. nor co-r.e. problem.

Figure 3. Another example program.

Figure 4. Example program with an assertion statement in line 3.

Figure 5. Encoding of termination argument validity.

Figure 6. Encoding of termination argument validity using the program.

Figure 7. Program prepared for abstract interpretation.

Figure 8. Example C loop over a linked-list data-structure with fields next and data.

Figure 9. Example program illustrating nontermination.

Figure 10. Example of multi-threaded terminating producer/consumer program.

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