Generalising Control Dependence 10th CREST Open Workshop Program - - PowerPoint PPT Presentation

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Generalising Control Dependence 10th CREST Open Workshop Program - - PowerPoint PPT Presentation

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Generalising Control Dependence 10th CREST Open Workshop Program Analysis and Slicing Sebastian Danicic Goldsmiths, University of London 25th January 2011 1 / 163 Co-Authors Richard W. Barraclough @UK PLC, Mark Harman Crest, University

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Generalising Control Dependence

10th CREST Open Workshop Program Analysis and Slicing Sebastian Danicic

Goldsmiths, University of London

25th January 2011

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Co-Authors

Richard W. Barraclough @UK PLC, Mark Harman Crest, University College London, UK ´ Akos Kiss University of Szeged, Hungary, Michael R. Laurence University of Sheffield, UK

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Control Dependence - a Brief History

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Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM

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Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM 1979 Weiser: PhD Thesis First use of Control Dependence in Slicing - although called something else.

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Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM 1979 Weiser: PhD Thesis First use of Control Dependence in Slicing - although called something else. 1987 J. Ferrante, K. J. Ottenstein, J. D. Warren, The program dependence graph and its use in optimization, TOPLAS. First to use the term Control Dependence

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Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM 1979 Weiser: PhD Thesis First use of Control Dependence in Slicing - although called something else. 1987 J. Ferrante, K. J. Ottenstein, J. D. Warren, The program dependence graph and its use in optimization, TOPLAS. First to use the term Control Dependence 1990 A. Podgurski, L. Clarke, A formal model of program dependences and its implications for software testing, debugging, and maintenance, TSE.

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Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM 1979 Weiser: PhD Thesis First use of Control Dependence in Slicing - although called something else. 1987 J. Ferrante, K. J. Ottenstein, J. D. Warren, The program dependence graph and its use in optimization, TOPLAS. First to use the term Control Dependence 1990 A. Podgurski, L. Clarke, A formal model of program dependences and its implications for software testing, debugging, and maintenance, TSE. 1996 G. Bilardi, K. Pingali, A framework for generalized control dependence, in: PLDI.

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SLIDE 9

Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM 1979 Weiser: PhD Thesis First use of Control Dependence in Slicing - although called something else. 1987 J. Ferrante, K. J. Ottenstein, J. D. Warren, The program dependence graph and its use in optimization, TOPLAS. First to use the term Control Dependence 1990 A. Podgurski, L. Clarke, A formal model of program dependences and its implications for software testing, debugging, and maintenance, TSE. 1996 G. Bilardi, K. Pingali, A framework for generalized control dependence, in: PLDI. 2007 V. P. Ranganath et al., A new foundation for control dependence and slicing for modern program structures, TOPLAS.

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Control Dependence - a Brief History

1977 D. E. Denning, P. J. Denning,Certification of programs for secure information flow Communications of the ACM 1979 Weiser: PhD Thesis First use of Control Dependence in Slicing - although called something else. 1987 J. Ferrante, K. J. Ottenstein, J. D. Warren, The program dependence graph and its use in optimization, TOPLAS. First to use the term Control Dependence 1990 A. Podgurski, L. Clarke, A formal model of program dependences and its implications for software testing, debugging, and maintenance, TSE. 1996 G. Bilardi, K. Pingali, A framework for generalized control dependence, in: PLDI. 2007 V. P. Ranganath et al., A new foundation for control dependence and slicing for modern program structures, TOPLAS. 2008 T. Amtoft, Slicing for modern program structures: a theory for eliminating irrelevant loops, IPL.

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Traditional (Ferrante et al.) Control Dependence

Node p controls node v if there is a path from p to end which does not pass through v but there is a successor of p from which all paths to end go through v.

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Traditional (Ferrante et al.) Control Dependence

Which nodes does p control? Node p controls node v if there is a path from p to end which does not pass through v but there is a successor of p from which all paths to end go through v.

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Traditional (Ferrante et al.) Control Dependence

Which nodes does p control? {q, v4, v5}, Node p controls node v if there is a path from p to end which does not pass through v but there is a successor of p from which all paths to end go through v.

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Traditional (Ferrante et al.) Control Dependence

Which nodes does p control? {q, v4, v5}, but not v3. Node p controls node v if there is a path from p to end which does not pass through v but there is a successor of p from which all paths to end go through v.

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Traditional (Ferrante et al.) Control Dependence

Which nodes does p control? {q, v4, v5}, but not v3. q controls v3. Node p controls node v if there is a path from p to end which does not pass through v but there is a successor of p from which all paths to end go through v.

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Traditional (Ferrante et al.) Control Dependence

Which nodes does p control? {q, v4, v5}, but not v3. q controls v3. So p transitively controls v3. Node p controls node v if there is a path from p to end which does not pass through v but there is a successor of p from which all paths to end go through v.

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Slicing - Data and Control Dependence

Slice at v5.

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 18 / 163

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 2 v5 is data dependent on v2. 19 / 163

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 2 v5 is data dependent on v2. 3 v5 is control dependent on p. 20 / 163

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 2 v5 is data dependent on v2. 3 v5 is control dependent on p. 4 p is data dependent on v1. 21 / 163

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 2 v5 is data dependent on v2. 3 v5 is control dependent on p. 4 p is data dependent on v1. 5 The set of red nodes is closed under

control and data dependence

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 2 v5 is data dependent on v2. 3 v5 is control dependent on p. 4 p is data dependent on v1. 5 The set of red nodes is closed under

control and data dependence

6 Remove the non-red nodes. 23 / 163

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Slicing - Data and Control Dependence

Slice at v5.

