CN113792878A - Automatic identification method for numerical program metamorphic relation - Google Patents
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Abstract
The invention provides an automatic identification method of numerical program Metamorphic Relation (MR), which comprises the following steps: (1) problem domain based input pattern analysis; (2) generating initial test input based on a random method; (3) output pattern mining based on Gene Expression Programming (GEP). The method decomposes the identification of the metamorphic relation into two steps of input mode analysis and output mode mining, and reduces the difficulty of metamorphic relation identification. According to the characteristics of a numerical program, identification ways of four input modes are provided, and the problem of constructing a metamorphic relation at will is avoided; an input mode which is meaningful for verification is established by applying domain knowledge, and the problems of huge solution space and low solution efficiency are solved; and a GEP automatic search output mode is introduced, so that the problems of high randomness and low effectiveness of an MR identification result are solved.
Description
Technical Field
The invention relates to the technical field of computer algorithms, in particular to an automatic identification method of numerical program metamorphic relations.
Background
Scientific calculation and industrial design software such as numerical simulation nuclear design and safety analysis software, aeronautical engine dynamic design simulation software and the like are adopted, and the problem that an expected result is difficult to construct or the construction cost is extremely high generally exists because complicated partial differential equations need to be solved and no analytic solution exists, so that the problem is called the Oracle problem of software testing. The traditional test method adopts a mode of directly comparing an actual result with an expected result to verify a tested program, and the Oracle problem makes it difficult to implement sufficient test on the software. Lack of adequate testing, software quality is difficult to guarantee.
Metamorphic Testing (MT) is one of the currently accepted effective methods for solving the Oracle problem, and it indirectly performs verification by examining whether Metamorphic Relations (MR) are satisfied between input and output when a program is executed multiple times. For example, assuming that the program under test P implements the sine function sin, the relationship given by the periodic, available degeneracy of sin: sin (x) ═ sin (x +2 pi), generating a random value x1As an initial Test Case (STC), x1+2 π gives x2As subsequent Test cases (FTC), P (x) was compared1) And P (x)2) Otherwise, P fails the test. A transmutation relationship is composed of an input mode and an output mode, such as x in the above example2＝x1+2 π and P (x)2)＝P(x1)。
The metamorphic relation is the core of metamorphic testing, and the current MR identification method can be divided into two categories, namely static and dynamic. The former adopts manual analysis and derivation MR based on the domain knowledge of the software to be tested, such as physical law, physical equation, numerical solution algorithm, program algorithm, etc.; the later finds a stable mode from the running data of the tested program through data mining or a method based on searching, thereby identifying the metamorphic relation, for example, assuming that MR is a quadratic polynomial, searching the polynomial coefficient by using a particle swarm algorithm, and the like.
The static identification technology has high accuracy, most of the static identification technology can reasonably explain the MR by using domain knowledge, but (1) even domain experts consider that obtaining the metamorphic relation is difficult; (2) mostly, metamorphic relations are obtained manually and randomly, the efficiency is low, the cost is high, and large-scale popularization is difficult.
The dynamic identification technique does not require domain knowledge, and the current method is to solve the input mode and the output mode as a whole, for example, if the tested program P only has one input parameter x and one output parameter y, the input mode is a linear relation x2＝ax1+ b, when the output modes are linear, MR is denoted as c1P(x1)+c2P(ax1+ b) + d is 0, and when the output mode takes a quadratic polynomial, MR is represented as
c1P2(x1)+c2P(x1)P(ax1+b)+c3P2(ax1+b)+d1P(x1)+d2P(ax1+b)+e＝0
Obviously, even if P has only one input parameter, the search space and the amount of computation for solving the quadratic polynomial MR are considerable.
Meanwhile, the effectiveness of the method is greatly influenced by the data set, and the generation of the data set is usually high in randomness or blindness due to lack of field expert guidance, so that a large amount of input/output meaningless to software verification exists in the data set, a solution space is difficult to compress effectively, and the solution efficiency is low. Since the search direction is random in the search-based method, which may cause the MR to be missed even if there is an MR, and the MR identification result has a large randomness, it is common to repeat the above process many times to improve the probability of finding an MR.
