1 Introduction
Model checking [2] is a “pushbutton” technology: it verifies a given program completely automatically, without any help from the human user. A fundamental challenge, however, is the infamous state space explosion problem: in a concurrent program, the number of control states grows exponentially with the number of parallel threads. This creates a barrier for the standard approach to model checking, which is based on a comprehensive traversal of the state space.
Causalitybased model checking [5, 6, 4] is inspired by the observation that it is not unusual for programs that are difficult to verify with a conventional model checker to have surprisingly short “paperandpencil” proofs. A likely explanation is that humans reason in terms of causal relationships. In most general terms, causality can be defined as the relation between two events, where the first event (the cause) is understood to be partly responsible for the second (the effect). Reasoning by causality is the usual style of constructing proofs: assume some situation (the effect) to be present, and derive all possible explanations (the causes). Consider the following assertion and its proof from Leslie Lamport’s paper [7] introducing the Bakery algorithm for mutual exclusion:
5mm5mm
Assertion 1.
If processors and are in the bakery and entered the bakery before entered the doorway, then .
Proof.
By hypothesis, had its current value while was choosing the current value of . Hence, must have chosen . ∎
The proof starts by assuming the situation where the event “ entered the bakery” precedes the event “ entered the doorway”, and, moreover, preserves its value between two events. The proof proceeds by deriving from this situation another necessary fact (notice the words “must have chosen”): .
Unlike standard model checking, causalitybased model checking is not based on a traversal of the state space but instead tracks the causal dependencies in the system. In concurrent programs, it is often the case that not many concurrent events depend on each other – most events are, in fact, independent, and precisely this allows concurrent programs to achieve better performance than sequential programs. Causalitybased model checking provides a formal proof system as well as an automatic method for constructing proofs or finding counterexamples employing the principles of causal reasoning.
2 The Causalitybased Model Checking Framework
We now introduce the main components of the causalitybased model checking framework.
Concurrent traces.
The basic building block, which we intend as a replacement of state in the standard model checking, is called a concurrent trace. Instead of a single momentary snapshot of the program computation, it represents a set of related computation events. Each event is labeled by a transition predicate, and describes a set of program transitions satisfying the predicate. Events are related by causal links, which are simply the ordering constraints between events. Causal links are also labeled by transition predicates; these constraints represent conditions which all events in scope of the causal link should satisfy. Finally, there is a conflict relation, which prohibits certain events to coincide in time.
A concurrent trace as a whole can be understood as a combination of existential and universal constraints, which collectively describe a set of program computations, such that the concurrent trace can be mapped to each of those computations respecting the constraints. E.g., a concurrent trace on the right represents the set of computations which satisfy the following formula:
In [6] we have extended the basic model of finite concurrent traces from [5] to infinite concurrent traces. An infinite concurrent trace consists of a stem and a cycle, each being a concurrent trace, with the semantics that the cycle occurs infinitely often after the stem occurs once.
Graphical Notation. We show event identities in circles, and labeling formulas in squares. Causal links are shown as solid lines with arrows, and conflicts as crossed zigzag lines. The cycle part of the trace is depicted in round brackets, superscripted with . We omit any of those when they are not important or would create clutter in the current context.
Representation of program properties.
Given a program property of interest, we start our analysis by representing the property violation as a concurrent trace. From our experience, we find out that violations of many useful properties can be quite naturally expressed as concurrent traces.
Suppose we are given a program , consisting of two processes and that require mutually exclusive access to their critical sections. Each process , , contains an endless loop with three sections: (“noncritical”), (“trying”), and (“critical”). We can easily represent violations of the desired properties of as concurrent traces (we denote the initial state of as ).
Mutual exclusion. Processes should not be in their critical sections simultaneously, which is described by the formula .
Strict precedence. When is trying to get access and is not in the critical section, then will be admitted to the critical section first: .
Bounded overtaking. We may want to give an upper bound on the amount of overtaking, where overtaking means that one process enters the critical section ahead of its rival. The property of 1bounded overtaking can be expressed as
.
Trace transformers.
Our analysis proceeds in steps, where each step takes some concurrent trace, and applies to it a so called trace transformer. Trace transformers are the proof rules describing necessary consequences from the current analysis situation represented as a concurrent trace. There is a number of general proof rules for safety and liveness analysis that were presented in our previous work [4, 5, 6]; we also envision that applicationspecific proof rules can be easily created for particular domains.
