Formal Methods lecture 01

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Formal Methods in Software Engineering
Formal Methods in Software EngineeringFormal MethodFormal method is branch of software engineering, in which we analyze software systems.Develop a program in a way that each step leads to a final solution, follow proper method to make sure that we do not take wrong steps.The Encyclopedia of Software Engineering defines formal methods in the following manner:Formal methods used in developing computer systems are mathematically based techniques for describing system properties. Such formal methods provide frameworks within which people can specify, develop, and verify systems in a systematic, rather than ad hoc manner.
Formal Method definitionA method is formal if it has a sound mathematical basis, typically given by a formal specification language. This basis provides a means of precisely defining notions like consistency, completeness, and more relevantly specification, implementation and correctness.Correctness, the property that an abstract model fulfills a set of well defined requirements.Consistency, to be consistent, facts stated in one place in a specification should not be contradicted in another place.Used to specify programs, what the system is suppose to do. Used for constructing programs.Used to verify the program.
Why Formal Methods are requiredHistory of softwareSoftwares encountered notorious bugs that were the cause of financial lose and deaths of many people.Famous bugs areTherac25 Computerized radiation therapy machine called the Therac25. Killed many people, controller could not stop radiation due to software bug.AT & T long distance breakdown bug 1990 Ill placed break statement in the code, caused the 1/3rd of entire American network to go down for 9 hours.
Why Formal Methods are requiredPatriot Missile Failure Gulf warThis missile hit the own American troops a software defect in 1991. Killing 28 people and many injuredPentium bug Software error in microcode of Pentium microprocessor, which resulted in error of floating point calculation problems. Intel had to take back all the Pentiums, and it caused huge loss.
Importance of formal methods in SoftwareThis is what we are going to study in formal methods.Methods to ensure that software is CorrectReliableThese two attributes deal with the software quality.To achieve software quality, we apply different techniques.TestingVerification Validation
TestingBlack box testingTest input versus outputInputTwo numbersOutput averageWhite box testingTest the structure of program.Loops testing, condition testing
"Testing can show the presence of errors, but not their absence."  Edsger Dijkstra
Testing contd..In testing we develop test cases and define scenarios, it is not possible to have all scenarios .Program to show equalness of two strings.isEqual(cat,dog) expected falseisEqual(testing,testing) expected trueisEqual(house,home) expected falseisEqual(house,mouse) expected false.
Testing Contd..Same is case with structural testing, white box testing.What is wrong with the following code?
How Formal Methods are developed?We develop models of system.With the help of models we will argue and prove correctness of models.What is a model?An item, a structure, a style,.Lets play Dijkstras game to define a model.
Dijkstras gameConsider the following game to be played by a single person with an urn/jar and as many w white balls and b black balls as he needs. To begin with, an arbitrary positive number of balls is put into the urn and as long as the urn contains two or more balls, the player repeats the following moves: He shakes the urn and, without looking, he takes two balls from the urn; If those two balls have the same color he throws one black ball into the urn, otherwise he returns one white ball into the urn Because each move decreases the total number of balls into the urn by 1, the game is guaranteed to terminate after a finite number of moves and it is not difficult to see that the game ends with exactly 1 ball in the urn. The question is: What can we say about the color of the final ball when we are given the initial contents of the urn?Dijkstras game contd..
Dijkstras game contd..What is the color of last ball?Difficult to answerLets play the same game with different number of balls.
Dijkstras game contd..One ball gameThe game will end without playingTwo balls gameThree different combinations of ballsOne Black, one whiteOne White ,one whiteOne Black, one blacktwo ball game
Dijkstras game contd..Dijkstras game contd..Mathematical models use functions.What is a function?Putting the balls in jar is a function
Conclusion from 2 and 3 balls gameDepends on parity of white balls, even or odd parity.Even number of white balls, last ball is black color.Odd number of balls, last ball is of white.If we play with 100 balls, then can we argue or prove our hypothesis?What is the color of last ball, given w white balls and b black balls?
Mathematical model and its proofF(b,w)=2 black out,1 black in b2+1, (b1,w) We reduce the number of black balls by 1 and we maintain the number of white balls.2 white out,1 black in w2,b+1We reduce the number of white balls by 2 and increase the number of black balls by 1.1 of each out,1 white in b1,w1+1,(w)we reduce the number of black balls by 1 and maintain the number of white balls.Total number of balls removed in each move is 1.Parity(even/odd number) of whit balls does not change.Yes we will say the parity of white balls determine the outcome of the game.Hence hypothesis is correct.
Formal method stepsWe will define state based model for our computer programs using formal methods.Define the specifications of the system(Formal specification).Define abstract model specifications.Define the states of system(steps of a model)Define invariant(condition)Define set of operations for model to function.System/model operation is associated with two conditionsPreconditionPost conditionModel verification and Implementation Make formal model and use tools to prove mechanically that formal execution model satisfies formal requirements.
Written RequirementInformal requirements expressed in English:A tank of cooling water shall be refilled when its low level sensor comes on. Refilling consists of adding 9 units of water to the tank.Notes:The maximum capacity of the tank is 10 units of water.From one reading of the water level to the next reading of the water level, 1 unit of water will be used.The low level sensor comes on when the tank contains 1 unit of water or less.A Simple Formal Methods Example88Assigning TypesThe above statement contains several descriptions, including two key notions: the water level in the tank and the water usage. Formally, these notions can be modeled as follows (statements 1 and 2):
1level is represented by a restricted integer type: a number between 0 and 10, inclusive2usage is represented as the integer constant 1
That is, level describes an amount of water that the tank may hold at any point in time and usage describes the amount of water used during one cycle.A Simple Formal Methods Example99Function DescriptionThe primary requirement is that 9 units of water will be added to the tank whenever the level is less than or equal to 1. This can be more precisely stated as (statement 3):
3Function fill takes, as input, a water level and returns, as output, a water level. Given an input of Lvl units of water, fill returns Lvl +9, if Lvl is one or less, otherwise it returns Lvl.
That is, we claim that fill(Lvl) accounts for any filling of water in the tank.A Simple Formal Methods Example1010PropertiesA common sense property of this system is that, at the next cycle, the new water level will be the current water level, plus any amount that was added, minus the amount that was used. That is, given Lvl as the current level of water, the level at the next cycle should be given by statement 4:
4 level = Lvl + fill(Lvl)  usage
A Simple Formal Methods Example1111Properties (continued)One approach to checking this specification is to ensure that each reference to a level of water is consistent with the definition of level, i.e., it should always be a number between 0 and 10. It turns out that the specification for fill given in 3 above is consistent with the definition of level if the following two logical statements are true:5FORALL levels Lvl(Lvl