Disclosure of Invention
The invention aims to provide an automatic identification method of numerical program metamorphic relation, which reduces the difficulty of MR identification, overcomes the problem of randomness of MR identification, avoids obtaining MR in a random mode and improves the solving efficiency.
The technical scheme of the invention is realized as follows:
the invention provides an automatic identification method of numerical program metamorphic relation, which comprises the following steps:
(1) problem domain based input pattern analysis;
(2) generating initial test input based on a random method;
(3) and mining based on the output mode of the GEP.
As a further improvement of the invention, the specific method is as follows:
(1) analyzing, deducing and identifying an input mode from four ways of model mathematical property, numerical algorithm property, input parameter data variation and existing mode compounding according to field background knowledge;
(2) randomly generating a batch of initial test inputs, and generating corresponding subsequent test inputs according to an input mode to form an input couple;
(3) using the input couple pair to drive the tested program to execute, and obtaining an output result couple pair;
(4) analyzing and deducing a function operator of the GEP symbolic expression by combining the elementary function and the field background knowledge;
(5) mining an output mode from the output result couple by adopting a GEP technology according to a preset function operator;
(6) the input mode and the output mode jointly form a metamorphic relation.
As a further improvement of the invention, the mathematical properties comprise monotonicity, periodicity, parity and symmetry of a physical model and a numerical solving method.
As a further improvement of the present invention, if monotonicity exists, the structure monotonically increases x1<x2Or decrement x1>x2Input doublet (x)1，x2) (ii) a If there is a periodicity T, then construct the input doublet (x)1，x2＝x1+ T); if parity exists, then construct input even pairs (x) of opposite sign1，x2＝(-1)x1) (ii) a If symmetry exists and symmetry is given about the line x ═ a, then an input doublet (x) is constructed1，x2＝2a-x1)。
As a further improvement of the invention, the algorithm property is analysis numerical algorithm parameters, including coordinate points and step length; the input mode is constructed by setting different parameter values.
As a further improvement of the invention, the input parameter data mutation is a mutation operation on the original data, and the input mode is constructed by applying a data mutation operator on the initial input.
As a further improvement of the present invention, the data mutation operator is at least one selected from INC (increment 1), DEC (decrement 1), ADD (plus a constant), SUB (minus a constant), DIV (reciprocal, constant if the value is 0), MUL (multiply constant), NEG (sign minus).
As a further improvement of the present invention, the compound input mode is a new input mode obtained by performing compound operation on an existing input mode.
The invention has the following beneficial effects: the method decomposes the identification of the metamorphic relation into two steps of input mode analysis and output mode mining, and reduces the difficulty of metamorphic relation identification. According to the characteristics of the numerical program, identification ways of four input modes are provided, and the problem of constructing metamorphic relations at will is avoided; an input mode which is meaningful for verification is established by applying domain knowledge, and the problems of huge learning space and low solving efficiency are solved; and a GEP automatic search output mode is introduced, so that the problems of high randomness and low effectiveness of an MR identification result are solved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic diagram of a metamorphic relation recognition algorithm of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Because most parameters of the numerical program are numerical variables, the automatic calculation of a dynamic method is facilitated, and a small number of non-numerical parameters can be mapped into integers to be converted into numerical values. The invention relates to an MR dynamic identification technology suitable for a numerical program.
Since the metamorphic relation is essentially an implication relation between the input mode and the output mode, a metamorphic relation identification method of 'separating the input mode from the output mode' is provided. The method comprises the steps of analyzing and deducing an input mode through the knowledge in the field of numerical programs, then randomly generating initial input, establishing subsequent input by using the input mode, forming an input couple pair for metamorphic testing, executing a tested program to obtain actual output, and then mining a hidden output mode between the initial output and the subsequent output by using a Gene Expression Programming (GEP) technology to further obtain the metamorphic relation.
