# What is partial order reduction?

by Stephen M. Walker II, Co-Founder / CEO

## What is partial order reduction?

Partial Order Reduction (POR) is a technique used in computer science to reduce the size of the state-space that needs to be searched by a model checking or automated planning and scheduling algorithm. This reduction is achieved by exploiting the commutativity of concurrently executed transitions that result in the same state.

The concept of partial order reduction is based on the observation that many systems, particularly concurrent systems, have behaviors that can be executed in different orders but still lead to the same state. By identifying these commutative operations, the algorithm can avoid redundant exploration of the state space, thereby reducing the computational resources required for model checking.

The method involves constructing a reduced state graph, which is a subset of the behaviors of the full state graph. The full state graph, which may be too large to fit in memory, is never constructed. The behaviors that are not present in the reduced graph do not add any new information, hence their exclusion does not affect the outcome of the model checking.

There are various versions of the method, including the stubborn set method and the ample set method. These methods have different conditions for preserving all deadlocks and may lead to smaller sets.

Partial order reduction has been applied in various logical formalisms such as Linear Temporal Logic (LTL), Computation Tree Logic (CTL), and process algebras. It can also be combined with other efficient model checking techniques.

In the context of software model checking, a dynamic approach to partial order reduction has been proposed. This approach involves initially exploring an arbitrary interleaving of the various concurrent operations and dynamically tracking interactions between these to identify backtracking points.

## How does partial order reduction work in practice?

Partial Order Reduction (POR) streamlines model checking by identifying and exploiting the commutativity of transitions in concurrent systems. It focuses on asynchronous systems where transitions occur in an interleaved fashion, rather than simultaneously.

The core of POR is the construction of a reduced state graph, a smaller representation of the system's behavior that omits redundant states without losing essential information. This graph is derived from the full state graph, which is impractical to construct in its entirety due to memory constraints.

The technique hinges on a dependency relation among system transitions to pinpoint commutative operations—those that can occur in varying orders yet arrive at the same state. By recognizing these operations, POR avoids unnecessary exploration of equivalent states.

Dynamic partial order reduction, a variant used in software model checking, begins with an exploration of an arbitrary sequence of concurrent operations. It then monitors the interactions among these operations to identify points where backtracking can optimize the search process.

Different implementations of POR, such as the stubborn set and ample set methods, offer various strategies for deadlock preservation and potentially smaller reduced state graphs.

POR is versatile and can be integrated with other model checking approaches, like symbolic model checking with binary decision diagrams (BDDs) for synchronous systems, enhancing their efficiency.

In hardware model checking, POR can be applied at individual time steps, which is essential for its practical use in verifying synchronous hardware designs.

POR improves the efficiency of verifying system properties by focusing on commutative operations, constructing reduced state graphs, and dynamically managing interactions between concurrent operations to minimize the state space for exploration.

## What are some examples of state-of-the-art partial order reduction techniques?

State-of-the-art partial order reduction (POR) techniques are advanced methods used to optimize the process of model checking by reducing the number of states that need to be explored. Here are some examples of such techniques:

### Dynamic Partial Order Reduction (DPOR)

DPOR is a technique that targets efficient state space exploration by dynamically determining which transitions can be executed in a different order without affecting the outcome of the model checking. It mitigates the combinatorial explosion of stateless exploration by tracking dependencies between transitions during runtime rather than relying on static analysis.

### Scalable Dynamic Partial Order Reduction

This is an extension of DPOR that aims to improve its scalability for systematic testing of large-scale concurrent systems. It involves techniques that can handle the vast state spaces encountered in practical applications, making it suitable for deployment in real-world scenarios.

### Stateful Dynamic Partial Order Reduction

This technique extends DPOR by maintaining some state information to further optimize the reduction process. It is particularly useful for complex systems where maintaining state information can lead to more significant reductions in the state space.

### Behavioral Analysis for POR

An improvement in POR can be achieved by using behavioral analysis to extract the independence relation among possible behaviors. This approach goes beyond static analysis of system model structures and can reveal more opportunities for reduction by understanding the behavior of the system.

### Cartesian Partial-Order Reduction

This method focuses on reducing the state space by considering the Cartesian product of independent state spaces. It is particularly useful when dealing with systems that can be decomposed into smaller, independent components.

### Probabilistic Dynamic Partial Order Reduction

This technique applies POR to probabilistic systems, where the goal is to reduce the state space while taking into account the probabilistic nature of the system's transitions. It allows for more efficient verification of systems where randomness plays a role.

These techniques represent the forefront of research and development in the field of model checking and state space reduction. They are designed to handle the complexity of modern software and hardware systems, ensuring that model checking remains a viable and efficient verification method as systems continue to grow in size and complexity.

## More terms

### What is an algorithm?

Algorithms are well-defined instructions that machines follow to perform tasks. They can solve problems, manipulate data, and achieve desired outcomes in various computing and AI domains.

### What is separation logic?

Separation logic is a formal method used in the field of computer science to reason about the ownership and sharing of memory resources within programs. It was developed by John C. Mitchell and others at Stanford University in the late 1990s as an extension to classical Hoare logic, with the goal of improving its ability to handle complex data structures, especially those that involve sharing and concurrency.