Automating SMT Strategy Creation with MCTS

Satisfiability Modulo Theories (SMT) solvers are important tools used in many areas, including software engineering, verification, security, and artificial intelligence. These solvers help determine if certain logical formulas can be satisfied. However, there is a challenge in finding one solver that works well for every problem type. Because of this, modern SMT solvers, like Z3, allow users to create their own Strategies for specific problems, making the solvers more effective for those particular cases.

Creating these custom strategies can be difficult and time-consuming. This paper discusses a new approach to automatically creating SMT strategies using a method called Monte Carlo Tree Search (MCTS). This approach treats strategy creation as a series of decisions. The search space for strategies is vast, and our method breaks it down into smaller parts, enabling more efficient exploration.

#Background

SMT solvers determine whether certain logical formulas are satisfiable based on specified theories. A strategy in this context is a way to select and use various Tactics for problem-solving. Tactics are specific steps or methods provided by the solver to break down and solve the logical problem.

For example, a tactic called "propagate-values" helps manage equalities in the logical formula. In contrast, other tactics handle different types of problems. Many default strategies used by solvers are optimized for traditional benchmark problems, but as the use of SMT solvers grows, users often face new and unique challenges. Default or custom strategies may not always be effective for these new problems. Historically, creating tailored strategies has required significant effort from human experts.

Several attempts have been made to automate strategy creation. Tools like StratEVO and FastSMT use different methods, such as evolutionary algorithms and imitation learning, to generate strategies. While these methods have shown promise, challenges still exist, including robustness and long training times.

#Monte Carlo Tree Search (MCTS)

MCTS is a search technique widely used in computer games and planning algorithms. It became well-known for its success in deep reinforcement learning systems like AlphaGo. MCTS is effective at solving complex symbolic and combinatorial problems by balancing exploration (trying new possibilities) and exploitation (using known successful paths).

MCTS operates through four main steps:

1. Selection: Starting from the root, it navigates through the search tree until it finds a leaf node.
2. Expansion: New nodes are added to the tree based on unexplored actions.
3. Rollout: The algorithm simulates the outcome using a random policy until a result is achieved.
4. Backup: The results are used to update the action values in the tree.

These steps allow MCTS to gather information about the value of different actions, guiding future decision-making.

#Strategy Synthesis Problem

The SMT strategy synthesis problem is focused on automatically finding the best strategy for a given set of problems. This is determined by the number of instances that can be solved within a certain timeout. Each strategy's Performance has to be evaluated based on a subset of instances, which is representative of a broader set.

Finding the best solution is impractical due to the vast number of strategies available. Instead, the goal is to discover a near-optimal solution within a reasonable time. Our work introduces a new method called Z3alpha, which applies MCTS to SMT strategy synthesis.

#Contributions

#Z3alpha: An MCTS-based Solver

Z3alpha is a novel framework for creating SMT strategies automatically. It uses MCTS to find effective strategies tailored to specific problem types. This marks the first application of MCTS in the context of SMT strategy synthesis.

#Layered and Staged MCTS

To improve efficiency and effectiveness, we developed two new techniques, layered search and staged search. The layered search breaks down the parameter tuning process into smaller problems. Each parameter is treated separately, simplifying the search process.

The staged search divides the search into two stages:

1. In the first stage, the focus is on finding simple strategies that do not involve branching.
2. The second stage combines these simple strategies into more complex ones.

This method allows the exploration of a greater strategy space and speeds up the evaluation process.

#Experimental Evaluation

We implemented Z3alpha with the leading SMT solver Z3 and conducted a variety of experiments across six important SMT logics. These experiments compared Z3alpha’s performance against state-of-the-art tools like FastSMT and the default strategies in Z3.

#Related Work

Various methods have been employed in the past to create SMT strategies. StratEVO is an example of a tool using genetic programming. FastSMT represents current leading methods in automated strategy generation but often struggles with robustness. MCTS has been applied to different problems in recent years, such as identifying optimal strategies in board games and addressing combinatorial problems.

Our approach differs from others by focusing specifically on SMT strategy synthesis and leveraging layered and staged methods to enhance the search efficiency.

#MCTS for SMT Strategy Synthesis

#Strategy Model

The process of creating a strategy for SMT solvers can be modeled as a decision-making process that transitions through various states. Each state represents a strategy string derived from a set of production rules. The goal is to maximize the reward over time through careful action selection.

#MCTS Framework

In Z3alpha, we apply MCTS to search for optimal SMT strategies. The selection phase uses the Upper Confidence Bounds applied for Trees (UCT) algorithm, while the rollout phase utilizes a random policy. During the backup phase, we focus on maximizing observed returns, steering the search toward the best-performing strategies.

#Layered Search Method

Layered search optimizes how tactic parameters are adjusted. Instead of treating each parameter adjustment as part of the main search tree, we treat each parameter as a separate multi-armed bandit problem. This isolation allows us to optimize parameters without adding complexity to the main search.

#Staged Search Method

Staged search splits the overall strategy search process into two parts. The first focuses on finding the best simple strategies, while the second combines them into a more intricate solution. This approach saves time because it enables the re-use of results from the first stage.

#Experimental Analysis

#Setup

We evaluated Z3alpha across six SMT logics, comparing its performance against FastSMT and the default Z3 strategy. Our experiments were designed to assess the effectiveness and robustness of our method.

#Benchmarks

We conducted experiments using multiple benchmark sets, including Sage2, AProVE, and others. Each benchmark set was used for both training and testing, allowing us to measure Z3alpha’s performance accurately.

#Performance Results

Across various experiments, Z3alpha consistently outperformed fast SMT and the default strategy in Z3. Notably, Z3alpha was able to solve a significantly greater percentage of instances compared to competing solvers, proving its effectiveness in strategy synthesis.

#Analysis of Results

The results showed that Z3alpha not only solves more instances but also does so with fewer branches than strategies generated by FastSMT, suggesting that our automated strategy is more interpretable and easier to understand.

#User-defined Tactics

In addition to built-in tactics, Z3 also allows users to define their own strategies. We tested Z3alpha's ability to create strategies that effectively incorporate these user-defined tactics. The results demonstrated that Z3alpha significantly outperformed traditional handcrafted strategies.

#Discussion

The introduction of Z3alpha marks a significant advancement in the automation of SMT strategy synthesis. The use of layered and staged search techniques allows for a more efficient search process, improving both the speed and effectiveness of solver strategies.

#Conclusion

In summary, Z3alpha offers a new method for automatically generating strategies in SMT solvers using MCTS. Through layered and staged search techniques, Z3alpha surpasses previous tools in performance, showcasing the potential benefits of automated customization in SMT strategy synthesis. The findings emphasize the need for continued research and development in this area to address the growing challenges within SMT applications.

Z3alpha represents a promising step forward in making SMT solvers more robust and capable of handling a wider variety of problems, ultimately supporting the broader adoption of user-controllable strategies in the SMT community.