GEP is a data mining method based on genotype and phenotype, which is proposed by Ferreira (Ferreira) for the purpose of using biogenetic evolution algorithm, and is commonly used in symbolic expression regression, i.e. searching symbolic expressions from input data x and output data y, so that y ═ f (x) fits best to the data. Research shows that when the mutation probability is 0.6-1, the GEP mining success rate reaches 20% -80%, and the GEP can be found as long as a mode exists in data, so that the stability of the identification result is ensured, and the method is very suitable for solving the problem of high randomness of the MR identification result.
As a result of analyzing domain knowledge, input patterns are obtained that are meaningful to verification, such as multiple x2＝3x1Sign negation x2＝-x1Reciprocal number x2＝1/x1And the blindness is avoided when the metamorphic test input doublet is established, and compared with a completely random method, the understanding space is reduced; meanwhile, for the output mode, given only the possible function operators, such as addition, power function, etc., the GEP searches the candidate form of f (x) formed by these operators, such as f (x) x2+ x or f (x) x, etc., so that f (x) is best even-to-match with the output. Research shows that the gene obtained after a series of random genetic operations such as deletion and modification of benign structure gene is a legal gene, and the validity of the identification result is ensured.
Specifically, in the method, the metamorphic relation identification comprises three steps: (1) problem domain based input pattern analysis; (2) generating initial test input based on a random method; (3) and mining based on the output mode of the GEP. The algorithm flow chart is shown in fig. 1.
The algorithm performs the following:
(1) analyzing, deducing and identifying an input mode from four ways of model mathematical property, numerical algorithm property, input parameter data variation and existing mode compounding according to field background knowledge;
(2) randomly generating a batch of initial test inputs, and generating corresponding subsequent test inputs according to an input mode to form an input couple;
(3) using the input couple pair to drive the tested program to execute, and obtaining an output result couple pair;
(4) analyzing and deducing a function operator of the GEP symbolic expression by combining the elementary function and the field background knowledge;
(5) mining an output mode from the output result couple by adopting a GEP technology according to a preset function operator;
(6) the input mode and the output mode jointly form a metamorphic relation.
According to the characteristics of the numerical program, the input mode can be analyzed from the following four aspects:
(1) mathematical properties. The monotonicity, the periodicity, the parity and the symmetry of the physical model and the numerical solution method are mainly considered. If monotonicity exists, the construct grows monotonically by x1<x2Or decrement x1>x2Input doublet (x)1，x2) (ii) a If there is a periodicity T, then construct the input doublet (x)1，x2＝x1+ T); if parity exists, then construct input even pairs (x) of opposite sign1，x2＝(-1)x1) (ii) a If symmetry exists and symmetry is given about the line x ═ a, then an input doublet (x) is constructed1，x2＝2a-x1)。
(2) The nature of the algorithm. The method mainly analyzes numerical algorithm parameters such as coordinate points, step lengths and the like, and constructs an input mode by setting different parameter values such as adjusting the quadrant where the initial coordinate changes, modifying step length to influence iteration times and the like.
(3) And (6) carrying out data mutation. Data mutation is a process of generating new data by performing mutation operation on original data values, an input mode in a metamorphic relation can be regarded as a mutation operation on original data, an input mode is constructed by applying a data mutation operator on initial input, and a common operator is shown in table 1.
TABLE 1 data mutation operator
| Serial number | Operator name | Description of the invention |
| 1 | INC | Self-increasing 1 |
| 2 | DEC | Self-decreasing by 1 |
| 3 | ADD | Adding a constant |
| 4 | SUB | Subtracting a constant |
| 5 | DIV | Reciprocal, if the value is 0, it is not changed |
| 6 | MUL | Multiplying by a constant |
| 7 | NEG | Sign gets negative |
(4) And (4) compounding input modes. If the input patterns are considered as functional relationships, the composition of the input patterns may be considered as a composition of functions. Performing composite operation on the existing input modes to obtain a new input mode, wherein if the two input modes are x respectively2＝(-1)x1And x2＝3x1Disclosure of the inventionOver-compounding to obtain x2＝(-1)*3x1。
Example 1
Take sine function sin as an example.
