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Testing, proof and automation an integrated approach

Testing, proof and automation an integrated approach
Testing, proof and automation an integrated approach

Testing,Proof and Automation.An Integrated Approach

Simon Burton John Clark John McDermid

Department of Computer Science.

University of York,

Heslington,York.

YO105DD,England.

+441904432749

burton,jac,jam@https://www.wendangku.net/doc/2d4034626.html,

Abstract

This paper presents a discussion on the complementary roles of testing and proof within automated software ver-i?cation and validation processes.We demonstrate how a combination of the two approaches can lead to greater lev-els of automation and integrity.In particular we discuss the use of automated counter-example generation to sup-port proof,and automated proof as a means of automating and checking test case generation.The high levels of au-tomation are made possible by identifying repeating struc-tures in the proofs,restricting the speci?cation to a subset of an otherwise expressive formal notation and exploit-ing a general–purpose theorem proving tool with built-in constraint solvers.

1Introduction

In the past,testing and proof have not been easy bed-fellows.Despite their shared goal of increased software quality,proof has been seen as being for the cognoscenti, testing for software engineering’s working class.The au-thors believe that this arti?cial dichotomy is harmful and that testing and proof can be used together to good ef-fect.Even without the bene?ts of formal re?nement,for-mal speci?cations can contribute greatly to the quality of a software product.They allow for a concise,unambigu-ous and explicit speci?cation of the desired behaviour of the system.As such,they are a good basis for automated test activities.Additionally,testing to generate counter-examples to proofs can save much effort and produce il-lustrative examples for debugging.

The use of formal speci?cations themselves is still seen by many as a barrier to the widespread industrial usage of formal methods.For the bene?ts of formal methods in V&V to be fully exploited in industry there is a need to“disguise”the formality in some way[15].Recent work[5]has shown that formal speci?cations and the cor-responding proof obligations for speci?cation validation can be generated from more intuitive engineering nota-tions with mathematical underpinnings.Such an approach not only enables engineers with the domain knowledge to use speci?cation notations they are comfortable with, but the translation to formal speci?cation has the effect of restricting the subset of the formal notation used and imposes regular structures on the proofs that need to be discharged to validate certain properties(such as com-pleteness and determinism)in the speci?cation.These restrictions,coupled with the subset of data types used for particular domains can be exploited to develop power-ful targeted heuristics for automating the V&V activities. The approaches discussed in this paper are assumed to be undertaken in the context of formal speci?cations gener-ated in this manner.

In the rest of the paper we describe how a combination

of testing,proof and restricted structures in the speci?-cation can be used to enhance both the integrity and au-tomation of several areas of the software veri?cation and validation process.This symbiotic relationship between testing and proof is made feasible by extending previous work on testing from formal speci?cations and making use of a?exible theorem proving tool with integrated con-straint solvers.

The paper is structured as follows.Section2discusses the role of testing in the automatic generation of counter-examples to proofs.Section3describes how proof can be used as a means of verifying automated test case gen-eration strategies and also as a means of performing the automation itself.We also show how more effective test-ing strategies can be developed based on the automatic generation of formally speci?ed test cases and how proof can be used as a testing oracle.Section4summarises the main contributions of the work and presents some conclu-sions.

2Testing and Proof

Proof conjectures can arise at various points in the V&V process.For example,to ensure that a speci?cation sat-is?es certain“healthiness”criteria such as completeness and determinism or to verify that a program is a correct re?nement of its formal speci?cation.In all cases,in-valid conjectures can waste a large amount of proof effort. Therefore,before a long and arduous manual proof is em-barked upon it is re-assuring to have a good degree of con-?dence in the validity of the https://www.wendangku.net/doc/2d4034626.html,e of constraint solving techniques to generate counter-examples not only saves proof effort but can provide illustrative information to use when tracking the fault.The generation of counter-examples to verify properties of a speci?cation couched in terms of proofs is a form of testing.Typically sample data are generated and then tested to see whether they break the speci?cation.If this is the case,a counter-example has been found.

