Formal Specifications

Formal Methods

Formal specification is part of a more general collection of techniques that are known as ‘formal methods’ These are all based on mathematical representation and analysis of software

Formal methods include– Formal specification– Specification analysis and proof– Transformational development– Program verification

Acceptance of Formal Methods

Formal methods have not become mainstream software development techniques as was once predicted– Other software engineering techniques have been successful

at increasing system quality. Hence the need for formal methods has been reduced

– Market changes have made time-to-market rather than software with a low error count the key factor. Formal methods do not reduce time to market

– The scope of formal methods is limited. They are not well-suited to specifying and analyzing user interfaces and user interaction

– Formal methods are hard to scale up to large systems

Use of Formal Methods Their principal benefits are in reducing the number of

errors in systems so their main area of applicability is critical systems:– Air traffic control information systems– Railway signalling systems– Spacecraft systems– Medical control systems

In this area, the use of formal methods is most likely to be cost-effective

Formal methods have limited practical applicability

Specification in the Software Process Specification and design are inextricably

mixed.

Architectural design is essential to structure a specification.

Formal specifications are expressed in a mathematical notation with precisely defined vocabulary, syntax and semantics.

Specification and Design

Architecturaldesign

Requirementsspecification

Requirementsdefinition

Softwarespecification

High-leveldesign

Increasing contractor involvement

Decreasing client involvement

Specification

Design

Specification in the Software Process

Requirementsspecification

Formalspecification

Systemmodelling

Architecturaldesign

Requirementsdefinition

High-leveldesign

Specification Techniques

Algebraic Approach– The system is specified in terms of its

operations and their relationships Model-Based Approach

– The system is specified in terms of a state model that is constructed using mathematical constructs such as sets and sequences

– Operations are defined by modifications to the system’s state

Formal Specification Languages

Z (“zed”) Notation

Formal specification language– most successful one -> easy to find faults, can prove correctness

Requires set theory, functions, and discrete math– also difficult to learn because of special symbols

Z specifications consists of 4 sections– given sets, data types, and constants

• sets that get defined in detail– state definition

• variable declarations & predicates that constrain values– initial state– operations

Use of Formal Specification

Formal specification involves investing more effort in the early phases of software development

This reduces requirements errors as it forces a detailed analysis of the requirements

Incompleteness and inconsistencies can be discovered and resolved!!!

Hence, savings as made as the amount of rework due to requirements problems is reduced

Development Costs with Formal Specification

Specification

Design andImplementation

Validation

Specification

Design andImplementation

Validation

Cost

Without formalspecification

With formalspecification

Interface Specification

Large systems are decomposed into subsystems with well-defined interfaces between these subsystems

Specification of subsystem interfaces allows independent development of the different subsystems

Interfaces may be defined as abstract data types or object classes

The algebraic approach to formal specification is particularly well-suited to interface specification

Sub-System Interfaces

Sub-systemA

Sub-systemB

Interfaceobjects

The Structure of an Algebraic Specification

sort < name >imports < LIST OF SPECIFICATION NAMES >

Informal descr iption of the sor t and its operations

Operation signatures setting out the names and the types ofthe parameters to the operations defined over the sort

Axioms defining the operations over the sort

< SPECIFICATION NAME > (Gener ic Parameter)

introduction

description

signature

axioms

Specification in Z Scenario: We maintain a membership list

and an associated phone database.[Person, Phone]|----PhoneDB-----------------------------------|members: P Person (‘set of’ person)|phones : Person Phone (relation)|-------------------------------------------------------|dom phones ⊆ members (invariant)|---------------------------------------------------

Z Operation: Assign a Phone Scenario: Someone would like a phone.

(Note: Missing precondition)|----Assign-----------------------------------| p? : Person; n? : Phone| PhoneDB|-------------------------------------------------------| phone’ = phone union { p? n? }| members’ = members|---------------------------------------------------

Example members {jim, sue} phones {(jim, 1231), (sue, 3956)} Assign(alice, 1231)

Cool Z property: Can calculate minimal preconditions!!

Simple analysis: Leave out preconditions and find minimum constraint to maintain invariants!

