Download - Formal Methods

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Lecture 16March 22, 2011

Formal Methods

Adapted from slides provided by Jason Hallstrom and Murali Sitaraman (Clemson)

Requirements vs. Specifications Requirements definition

Intended for customers in addition to software developers

Informal descriptions are necessary

Specification For use by members of a software

development team Formal (mathematical) descriptions are

necessary

Interface Specification Serves as a contract between component

users (clients) and developers (implementers)

Typically describes the demands on users and responsibilities for implementers

Should present the essentials in “user-oriented” terms (abstraction) and hide the inessentials (information hiding)

Informal Specification:Examples

C++ STL Template specifications Java util component specifications

http://doc.java.sun.com/DocWeb/api/java.util.Stack http://doc.java.sun.com/DocWeb/api/java.util.Queu

e

Questions for discussion Do they support information hiding? Do they support abstraction? Can they generalize? Is it possible to make them unambiguous?

http://doc.java.sun.com/DocWeb/api/java.util.Stack

http://doc.java.sun.com/DocWeb/api/java.util.Stack

http://doc.java.sun.com/DocWeb/api/java.util.Queue

http://doc.java.sun.com/DocWeb/api/java.util.Queue

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Informal Specifications Straightforward descriptions

Push pushes an object on a stack How much do they help?

Use of metaphors A Queue is like a line at a fast food restaurant Do they generalize?

Use of implementation details Push behaves like AddElement method on Vector Is this appropriate for a user-oriented cover story?

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Informal Specifications See Bertrand Meyer’s article on

Formal Specifications in IEEE Computer

Problems with even very carefully designed informal specs Contradiction Noise …

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Formal Interface Specification Communicates precisely the demands

and responsibilities to component users and developers

Allows for independent development of client and implementation components in parallel in a team environment

Minimizes integration costs

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Reasoning Benefits Formal Specifications make it

possible to formally reason about correctness of software

Such reasoning may be manual or mechanical (i.e. with automate support)

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Languages for Formal Specification ANNA (and SPARK) for Ada JML for Java Larch/C++ for C++ Spec# for C3 … Eiffel RESOLVE … VDM Z

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Specification Language Summary Some specification languages are designed

for particular programming languages

Some are general purpose

Some specification languages are integrated with programming constructs

A few additionally integrate the ability to perform formal mathematical reasoning

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Introduction to Mathematical Reasoning

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Motivating Example What does the following code do to

Integer I, where Foo1 and Bar1 are functions that modify their argument?

I = Foo1(I);I = Bar1(I);

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Motivating Example Or, what does this code do to

integers I and J?

I = Foo2(I,J);J = Bar2(I,J);I = Bar2(I,J);

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Motivating Example Now, what does this code do to Integer I?

I = Next(I);I = Prev(I);

How sure are we?

Have to account for bounds in our analysis

Summary: … Need formal descriptions beyond just names

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Motivating Example What does this code do to Integers I

and J?

I = Sum (I,J);J = Difference (I,J);I = Difference (I,J);

How sure are we?

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Specification of Integer Operations Think of ints as integers in math

Constraints, for all Integers I: Min_Int <= I <= Max_Int

Operation Next (I:Integer): Integer; requires I < Max_Int; ensures Next = I + 1;

Operation Prev (I:Integer): Integer; requires I > Min_Int; ensures Prev = I – 1;

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Specification of Integer Operations Can parameter values change?

Depending on the language Depending on how parameters are passed in

Need to make it clear with a specification whether or not a parameter can be modified

Operation Next (preserves I: Integer): Integer;requires I < Max_Int;ensures Next = I + 1;

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Specification of Integer Operation

Operation Next (preserves I: Integer): Integer;requires I < Max_Int;ensures Next = I + 1;

Operation Next (I: Integer): Integer;requires I < Max_Int;ensures Next = I + 1;

Operation Increment (updates I: Integer): Integer;requires I < Max_Int;ensures I = #I + 1;

Ambiguous Specification

Clear Specification – I unchanged

Clear Specification – I modified

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Exercise Specify Decrement Operation

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Meaning of Specifications Requirements and guarantees

Requires clauses are preconditions Ensures clauses are postconditions

Callers are responsible for requirements Caller of Increment is responsible for making

sure input I < Max_Int

Guarantees hold only if callers meet their requirements

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Using a Specification A specification can be implemented various ways

Have to judge if code meets specification

Example – is the code correct? Spec

Operation Do_Nothing (updates I:Integer);requires …ensures I = #I;

CodeIncrement (I);Decrement (I);

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Comparing Specifications Are these two specifications the same?