Constraint solving in general is known to be intractable [13].However,in practical situations,one never needs to solve“general”constraints but a particular subset that have restricted structures and particular input space char-acteristics.These properties can be exploited to automate the search for counter-examples.The authors use the Z [16]type checker and theorem prover CADi[19]to au-tomate this task.In this sense our usage of CADi is sim-ilar to that of the Nitpick Z-based speci?cation checker [12]that used model-checking techniques to generate counter-examples to speci?cation assertions.However, CADi has the additional?exibility that general purpose proof tactics can be written(using a lazy functional nota-tion[20]),that can be invoked interactively from within the tool and applied to any proof obligation on the screen. Proof tactics have been written that attempt a best effort at automatically proving conjectures of certain types(e.g. completeness checks).If the proof fails or is inconclusive, the tactics then perform some simpli?cation to transform the conjecture into a suitable form for the integrated con-straint solvers.A number of constraint solvers can then be invoked to attempt counter-example generation,these include a model-checker(SMV[3])and a simulated an-nealing based heuristic search[6].The amount of simpli-?cation required before the constraint solvers can be ef?-ciently applied will depend on the structure of the proof obligations.

Such automated proof tactics have been used to good effect when a large number of similar proof conjectures were needed to be solved[5].A situation which would have otherwise been time consuming if done manually and could have led to“reviewer blindness”leading to missed error cases.Different constraint solvers have been found to be effective for different input domains.For ex-ample model-checking is only practical for discrete input domains,whereas optimisation-based search techniques are also suited to in?nite state spaces and non-linear con-straints.

3Proof and Testing

Formal speci?cations are a good basis for testing.They allow for a concise and unambiguous representation of the requirements and are amenable to proof and automated analysis.Test generation techniques for model-based for-mal speci?cations[14,8,17]such as Z[16]or VDM-SL [10]are typically based on the principle of partitioning the speci?cation into equivalence classes[9].Equivalence classes are partitions of the speci?cation input space that are assumed,for the purpose of testing,to represent the same behaviour in the speci?cation.Such techniques are

amenable to automation and tool support.However,as in all cases where automation is introduced,and especially for high integrity systems,the integrity of such tools is of great importance.For automated testing to be able to pro-vide con?dence in the conformance of the software to its speci?cation,the test generation strategies must be both veri?ed and validated.In other words,they must not only be shown to be correctly implemented but must also be shown to be adept at?nding errors in the implementation.

3.1Veri?cation of Automated Testing

Strategies

There are various criteria that can be used when verifying that test partitioning strategies have been correctly imple-mented.For example,the tests can be shown to com-pletely cover the valid input space of the original speci-?cation.If this were not the case,important parts of the implementation,that could possibly contain faults might remain untested.If the resulting tests are represented us-ing the same formal notation as the original speci?cation, these veri?cation activities can be performed using proof. The completeness of the generated tests()with re-spect to the original speci?cation()can be veri?ed by proving a conjecture of the following form:

An example partitioning strategy identi?es expressions in the speci?cation of the form and partitions these into the following test cases,and

[8]where and could be complex predicates themselves.The conjecture used to prove that these par-titions preserve the valid input state-space of the original speci?cation would therefore take the following form:

This theorem can be proven in a few simple steps.The same proof steps can be used regardless of the structure of the expressions represented by and.In general, a proof can be derived for each partitioning strategy and used to verify the outcome each time that strategy is https://www.wendangku.net/doc/2d4034626.html,ing the proof tactic mechanism in CADi the au-thors have automated these proofs for a number of com-mon partitioning strategies.Whenever a strategy is ap-plied,the corresponding correctness proof can be auto-matically invoked on the result.This ensures that,what-ever the means of test generation,the result can always be shown to be valid or otherwise.The tool can be instructed to record the individual proof steps taken in applying a proof tactic and these can be printed in a form amenable to human scrutiny.Therefore,if the tool cannot be trusted, a rigorous argument can be developed to support the va-lidity of the proof steps.

Given a formal de?nition of a testing strategy as an equivalence(e.g.Theorem2above),the derivation of the test cases themselves can also be automated using general purpose proof tactics.The principle is similar in operation to the use of Disjunctive Normal Form(DNF)to simplify an expression into a disjunction of conjuncts that can each be used as separate test cases.Where conversion to DNF uses simple logic rewrite rules to distribute disjunctions, more targeted equivalences can be formulated based on common testing heuristics.

Test partitioning based on the formal speci?cation of the testing heuristics has been implemented using CADi proof tactics.Generic partitioning strategies are speci?ed as equivalences in the form of Theorem1.A proof tactic is invoked upon the predicate to be partitioned to instantiate the generic equivalence with the operands of the predi-cate and simplify the whole speci?cation to reveal a dis-junction of partitions.Each test case is equivalent to the original speci?cation where the input space has been con-strained according to one of the partitions.The complete-ness of the partitioning strategy is left as a side conjecture to prove to ensure that the partitioning was valid.This can be automated by extending the partitioning tactic with the general purpose proof for the strategy as described above. The following simple example demonstrates how test par-titions are derived.The example speci?cation calculates the square root()of a positive integer()and the test partitions are generated using boundary value analysis of the operator(based on the premise that errors often oc-cur on or around the boundary[2]).

