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 propertiescalled liveness propertiessuch 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 programin 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 valuethus 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 "or2w 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^{1,68,19,37}). 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 whichin effectthe validity condition for disjunctive termination arguments is almost guaranteed to hold by construction. In some casesfor 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 4meaning 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 decreaseshere 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 argumentinstead 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 toolsto 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 soundmeaning 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^{2}, for example) when they are required. Some preliminary efforts in this direction have been made,^{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 lockbut 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 terminationtermination 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 proverscurrent 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 applicationssuch as detecting livelock at runtime or Wang's tiling problemwill 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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Byron Cook is a Principal Researcher at Microsoft's research laboratory at Cambridge University, and a professor of computer science at Queen Mary, University of London, England.

Andreas Podelski is a professor of computer science at the University of Freiburg, Germany.

Andrey Rybalchenko is a professor of computer science at the Technische Universität München, Germany.

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}

c. The program does terminate

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.

Sidebar: Turing's Classic Method and Disjunctive Well-Foundness

Formally proving program termination amounts to proving the program's transition relation R to be well-founded. If (S, ) is a well-order then > is a well-founded relation. Furthermore, any map f into S defines a well-founded relation, by lifting > via f, that is, {(s, t) | f(s) > f(t)}. Turing's method^{39} of proving a program's transition relation R well-founded amounts to finding a map f into a well-order, which defines a termination argument T = {(s, t) | f(s) > f(t)}. To prove the validity of T we must show RT. From the well-foundedness of T and the fact that every sub-relation of a well-founded relation is well-founded follows that R is well-founded.

In this article we are using the phrase disjunctive termination argument to refer to a disjunctively well-founded transition invariant.^{31} This is a finite union T_{1} ...T_{n} of well-founded relations that contains R+, which is the transitive closure of the transition relation of the program, as a superset, such as, R+ T_{1} ... T_{n}.

Usually, each T_{1},...,T_{n} will be constructed as above via some map into a well-order. Note that the non-reflexive transitive closure (the + in R+) is crucial. It is not sufficient to show that RT_{1} ... T_{n}, as the union of well-founded relations is not guaranteed to be well-founded. It is the transitive closure that makes checking the subset inclusion more difficult in practice.

The recent approaches for proving termination for general programs^{3,4,9,12,14,32} are based on the proof rule of disjunctively well-founded transition invariants. 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 expressed by the proof rule appears implicitly already in previously developed termination algorithms for rewrite systems and logic and functional programs, see refs^{10,15,17,24}

Here, we give a brief summary of implementation strategies based on disjunctive termination arguments deployed by the recent termination checkers:

Refinement:^{9,14} In Cook et al.,^{14} the termination argument begins with ø. We first attempt to prove that R+ ø. When this proof fails, rank function synthesis is applied to the witness, thus giving a refinement T_{1} to the argument, which is then rechecked R+ ø T_{1}. This process is repeated until a valid argument is found or a real counterexample is found.

In Chawdhary et al.,^{9} the termination argument T is constructed following the structure of the transition relation R = R_{1} ... R_{m} by using a ranking function synthesis procedure, which is used to compute a well-founded overapproximation WF(X) of a binary relation X. The initial candidate T = WF(R_{1}) ... WF (R_{m}) is extended with WF (WF (R_{i}) ° R_{j}) and so on until the fixpoint is reached.

Variance analysis:^{3,32} As described in some detail in this article, the approach from Berdine et al.^{3} and Podelski et al.^{32} uses program transformations and abstract interpretation for invariants to compute an overapproximation T_{1}; T_{2},...,T_{n} such that R+ T_{1}T_{2}... T_{n}. It then uses rank function synthesis to check that each T_{i} is well-founded.

In contrast to the refinement-based methods, variance analysis always terminates, but may return "don't know" in cases when a refinement-based method succeeds.

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