Xuran Cai

Xuran Cai, Liqian Chen, Hongfei Fu

In numeric-intensive computations, it is well known that the execution of floating-point programs is imprecise as floating-point arithmetic incurs round-off errors. Although round-off errors are small for a single floating-point operation, the aggregation of such errors may be dramatic and cause catastrophic program failures. Therefore, to ensure the correctness of floating-point programs, round-off error needs to be carefully taken into account. In this work, we consider polynomial invariant generation for floating-point programs, aiming at generating tight invariants under the perturbation of round-off errors. Our contribution is a novel framework for applying polynomial constraint solving to address the invariant generation problem, which is also the first polynomial constraint solving based approach that handles floating-point errors to our best knowledge. In our framework, we propose a novel combination of round-off error analysis and polynomial constraint solving, aiming to circumvent the cost of handling a large number of error variables in the floating-point model. Experimental results over a variety of challenging benchmarks show that our framework outperforms SOTA approaches in both time efficiency and the precision of generated invariants.

Xuran Cai, Amir Goharshady

In this work, we focus on the Partial Constraint Satisfaction Problem (PCSP) over control-flow graphs (CFGs) of programs. PCSP serves as a generalization of the well-known Constraint Satisfaction Problem (CSP). In the CSP framework, we define a set of variables, a set of constraints, and a finite domain $D$ that encompasses all possible values for each variable. The objective is to assign a value to each variable in such a way that all constraints are satisfied. In the graph variant of CSP, an underlying graph is considered and we have one variable corresponding to each vertex of the graph and one or several constraints corresponding to each edge. In PCSPs, we allow for certain constraints to be violated at a specified cost, aiming to find a solution that minimizes the total cost. Numerous classical compiler optimization tasks can be framed as PCSPs over control-flow graphs. Examples include Register Allocation, Lifetime-optimal Speculative Partial Redundancy Elimination (LOSPRE), and Optimal Placement of Bank Selection Instructions. On the other hand, it is well-known that control-flow graphs of structured programs are sparse and decomposable in a variety of ways. In this work, we rely on the Series-Parallel-Loop (SPL) decompositions as introduced by~\cite{RegisterAllocation}. Our main contribution is a general algorithm for PCSPs over SPL graphs with a time complexity of \(O(|G| \cdot |D|^6)\), where \(|G|\) represents the size of the control-flow graph. Note that for any fixed domain $D,$ this yields a linear-time solution. Our algorithm can be seen as a generalization and unification of previous SPL-based approaches for register allocation and LOSPRE. In addition, we provide experimental results over another classical PCSP task, i.e. Optimal Bank Selection, achieving runtimes four times better than the previous state of the art.