The speci?cation is given as the following Z schema1

2This can be roughly interpreted as:there exist some values for and that satisfy the speci?cation and can therefore be used as suitable test data.

Once the test partitions have been produced,satis-fying test data can be generated by“solving”the ex-istential quanti?cations using the constraint solvers in CADi.The partitioning method described above sup-ports work by Stocks and Carrington[17,18]3who pro-posed a framework for the derivation and speci?cation of test cases based on the Z notation(the Test Template Framework).The method of test case derivation described here complements that work by providing a mechanism for automatically applying the test heuristics to reveal the test partitions that can then be structured using the Test Template Framework.

3.2Validation of Automated Testing Strate-

gies

Mutation testing[7]is a fault-based testing technique that deliberately injects faults into a program in order to as-sess a test set’s adequacy at detecting those faults.Based on the number of injected faults detected(mutation score), conclusions about the general fault?nding ability(muta-tion adequacy)of the test set can be formed.Mutation testing provides a means of validating the test strategies discussed in the previous section.Automatic test case generation,as described above,can provide a statistically signi?cant number of test cases for various strategies.The mutation adequacy of each of these strategies can then be assessed to compare their relative effectiveness at detect-ing faults[1].

Expressing a test case speci?cation as a formal speci?-cation from which the test data are generated also opens up the possibility for some additional manipulation to in-crease the mutation score of the data.Mutation techniques can be applied at the speci?cation level to create speci?-cations that represent an abstract description of potential faults in the implementation(as?rst suggested by Budd and Gopal[4]).If test data can be generated from the original speci?cation that identify(kill)the mutants,that data is also likely to achieve a relatively high score at the program level.The data generated from the speci?-cation would have been“hardened”in some sense against

the likelihood of encountering co-incidental correctness in the implementation.

In general,the number of mutants that can be gener-ated for an expressive formal speci?cation notation such as Z would be extremely large.However,in practice,only a subset of the notation would be used for any particular application domain.In this case,the number of possi-ble mutants would be limited.From within this subset more selective choices of which mutation strategies to ap-ply can be made by analysing the mutation score of par-ticular testing strategies.Hardened test data can then be generated from the test cases by strengthening the predi-cate of the test case to improve the probability that data are generated to kill the chosen set of speci?cation mutants. In some cases,one set of test data could be generated to kill a number of mutants.However,where the hardening predicates are inconsistent,several sets of test data may need to be generated.Take as an example,the following simple test case for a system which averages two num-bers:

The test data can be hardened against the mutation where the is replaced by a by adding an inequality to the test case.

The hardening predicate in this case was

.This represents a necessary condition for de-tecting the mutant but,in general,will not always be suf-?cient.Depending on other mutations that may arise else-where in the implementation,the new test case can not be guaranteed to produce data which kills the mutant,but is more likely to do so than without the hardening predi-cate.Mutation analysis was brie?y mentioned by Stocks and Carrington[17]as an alternative testing heuristic to domain propagation in their Test Template Framework. However,the authors believe there is still scope for re-search in investigating effective mutation strategies for Z-based test sets and whether mutation analysis can be combined with standard domain partitioning to provide more effective test sets.Therefore,future work will evalu-ate various criteria for designing the hardening predicates and their relative ef?cacy at increasing mutation scores when applying the tests to the implementation.If harden-ing predicates could be automatically generated based on a known set of speci?cation mutations,it may be possi-ble to use the feedback from traditional mutation testing approaches(for assessing the effectiveness of test sets)to automatically select the most effective test strategies for particular types of program.

3.3Proof as a Testing Oracle

Test data generated from formal speci?cations are typi-cally not at the same level of abstraction as is needed to test the implementation.Some re?nement will be needed to exercise the implementation with the test in-puts.For implementations which do not preserve the structure of the original speci?cation this re?nement may be dif?cult.In addition,some speci?cations may be non-deterministic,eliminating the possibility of pre-calculating expected test results.