A trace transformer is an ordered set of trace productions of the form , where all productions share the same lefthand side . A trace production is a generalization of a graph production [8] and describes a formal rule to transform one concurrent trace into another. Collectively, all productions of a trace transformer encode some case distinction, where the union of trace languages of individual cases is contained in the language of the original trace. Below we show several examples of trace transformers.
trace transformer considers alternative orderings of two concurrent events and .
, given two causally related and conflicting events and in a concurrent trace, and a state predicate , such that the label of implies , and the label of implies , introduces a new “bridging” event in between. This condition can be interpreted as a contradiction between events and ( “ends” in the region , while “starts” in the region ).
makes a case distinction about the program behavior at infinity: for a given predicate either all events in the cycle part satisfy it, or a violating event should happen infinitely often.
Tableau construction.
The final component of our framework is a collection of datastructures for the organization of proof or counterexample search. Each datastructure comes with the corresponding algorithm for its automatic construction; the datastructures and algortihms have different properties which are detailed in Figure 1.
The conceptually simplest datastructure, called trace unwinding, is a forest of nodes each labeled with a concurrent trace. The forest roots represent concurrent traces that encode possible violations of the property of interest. The exploration algorithm proceeds by picking some forest leaf, and employing an applicable trace transformer, producing a number of further nodes. The exploration stops when all forest leaves are found to be contradictory.
The most involved datastructure, abstract trace tableau, contains, besides concrete trace unwinding, an abstract looping trace tableau, which may have covering edges between tableau nodes. A covering condition is an extension of subgraph isomorphism to concurrent traces: a concurrent trace, which is a subgraph of another trace, represents a situation which was encountered in the analysis before. This tableau is also allowed to contain causal loops, i.e. infinite repetitions of a sequence of trace productions, which together imply the impossibility of a computation satisfying them. The abstract trace tableau is used to track premises of already applied proof rules, and, thus, simplifies coverings.
3 Two Examples
We illustrate the framework with two examples taken from [5] and [6]: one for the analysis of a safety property (reachability), and another for the analysis of a liveness property (termination).
Safety.
Consider the synchronized system shown in the top part of Figure 2: the example was introduced by Esparza and Heljanko in [3] to illustrate the exponential succinctness of Petri net unfoldings. There are processes, and we want to check whether the global transition is executable. Note that the state space of this system is exponential with respect to : the system contains reachable states. Thus, approaches based on state space exploration will suffer from the state space explosion problem. The authors of [3] show that the Petri net unfolding of the example system contains places, i.e., a linear size unfolding can represent succinctly the exponential state space.
We use the same example to demonstrate that the trace unwinding of the example system never exceeds nodes, but a constant size unwinding of just nodes also suffice, which we show in the bottom part of Figure 2. Node 1, the root of the unwinding, captures all system traces where is executed. One of it’s preconditions is that the first process should be at location ; but the initial condition says that the system is at location : a contradiction. Thus, a transition that goes from to is necessary, and we insert the only such transition, , into the trace of node 2. Formally, this is done by applying the trace transformer , this is where the link label comes from: it enforces to select the last transition that goes into location . By a similar reasoning we conclude that transition is also necessary, and include it into the trace of node 3. Notice that these events occur concurrently, i.e. no specific order between them is specified till now.
In the next iteration, when we try to put them in some linear order, we find out that these transitions contradict each other: goes to location , goes to location , but both can start only at location . Therefore, we perform a case split between two possible linearizations using the transformer , and obtain the traces of nodes 4 and 5. In node 4 the contradiction between now linearly ordered transitions and is enforced by the trace, and we again apply the transformer . It inserts an event between transitions and which needs to change the location from (the postcondition of ) to (the precondition of
). Formally, this requirement is captured by the Craig interpolant between between the post and preconditions of
and , respectively, which happens to be . As there is no transition that goes from a location different from to , this trace is declared as contradictory, and the left branch of the unwinding is closed. For the right branch of the unwinding, consisting of nodes 5 and 7, we proceed in the same way, and also close it as contradictory. Thus, the unwinding is complete, and we conclude that transition is not executable.The unwinding of Figure 2 has constant size, independent on the number of processes in the example system. In the worst case it could have linear size: for that to happen, the other necessary transitions through would have to be introduced into concurrent before transitions and
are introduced. In fact, a simple heuristic is able to select transitions
and first, and it will always produce the trace unwinding of constant size.Liveness.
Producer 1  Producer 2  Consumer 1  Consumer 2 
while (p1>0) if(*) q1++; else q2++; p1;