Spec 1:Operation Do_Nothing (preserves I: Integer);

requires …

Spec 2:Operation Do_Nothing (updates I: Integer);

requires …ensures I = #I;

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Methods for Checking Correctness Testing

Tracing or Inspection

Mathematical Reasoning

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Mathematical Reasoning Goal: To prove correctness

Method: The rest of this presentation

Consequences: Can provide correctness on all valid

inputs Can show the absence of bugs

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Mathematical Reasoning:Example – Prove Correctness

Spec:Operation Do_Nothing (updates I:

Integer);requires I < Max_Int;ensures I = #I;

Code:Increment(I);Decrement(I);

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Assume Confirm0

Increment (I);1

Decrement (I);2 I2 = I0

Establish the goals in state-oriented terms using a table

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Assume Confirm0 I0 < Max_Int

and …Increment (I);

1Decrement (I);

2 I2 = I0

Assume the requires clause at the beginning (Why?)

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and …Increment (I);

1 I1 = I0 + 1Decrement (I);

2 I2 = I1 - 1 I2 = I0

Assume calls work as advertised

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Prove the goal(s) using assumptions

Prove I2 = I0 I2 = I1 -1 (assumption in State 1) = (I0 + 1) – 1 (assumption in

state 1) = I0 (simplification)

More proof needed …

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and …I0 < Max_Int

Increment (I);1 I1 = I0 + 1 I1 > Min_Int

Decrement (I);2 I2 – I1 - 1 I2 = I0

More assertions to be confirmed (Why?)

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Basics of Mathematical Reasoning Suppose you are verifying code for some operation P

Assume its required clause in state 0 Confirm its ensures clause at the end

Suppose that P calls Q Confirm the requires clause of Q in the state before Q is

called. Why? Because caller is responsible

Assume the ensures clause of Q in the state after Q. Why?

Because Q is assumed to work

Prove assertions to be confirmed

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Mathematical Reasoning:Example 2 – Prove Correctness

Spec:Operation Do_Nothing (updates I:

Integer);ensures I = #I;

Code:If (I < Max_Int()) then

Increment(I);Decrement(I);

end;

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These specs are the same

Spec:Operation Do_Nothing (updates I: Integer);ensures I = #I;

Spec:Operation Do_Nothing (restores I: Integer);

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Condition Assume Confirm0

If (I < Max_Int())1

Increment (I);2

Decrement (I);3

End;4 I4 = I0

Establish the goals in state-oriented terms using a table

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If (I < Max_Int())1 I0 < max_int

Increment (I);2 I0 < max_int

Decrement (I);3 I0 < max_int

End;4 I4 = I0

Establish the conditions

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If (I < Max_Int())1 I0 < max_int

Increment (I);2 I0 < max_int

Decrement (I);3 I0 < max_int

End;4.1

not(I0 < max_int)

I4 = I0 I4 = I0

4.2

I0 < max_int I4 = I3 I4 = I0

Establish sub-goals for different conditions

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If (I < Max_Int())1 I0 < max_int I1 = I0

Increment (I);2 I0 < max_int I2 = I1 + 1

Decrement (I);3 I0 < max_int I3 = I2 - 1

End;4.1

not(I0 < max_int)

I4 = I0 I4 = I0

4.2

I0 < max_int I4 = I3 I4 = I0

Fill in other assumptions and obligations as before

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Prove the subgoal(s)

4.1 Case: not(I0 < max_int) Prove I4 = I0 True from assumption

4.2 Case: (I0 < max_int) Prove I4 = I0

Prove: I3 = I0 (assumption in state 4) Prove: (I2 – 1) = I0 (assumption in state 3) …

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For the condition (I0 < max_int), additional proofs are needed

These proofs of assertion to be confirmed in States 1 and 2 are left as exercises.