An alternative to the structured decomposition of the speci?cation into test cases and expected results,as dis-cussed above,is to use a“generate and test”approach. Test inputs are chosen via any means(e.g.randomly) and the results of applying the inputs to the implemen-tation are then checked for conformance with the speci-?cation.This approach can also be used in conjunction with the partition-based testing.A statistically signi?cant number of samples can be chosen from each test parti-tion to increase the con?dence in the equivalence class hypothesis used to generate the test cases.In either case, the process of checking the test inputs and outputs against the speci?cation requires a test“oracle”.If enough re-?nement information is known to transform the concrete inputs and outputs of the system into their equivalent in the abstract speci?cation,the formal speci?cation and au-tomated proof tactics can be exploited to form an auto-mated oracle.The speci?cation is instantiated with the inputs and outputs and a proof tactic is used to simplify the expression to True(test passed)or False(test failed). Such simpli?cation is ideally suited to automated theo-rem provers as it typically involves applying many“one-point”simpli?cations until the expression is reduced to either True or False.

4Conclusions

In this paper we have shown how judicious use of testing and proof to support one another can lead to signi?cant bene?ts for the software V&V process,both in terms of increased automation and integrity.The use of counter-example generation can save much wasted proof effort and the use of proof to support test case design can be used to demonstrate the correctness of the test partition-ing techniques as well as offering a means of automation in itself.

The high level of automation is made possible because of the combination of restricting the subset of the formal notation used,the ability to predict the structure of the proofs that are required(and therefore the ability to re-use proof tactics many times)and the use of a powerful theorem proving tool with integrated constraint solving abilities.In the authors’experience in aerospace applica-tions,these restrictions did not need to be contrived but occurred naturally as a property of the domain and the types of proof that were performed.

Some of the techniques described here(e.g.counter-example generation and automated test case and data gen-eration)have already been applied to a large industrial case study[5].Other techniques,(e.g.automated proof as a testing oracle and application of mutation testing con-cepts)require more research to fully explore their poten-tial.In particular,the use of mutation testing techniques, both at the code and speci?cation level appears a promis-ing method of automatically generating effective and ef-?cient test criteria for Z-based testing of particular appli-cation domains.This is an area of research that is made possible by the automated framework described in this pa-per and will be the focus of future work.

5Acknowledgements

This work was funded by the High Integrity Systems and Software Centre,Rolls-Royce Plc.

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一年级复韵母拼音练习

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小学一年级上册复韵母练习题

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四、读读这首古诗,然后,背一背,做一做。 长cháng 歌g y 行xíng 百bǎi 川chuün 东dōng 到dào 海hǎi ,何h ? 时shí复f ù 西x ī 归guī 。 少shào 壮zhuàng 不b ù 努n ǔ 力l ì,老lǎo 大d à 徒t ú 伤shüng 悲byi 。 找出学过的复韵母,用红笔圈出来。 ie ?e er 五、你能根据下面的图画写出复韵母 吗? 六、请你把下面音节补充完整, 七、连一连? f yi j ī xu ? xi ào xi ǎo h ? yu a er 月儿 小河 飞机 学校 ?e j q x ( ) ( ) ( ) ie j q x ( ) ( ) ( )

八、把音节补充完整 y m ? h j 要门爷回家 你栽树,他培土, 我去提水。 九、图画音节连一连。 wū guī cì wei xǐ qua bái cài 十、知道这些动物爱吃什么吗?找一找连一连。 cǎo yú luó bo gǔ tou

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1正确认读三个复韵母aieiui,认清形读准音 2记清标声调的规则,特别是ui的声调,应该加在i的头上 3能在四线格里正确美观的书写三个复韵母aieiui 教学过程: 一复习导入 抽读声母的卡片 二新授认识复韵母 师:刚才我们复习了我们已经学过的声母,同学们读得很好,现在下面的音节我相信大家也能读的很棒! (出示:声母和单韵母组成的音节) 师:刚才大家读了声母和单韵母组成的音节,拼读的非常正确,许多单韵母都和声母手拉手变成了好朋友,其实我们大家不知道, 单韵母和单韵母之间,它们也有很深的感情! (出示单韵母读一读) 师:这就是我们学习过的六个单韵母兄弟,它们团结友爱,是相亲相爱的一家人,于是它们中间的小i就站出来对其他的五个单韵母说:“我在我们单韵母家族是最受欢迎的了,我和小a 小e 小u的关系最好!”于是它们几个就相互的手拉着手,你们看(课件演示:ai ei ui) 它们又组成了一种新的韵母,象这种单韵母和单韵母互相配对组合的韵母,我们就叫它们:(老师板书:复韵母带读学生读)

一年级拼音(声母与复韵母)

仅供参考小学教育资料 姓名:__________________ 班级:__________________ 第1 页共6 页

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