Behavioural Specification

Algebraic specification can be cumbersome when the object operations are not independent of the object state

Model-based specification exposes the system state and defines the operations in terms of changes to that state

Abstract State Machine Language (AsmL)

AsmL is a language for modeling the structure and behaviour of digital systems

AsmL can be used to faithfully capture the abstract structure and step-wise behaviour of any discrete systems, including very complex ones such as:Integrated circuits, software components, and devices that combine both hardware and software

Abstract State An AsmL model is said to be abstract because it encodes

only those aspects of the system’s structure that affect the behaviour being modeledThe goal is to use the minimum amount of detail that accurately reproduces (or predicts) the behaviour of the system

Abstraction helps us reduce complex problems into manageable units and prevents us from getting lost in a sea of details

AsmL provides a variety of features that allow you to describe the relevant state of a system

in a very economical, high-level way

Abstract State Machine and Turing Machine

An abstract state machine is a particular kind of mathematical machine, like the Turing machine (TM)

But unlike a TM, ASMs may be defined a very high level of abstraction

An easy way to understand ASMs is to see them as defining a succession of states that may follow an initial state

State Transitions The behaviour of a machine (its run) can always

be depicted as a sequence of states linked by state transitions

• Moving from state A to state B is a state transition

paint in green

paint in redA B

Configurations

Each state is a particular “configuration” of the machine

The state may be simple or it may be very large, with complex structure

But no matter how complex the state might be, each step of the machine’s operation can be seen as a well-defined transition from one particular state to another

Evolution of State Variables

paint in green

paint in redA B

We can view any machine’s state as a dictionary of

(Name, Value) pairs, called state variables

(Colour, Red) is a variable, where “Colour” isthe name of variable, “Red” is the value

Evolution of State Variables

Names are given by the machine’s symbolic vocabulary

Values are fixed elements, like numbers and strings of characters

The run of a machine is a series ofstates and state transitions that

results from applying operations to each state in succession

ExampleDiagram shows the run of a machine that models how orders mightbe processed

S1

Mode = “Initial”

Orders = 0

Balance = $0

Initialise Process All Orders

S3

Mode = “Final”

Orders = 0

Balance = $500

S2

Mode = “Active”

Orders = 2

Balance = $200

Each transition operation: • can be seen as the result of invoking the machine’s

control logic on the current state • calculates the subsequence state as output

Control Logic

The machine’s control logic behaves like a fix set of transition

rules that say how state may evolve

We can think of the control logic as a text that precisely specifies, for any given state, what the values of the machine’s variables will be in the following step

Typical form of the operational text is:“ if condition then update ”

Control Logic as a Black Box

The two dictionaries S1 and S2 have the same set of keys, but the values associated with each variable name may differ between S1 and S2

The Machine’s Control Logic

…

if mode = “Initial”

then mode := “Active”

mode “Initial”

orders 0

balance $0

Mode “Active”

orders 2

balance $200

input

output

• The machine control logic is a black box that takes as input a state dictionary S1 and gives as output a new dictionary S2

Run of the Machine The run of the machine can be seen as what happens

when the control logic is applied to each state in turn The run starts form initial state

S1 S2 S3 …S1 is given to the black box yielding S2, processing S2 results in S3,

and so on …

When no more changes to state are possible, the run is complete

Update Operations We use the symbol

“: =” (reads as “gets”) to indicate the value that a name will have in the resulting state

For example: mode:=“Active”

Update can be seen only during the following step (this is in contrast to Java, C, Pascal, …)

All changes happen simultaneously, when you moving from one step to another. Then, all updates happen at once. (atomic transaction)

ProgramsExample 1. Hello, world

Main()step WriteLine(“hello, world!”)

ASML uses indentations to denote block structure, and blocks can be places inside other blocks

Statement block affect the scope of variables

White space includes blanks and new-line character, ASML does not recognize tab character for indentation!!!!!!!

An operation names run() gives the top-level operational definition of the model (Main() is like main() in Java and C )

Example 2. Reading a Filevar F as File? = undefvar Fcontents as String = “”var Mode as String = “Initial”Main() step until fixpoint if Mode = “Initial” then F :=open(“mfile.txt”) Mode :=“Reading”

if Mode = “Reading” and length(FContents) =0 then FContents :=fread (F,1)

if Mode = “Reading” and length(FContents) =1 then FContents := FContents + fread (F,1)

if Mode = “Reading” and length(FContents) >1 then WriteLine (FContents) Mode :=“Finished”

Example 2. Graph Representation

S1

F= undef

Fcontents =“”

Mode = Initial

Step 1 Step 2

S3

F= <open file 1>

Fcontents =“a”

Mode = Reading

S2

F= <open file 1>

Fcontents =“”

Mode = Reading

S4

F= <open file 1>

Fcontents =“ab”

Mode = Reading Step 4Step5

S5

F= <open file 1>

Fcontents =“ab”

Mode = Finished

Step 3

Key Points Formal system specification complements

informal specification and modeling techniques. Formal specifications are precise and

unambiguous. They remove areas of doubt in a specification, but still depend on interpretation of terms, inputs and outputs.

Formal specification forces a very detailed analysis of the system requirements at an early stage. Correcting errors at this stage is cheaper.

Key Points Formal specification techniques are most applicable in

the development of critical systems and standards. Algebraic techniques are suited to interface specification

where the interface is defined as a set of object classes. Model-based techniques model the system using sets and

functions. This simplifies some types of behavioural specification.

Its not for the faint of heart, you’ll need some special training to go down this path.