Complexity of Bradley-Manna-Sipma Lexicographic Ranking Functions

In this paper we turn the spotlight on a class of lexicographic ranking functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper, and establish for the first time the complexity of some problems involving the inference of such functions for linear-constraint loops (without precondition). We show that finding such a function, if one exists, can be done in polynomial time in a way which is sound and complete when the variables range over the rationals (or reals). We show that when variables range over the integers, the problem is harder -- deciding the existence of a ranking function is coNP-complete. Next, we study the problem of minimizing the number of components in the ranking function (a.k.a. the dimension). This number is interesting in contexts like computing iteration bounds and loop parallelization. Surprisingly, and unlike the situation for some other classes of lexicographic ranking functions, we find that even deciding whether a two-component ranking function exists is harder than the unrestricted problem: NP-complete over the rationals and $\Sigma^P_2$-complete over the integers.Comment: Technical report for a corresponding CAV'15 pape

Ranking Templates for Linear Loops

We present a new method for the constraint-based synthesis of termination arguments for linear loop programs based on linear ranking templates. Linear ranking templates are parametrized, well-founded relations such that an assignment to the parameters gives rise to a ranking function. This approach generalizes existing methods and enables us to use templates for many different ranking functions with affine-linear components. We discuss templates for multiphase, piecewise, and lexicographic ranking functions. Because these ranking templates require both strict and non-strict inequalities, we use Motzkin's Transposition Theorem instead of Farkas Lemma to transform the generated $\exists\forall$-constraint into an $\exists$-constraint.Comment: TACAS 201

The Hardness of Finding Linear Ranking Functions for Lasso Programs

Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem is known to be in coNP when variables range over the integers and in PTIME for the rational numbers, or real numbers. Here we show that deciding whether a linear-constraint loop with a precondition, specifically with partially-specified input, has a linear ranking function is EXPSPACE-hard over the integers, and PSPACE-hard over the rationals. The precise complexity of these decision problems is yet unknown. The EXPSPACE lower bound is derived from the reachability problem for Petri nets (equivalently, Vector Addition Systems), and possibly indicates an even stronger lower bound (subject to open problems in VAS theory). The lower bound for the rationals follows from a novel simulation of Boolean programs. Lower bounds are also given for the problem of deciding if a linear ranking-function supported by a particular form of inductive invariant exists. For loops over integers, the problem is PSPACE-hard for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of natural numbers as invariants.Comment: In Proceedings GandALF 2014, arXiv:1408.5560. I thank the organizers of the Dagstuhl Seminar 14141, "Reachability Problems for Infinite-State Systems", for the opportunity to present an early draft of this wor

Unrestricted Termination and Non-Termination Arguments for Bit-Vector Programs

Proving program termination is typically done by finding a well-founded ranking function for the program states. Existing termination provers typically find ranking functions using either linear algebra or templates. As such they are often restricted to finding linear ranking functions over mathematical integers. This class of functions is insufficient for proving termination of many terminating programs, and furthermore a termination argument for a program operating on mathematical integers does not always lead to a termination argument for the same program operating on fixed-width machine integers. We propose a termination analysis able to generate nonlinear, lexicographic ranking functions and nonlinear recurrence sets that are correct for fixed-width machine arithmetic and floating-point arithmetic Our technique is based on a reduction from program \emph{termination} to second-order \emph{satisfaction}. We provide formulations for termination and non-termination in a fragment of second-order logic with restricted quantification which is decidable over finite domains. The resulted technique is a sound and complete analysis for the termination of finite-state programs with fixed-width integers and IEEE floating-point arithmetic

Bounding the Computational Complexity of Flowchart Programs with Multi-dimensional Rankings

Proving the termination of a flowchart program can be done by exhibiting a ranking function, i.e., a function from the program states to a well-founded set, which strictly decreases at each program step. A standard method to automatically generate such a function is to compute invariants for each program point and to search for a ranking in a restricted class of functions that can be handled with linear programming techniques. Our first contribution is to propose an efficient algorithm to compute ranking functions: It can handle flowcharts of arbitrary structure, the class of candidate rankings it explores is larger, and our method, although greedy, is provably complete. Our second contribution is to show how to use the ranking functions we generate to get upper bounds for the computational complexity (number of transitions) of the source program, again for flowcharts of arbitrary structure. This estimate is a polynomial, which means that we can handle programs with more than linear complexity. We applied the method on a collection of test cases from the literature. We also point out important extensions, mainly to do with the scalability of the algorithm and, in particular, the integration of techniques based on cutpoints

Synthesis for Polynomial Lasso Programs

We present a method for the synthesis of polynomial lasso programs. These programs consist of a program stem, a set of transitions, and an exit condition, all in the form of algebraic assertions (conjunctions of polynomial equalities). Central to this approach is the discovery of non-linear (algebraic) loop invariants. We extend Sankaranarayanan, Sipma, and Manna's template-based approach and prove a completeness criterion. We perform program synthesis by generating a constraint whose solution is a synthesized program together with a loop invariant that proves the program's correctness. This constraint is non-linear and is passed to an SMT solver. Moreover, we can enforce the termination of the synthesized program with the support of test cases.Comment: Paper at VMCAI'14, including appendi