while (p2>0) if(*) q1++; else q2++; p2;

while (true) await(q1>0); skip; //step 1 skip; //step 2 q1;

while (true) await(q2>0); skip; //step 1 skip; //step 2 q2;

Consider the ProducerConsumer example presented in Figure 3, which is a simplified model of the MapReduce architecture from distributed processing: producers model the mapping step for separate data sources, consumers model the reducing step for different types of input data. The natural requirement for this architecture is that the processing terminates for any finite amount of input data.
Our analysis starts with the assumption (by way of contradiction) that there exists some infinite computation. The assumption is expressed as the concurrent trace of node 1 in Figure 4: infinitely often some transition should occur. The transition is so far unknown, and therefore characterized by the predicate . Our argument proceeds by instantiating this unknown transition with the transitions of the program, resulting in one new trace per transition.
For example, transition of Producer 1, gives us the trace of node 2. The consequence of the decision that occurs infinitely often is that must also occur infinitely often: starts at location 1, and only can bring Producer 1 to this location. The requirement that both and occur infinitely often is expressed as the trace of node 3, obtained from the trace of node 1 by the trace transformer. The trace of node 3 is terminating: is decreased infinitely often and is bounded from below; it is therefore a ranking function. An infinite computation might exist only if some transition increases (satisfies the predicate ) is executed infinitely often. This situation is expressed by the trace of node 4, obtained by the application of the trace transformer. Since there is no transition in the program transition relation that satisfies , we arrive at a contradiction.
Let us explore another instantiation of the unknown event in the trace of node 1, this time with transition of Consumer 1: we obtain the trace of node 5. Again, exploring causal consequences, local safety analysis gives us that events , , and should also occur infinitely often in the trace: we insert them, and get the trace of node 6. Termination analysis for that trace gives us the ranking function : it is bounded from below by event and decreased by event . Again, we conclude that the event increasing should occur infinitely often, and introduce it in the trace of node 7.
Next, we try all possible instantiations of the event characterized by the predicate : there are two transitions that satisfy the predicate, namely and . We explore the instantiation with in the trace of node 8, and see that transitions and should occur infinitely often (node 9). At this point, we realize that the trace of node 9 contains as a subgraph the trace of node 2, namely the transition . We cover node 9 with node 2, and avoid repeating the analysis done for nodes 2–4. The remaining tableau branches are analyzed similarly. The resulting tableau for the case of two producers and two consumers will have the shape shown in the bottom left part of Figure 4. It can be interpreted as a partially ordered ranking function, which shows that all threads satisfying the function components and terminate unconditionally, while the threads that satisfy the function components and terminate under the condition that both of the previous components terminate. Notice also that the tableau is of quadratic size with respect to the number of threads.
4 Conclusions
Causalitybased model checking has significant advantages over standard statebased model checking. As illustrated by the two examples, the complexity of the verification problem can be substantially lower; in particular, the complexity of verifying multithreaded programs with locks reduces from exponential to polynomial (see [5], Theorem 4). The efficiency of causalitybased model checking is also reflected in the experimental results obtained with our tool implementation Arctor (cf. [6]). In our experience, Arctor scales to many, even hundreds, of parallel threads in benchmarks where other tools can only handle a small number of parallel threads or no parallelism at all: see Table 1 for experimental results obtained with Arctor on the ProducerConsumer example.
Terminator  T2  AProVE  Arctor  

Threads  Time(s)  Mem.(MB)  Time(s)  Mem.(MB)  Time(s)  Mem.(MB)  Time(s)  Mem.(MB)  Vertices 
1  3.37  26  2.42  38  3.17  237  0.002  2.3  6 
2  1397  1394  3.25  44  6.79  523  0.002  2.6  11 
3  MO  U(29.2)  253  U(26.6)  1439  0.002  2.6  21  
4  MO  U(36.6)  316  U(71.2)  1455  0.003  2.7  30  
5  MO  U(30.7)  400  U(312)  1536  0.007  2.7  44  
20  MO  Z3TO  MO  0.30  4.2  470  
40  MO  Z3TO  MO  4.30  12.7  1740  
60  MO  Z3TO  MO  20.8  35  3810  
80  MO  Z3TO  MO  67.7  145  6680  
100  MO  Z3TO  MO  172  231  10350 
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