(1) From periodically available input patterns r1：x2＝x1+2 π, r is obtained from symmetry2：x2＝π-x1R is obtained from the properties of an odd function3：x2＝-x1；
(2) In [0,1 ]]Range-random generation of 100 initial test inputs IstcRandom (0,1,100) according to r1Generating a corresponding subsequent test input Iftc＝{Istc+2 π } form an input doublet (I)stc，Iftc) Other input modes and so on;
(3) driving a tested program P with an input doubletsinExecuting to obtain an output result even pair (P)sin(Istc)，Psin(Iftc))；
(4) The function operator of the preset GEP is Op { +, -,/, x { +, -,/, x { + }2}；
(5) According to Op, an output mode is mined from an output result couple by adopting a GEP technology, sum of variance (SSE) is used as a fitness function to evaluate The quality of The output mode, The smaller The numerical value is, The better The quality is, The main parameters of The GEP are The population size 100, The variation probability 0.05 and The evolution algebra 200, and The result is shown in The following table 2;
TABLE 2
(5) The metamorphic relation is formed by the input mode and the output mode together, as shown in table 3.
TABLE 3
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (10)
1. An automatic identification method for numerical program metamorphic relations is characterized by comprising the following steps:
(1) problem domain based input pattern analysis;
(2) generating initial test input based on a random method;
(3) and mining based on the output mode of the GEP.
2. The method for automatically identifying numerical program degenerating relations of claim 1, wherein the specific method is as follows:
(1) analyzing, deducing and identifying an input mode from four ways of physical model mathematical properties, numerical algorithm properties, input parameter data variation and existing mode compounding according to field background knowledge;
(2) randomly generating a batch of initial test inputs, and generating corresponding subsequent test inputs according to an input mode to form an input couple;
(3) using the input couple pair to drive the tested program to execute, and obtaining an output result couple pair;
(4) analyzing and deducing a function operator of the GEP symbolic expression by combining the elementary function and the field background knowledge;
(5) mining an output mode from the output result couple by adopting a GEP technology according to a preset function operator;
(6) the input mode and the output mode jointly form a metamorphic relation.
3. The method of claim 2, wherein the mathematical properties include fundamental properties of monotonicity, periodicity, parity, symmetry, etc. of the physical model and the numerical solving method.
4. The method of automatically identifying a metamorphic relation of a numerical program according to claim 3,wherein if monotonicity exists, the structure monotonically increases by x1<x2Or decrement x1>x2Input doublet (x)1，x2) (ii) a If there is a periodicity T, then construct the input doublet (x)1，x2＝x1+ T); if parity exists, then construct input even pairs (x) of opposite sign1，x2＝(-1)x1) (ii) a If symmetry exists and symmetry is given about the line x ═ a, then an input doublet (x) is constructed1，x2＝2a-x1)。
5. The method of claim 2, wherein the algorithmic property is analyzing a numerical algorithm parameter, including coordinate points, step length; the input mode is constructed by setting different parameter values.
6. The method of claim 2, wherein said input parameter data mutation is a mutation operation on original data, and said input mode is constructed by applying a data mutation operator to the original input.
7. The method of claim 6, wherein the data mutation operator is at least one selected from INC (1 self increment), DEC (1 self decrement), ADD (plus a constant), SUB (minus a constant), DIV (reciprocal, constant if the value is 0), MUL (multiplication by a constant), NEG (negative sign).
8. The method of claim 2, wherein the compound input mode is a new input mode obtained by performing compound operation on an existing input mode.
9. The method of claim 2, wherein the input pattern and the output pattern are arithmetic symbolic expressions.
10. The method of claim 2, wherein the function operators comprise constant functions such as addition, subtraction, multiplication and division, elementary functions such as power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function, and all functions obtained by performing finite quartic operation or function composition on the functions.
