Termination of Linear Programs with Nonlinear Constraints

Tiwari proved that termination of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvectors computation. Braverman proved that it is also decidable over the integers. In this paper, we consider the termination of loops with polynomial loop conditions and linear updates over the reals and integers. First, we prove that the termination of such loops over the integers is undecidable. Second, with an assumption, we provide an complete algorithm to decide the termination of a class of such programs over the reals. Our method is similar to that of Tiwari in spirit but uses different techniques. Finally, we conjecture that the termination of linear programs with polynomial loop conditions over the reals is undecidable in general by %constructing a loop and reducing the problem to another decision problem related to number theory and ergodic theory, which we guess undecidable.Comment: 17pages, 0 figure

Discovering non-linear ranking functions by solving semi-algebraic systems

Abstract. Differing from [6] this paper reduces non-linear ranking function discovering for polynomial programs to semi-algebraic system solving, and demonstrates how to apply the symbolic computation tools, DISCOVERER and QEPCAD, to some interesting examples

discovering non-linear ranking functions by solving semi-algebraic systems

Formal Methods Europe, Macao Polytech Inst, United Natl UnivDiffering from 6 this paper reduces non-linear ranking function discovering for polynomial programs to semi-algebraic system solving, and demonstrates how to apply the symbolic computation tools, DISCOVERER and QEPCAD, to some interesting exa

Machine learning for function synthesis

Function synthesis is the process of automatically constructing functions that satisfy a given specification. The space of functions as well as the format of the specifications vary greatly with each area of application. In this thesis, we consider synthesis in the context of satisfiability modulo theories. Within this domain, the goal is to synthesise mathematical expressions that adhere to abstract logical formulas. These types of synthesis problems find many applications in the field of computer-aided verification. One of the main challenges of function synthesis arises from the combinatorial explosion in the number of potential candidates within a certain size. The hypothesis of this thesis is that machine learning methods can be applied to make function synthesis more tractable. The first contribution of this thesis is a Monte-Carlo based search method for function synthesis. The search algorithm uses machine learned heuristics to guide the search. This is part of a reinforcement learning loop that trains the machine learning models with data generated from previous search attempts. To increase the set of benchmark problems to train and test synthesis methods, we also present a technique for generating synthesis problems from pre-existing satisfiability modulo theories problems. We implement the Monte-Carlo based synthesis algorithm and evaluate it on standard synthesis benchmarks as well as our newly generated benchmarks. An experimental evaluation shows that the learned heuristics greatly improve on the baseline without trained models. Furthermore, the machine learned guidance demonstrates comparable performance to CVC5 and, in some experiments, even surpasses it. Next, this thesis explores the application of machine learning to more restricted function synthesis domains. We hypothesise that narrowing the scope enables the use of machine learning techniques that are not possible in the general setting. We test this hypothesis by considering the problem of ranking function synthesis. Ranking functions are used in program analysis to prove termination of programs by mapping consecutive program states to decreasing elements of a well-founded set. The second contribution of this dissertation is a novel technique for synthesising ranking functions, using neural networks. The key insight is that instead of synthesising a mathematical expression that represents a ranking function, we can train a neural network to act as a ranking function. Hence, the synthesis procedure is replaced by neural network training. We introduce Neural Termination Analysis as a framework that leverages this. We train neural networks from sampled execution traces of the program we want to prove terminating. We enforce the synthesis specifications of ranking functions using the loss function and network design. After training, we use symbolic reasoning to formally verify that the resulting function is indeed a correct ranking function for the target program. We demonstrate that our method succeeds in synthesising ranking functions for programs that are beyond the reach of state-of-the-art tools. This includes programs with disjunctions and non-linear expressions in the loop guards