# Generating exact nonlinear ranking functions by symbolic-numeric hybrid method

@article{Shen2013GeneratingEN, title={Generating exact nonlinear ranking functions by symbolic-numeric hybrid method}, author={Li-Yong Shen and Min Wu and Zhengfeng Yang and Zhenbing Zeng}, journal={Journal of Systems Science and Complexity}, year={2013}, volume={26}, pages={291-301} }

This paper presents a hybrid symbolic-numeric algorithm to compute ranking functions for establishing the termination of loop programs with polynomial guards and polynomial assignments. The authors first transform the problem into a parameterized polynomial optimization problem, and obtain a numerical ranking function using polynomial sum-of-squares relaxation via semidefinite programming (SDP). A rational vector recovery algorithm is deployed to recover a rational polynomial from the numerical…

## 13 Citations

### Proving total correctness and generating preconditions for loop programs via symbolic-numeric computation methods

- Computer Science, MathematicsFrontiers of Computer Science
- 2014

We present a symbolic-numeric hybrid method, based on sum-of-squares (SOS) relaxation and rational vector recovery, to compute inequality invariants and ranking functions for proving total…

### SVMRanker: a general termination analysis framework of loop programs via SVM

- Computer ScienceESEC/SIGSOFT FSE
- 2020

SVMRanker is presented, a general framework for proving termination of programs, which is able to synthesize different types of ranking functions for programs with both linear and polynomial updates, based on Support-Vector Machines (SVM).

### Synthesizing Nested Ranking Functions for Loop Programs via SVM

- Computer ScienceICFEM
- 2019

This paper considers nested ranking functions for loop programs and shows that the existence problem of a nested ranking function is equivalent to the existenceProblem of a hyperplane separating classes of data, which allows us to leverage Support-Vector Machines techniques for the synthesis of nestedranking functions.

### Non-polynomial Worst-Case Analysis of Recursive Programs

- Computer Science, MathematicsCAV
- 2017

This work applies ranking functions to recursion, resulting in measure functions, and can synthesize bounds of various forms including O(n log n) and O(nr), where r is not an integer.

### Ranking Function Detection via SVM: A More General Method

- Computer ScienceIEEE Access
- 2019

This paper proposes an approach to the synthesis of ranking functions for loops via support vector machine (SVM), and transforms the ranking function detection problem into a binary classification problem.

### Termination Analysis of Probabilistic Programs Through Positivstellensatz's

- Computer ScienceCAV
- 2016

The approach synthesizes polynomial ranking-supermartingales through Positivstellensatz’s, yielding an efficient method which is not only sound, but also semi-complete over a large subclass of programs.

### Synthesis of ranking functions via DNN

- Computer ScienceNeural Comput. Appl.
- 2021

This paper proposes a new approach to synthesis of non-polynomial ranking functions for loops via deep neural network (DNN), and builds a ranking function template by DNN structure which can be learned by the train-set and verified by a new verification method.

### Automated Recurrence Analysis for Almost-Linear Expected-Runtime Bounds

- Computer Science, MathematicsCAV
- 2017

The approach can infer the asymptotically optimal expected-runtime bounds for recurrences of classical randomized algorithms, including RANDOMIZED-SEARCH, QUICK-SORT,QUICK-SELECT, COUPONCOLLECTOR, where the worst-case bounds are either inefficient, quadratic as compared to linear or almost-linear, or ineffective.

### Efficient Algorithms for Checking Fast Termination in VASS

- Computer Science, MathematicsArXiv
- 2017

It is shown that singularities in the normal are the key reason for asymptotic bounds such as exponential and non-elementary for VASS, and that the asymPTotic complexity bound is always polynomial and of the form ${\Theta}(n^k)$, for some k $\leq$ d.

### Average-case analysis of programs: Automated recurrence analysis for almost-linear bounds

- Computer Science
- 2020

The approach can infer the asymptotically optimal average-case bounds for classical randomized algorithms, including RANDOMIZED-SEARCH, QUICKSORT,QUICK-SELECT, COUPON-COLLECTOR, where the worstcase bounds are either inefficient, quadratic as compared to linear or almost-linear of average- case, or ineffective.

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