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Citations (10)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20030195706A1 (en) * | 2000-11-20 | 2003-10-16 | Michael Korenberg | Method for classifying genetic data |
| US6941287B1 (en) * | 1999-04-30 | 2005-09-06 | E. I. Du Pont De Nemours And Company | Distributed hierarchical evolutionary modeling and visualization of empirical data |
| US20070196851A1 (en) * | 2006-02-14 | 2007-08-23 | Andrei Yakovlev | System and method for selecting differentially dependent genes from gene expression data |
| WO2009061703A2 (en) * | 2007-11-08 | 2009-05-14 | Ntt Docomo, Inc. | Automated test input generation for web applications |
| US20100138026A1 (en) * | 2008-03-08 | 2010-06-03 | Tokyo Electron Limited | Method and apparatus for self-learning and self-improving a semiconductor manufacturing tool |
| CN102170378A (en) * | 2011-04-22 | 2011-08-31 | 北京科技大学 | Web service test method with no need of anticipation |
| US20120222013A1 (en) * | 2011-02-28 | 2012-08-30 | Microsoft Corporation | Modeling software behavior using learned predicates |
| CN107967928A (en) * | 2017-10-18 | 2018-04-27 | 武汉忆数存储技术有限公司 | A kind of flash chip life-span prediction method based on mathematical model |
| CN112102876A (en) * | 2020-09-27 | 2020-12-18 | 西安交通大学 | Method for automatically modeling gene circuit and transcription regulation and control relation |
| CN112270398A (en) * | 2020-10-28 | 2021-01-26 | 西北工业大学 | Cluster behavior learning method based on gene programming |
-
2021
- 2021-08-18 CN CN202110950800.XA patent/CN113792878B/en active Active
Patent Citations (10)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6941287B1 (en) * | 1999-04-30 | 2005-09-06 | E. I. Du Pont De Nemours And Company | Distributed hierarchical evolutionary modeling and visualization of empirical data |
| US20030195706A1 (en) * | 2000-11-20 | 2003-10-16 | Michael Korenberg | Method for classifying genetic data |
| US20070196851A1 (en) * | 2006-02-14 | 2007-08-23 | Andrei Yakovlev | System and method for selecting differentially dependent genes from gene expression data |
| WO2009061703A2 (en) * | 2007-11-08 | 2009-05-14 | Ntt Docomo, Inc. | Automated test input generation for web applications |
| US20100138026A1 (en) * | 2008-03-08 | 2010-06-03 | Tokyo Electron Limited | Method and apparatus for self-learning and self-improving a semiconductor manufacturing tool |
| US20120222013A1 (en) * | 2011-02-28 | 2012-08-30 | Microsoft Corporation | Modeling software behavior using learned predicates |
| CN102170378A (en) * | 2011-04-22 | 2011-08-31 | 北京科技大学 | Web service test method with no need of anticipation |
| CN107967928A (en) * | 2017-10-18 | 2018-04-27 | 武汉忆数存储技术有限公司 | A kind of flash chip life-span prediction method based on mathematical model |
| CN112102876A (en) * | 2020-09-27 | 2020-12-18 | 西安交通大学 | Method for automatically modeling gene circuit and transcription regulation and control relation |
| CN112270398A (en) * | 2020-10-28 | 2021-01-26 | 西北工业大学 | Cluster behavior learning method based on gene programming |
Non-Patent Citations (6)
| Title |
|---|
| MENG L等: "Metamorphic Relation Generation for Physics Burnup Program Testing", INTERNATIONAL JOURNAL OF PERFORMABILITY ENGINEERING, pages 1 - 5 * |
| TAHIR JAMEEL等: "Test oracles based on metamorphic relations for image processing applications", 2015 IEEE/ACIS 16TH INTERNATIONAL CONFERENCE ON SOFTWARE ENGINEERING, ARTIFICIAL INTELLIGENCE, NETWORKING AND PARALLEL/DISTRIBUTED COMPUTING (SNPD) * |
| 元昌安等: "基于基因表达式编程的智能模型库系统的实现", 四川大学学报(工程科学版), 28 May 2005 (2005-05-28) * |
| 惠战伟等: "一种基于蜕变关系的测试与失效测试用例定位模型", 万方知识数据服务平台 * |
| 闫仕宇等: "基于蜕变测试的热传导程序的验证测试研究", 核科学与工程 * |
| 陈月辉等: "利用遗传编程的符号表达式逼近任意非线性函数", 山东建材学院学报 * |
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