1 First add start and end. 2 v5 is data dependent on v2. 3 v5 is control dependent on p. 4 p is data dependent on v1. 5 The set of red nodes is closed under

control and data dependence

6 Remove the non-red nodes. 7 Finally, ‘rewire’ the graph. 24 / 163

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Rewiring

To rewire, add an edge between two red nodes if there is a path with no intervening red nodes. We label it with the same label as the initial edge.

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The Induced Graph

To rewire, add an edge between two red nodes if there is a path with no intervening red nodes. We label it with the same label as the initial edge.

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Slicing

So slicing involves computing a set closed under control and data and then building the induced graph.

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Slicing

Notice: non-termination may not be preserved.

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Slicing

Notice: non-termination may not be preserved. This is because traditional control dependence is ‘non-termination insensitve’.

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Slicing

Notice: non-termination may not be preserved. This is because traditional control dependence is ‘non-termination insensitve’. We prefer to call it weak.

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Slicing

What if you want slices to preserve non-termination?

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Slicing

What if you want slices to preserve non-termination? We need q to be included too.

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Slicing

but q does not control anything using the traditional definitions of Ferrante and Ottenstein (1987) and previously Weiser (1981).

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Slicing

Podgurski and Clarke (1990) introduced a form of control dependence which solved this problem.

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Slicing

Podgurski and Clarke (1990) introduced a form of control dependence which solved this problem. q controls end using their definition.

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Slicing

the slice produced using Podgurski and Clarke’s control dependence preserves non-termination.

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Slicing

Theirs is a ‘non-termination sensitive’ or as we prefer, strong form of control dependence.

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Two Forms of Slice

The weak slice and the strong slice

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Slicing

but Podgurski and Clarke’s definition only works if end is reachable from every node.

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Slicing

but Podgurski and Clarke’s definition only works if end is reachable from every node. This is not the case in reactive systems.

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Slicing Reactive Systems

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Slicing Reactive Systems

In reactive systems we have intentionally non-terminating programs.

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Slicing Reactive Systems

In reactive systems we have intentionally non-terminating programs. Here we have a ‘deliberate’ infinite loop.

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Slicing Reactive Systems

In reactive systems we have intentionally non-terminating programs. Here we have a ‘deliberate’ infinite loop. This is a problem.

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Slicing Reactive Systems

The red set is closed under traditional control dependence and also under Podgurski and Clarke’s control dependence.

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Slicing Reactive Systems

The red set is closed under traditional control dependence and also under Podgurski and Clarke’s control dependence. But ...

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Slicing Reactive Systems

The induced graph isn’t even a legal CFG. v2 is a non-predicate of out degree greater than one.

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Slicing Reactive Systems

Ranganath et al. (2007) noticed that we need new forms of control dependence to solve this problem.

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Slicing Reactive Systems

They introduced NTSCD − − − − − → and DOD − − − → which produced strong slices for reactive systems. (A generalisation of Podgurski and Clarke’s definition).

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Slicing Reactive Systems

Later Amtoft (2008) produced WOD − − − → which gives rise to weak slices of reactive systems. (A generalisation of Ferrante et al.’s definition).

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Contributions of our Work

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007)

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Contributions of our Work

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way?

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Contributions of our Work

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way? yes!

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Contributions of our Work

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way? yes! Are there underlying semantic properties captured by all these different forms of control dependence?

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Contributions of our Work

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way? yes! Are there underlying semantic properties captured by all these different forms of control dependence? yes!

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Categorisation of the Different Forms of Control Dependence

Weak (Non-termination sensitive):

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

WOD

− − − → (Amtoft 2007)

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Categorisation of the Different Forms of Control Dependence

Weak (Non-termination sensitive):

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

WOD

− − − → (Amtoft 2007) Strong (Non-termination sensitive):

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

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Weak Commitment-Closedness

We do not give yet another definition of control dependence.

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Weak Commitment-Closedness

We do not give yet another definition of control dependence. Instead we give a property of sets closed under non-termination insensitive control dependence.

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Weak Commitment-Closedness

We do not give yet another definition of control dependence. Instead we give a property of sets closed under non-termination insensitive control dependence. The sets are Weak commitment-closed

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Weak Commitment-Closedness

We do not give yet another definition of control dependence. Instead we give a property of sets closed under non-termination insensitive control dependence. The sets are Weak commitment-closed This definition works for all directed graphs and is hence more general.

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Definition: S-Weakly Committing Nodes

A node is S-weakly committing if on every path from it we reach the same element of S first.

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Definition: S-Weakly Committing Nodes

A node is S-weakly committing if on every path from it we reach the same element of S first. Trivially, all elements of S are S-weakly committing.

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Definition: S-Weakly Committing Nodes

A node is S-weakly committing if on every path from it we reach the same element of S first. Trivially, all elements of S are S-weakly

  • committing. v1 is S-weakly committing, since we always reach v2 first.

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Definition: S-Weakly Committing Nodes

A node is S-weakly committing if on every path from it we reach the same element of S first. Trivially, all elements of S are S-weakly

  • committing. v1 is S-weakly committing, since we always reach v2 first. So

is v4.

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SLIDE 66

Definition: S-Weakly Committing Nodes

A node is S-weakly committing if on every path from it we reach the same element of S first. Trivially, all elements of S are S-weakly

  • committing. v1 is S-weakly committing, since we always reach v2 first. So

is v4. Nodes p and q are not weakly committing.

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Definition: Weakly Commitment-closed Sets

A set S is weakly commitment-closed if all nodes not in S are S-weakly committing.

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Definition: Weakly Commitment-closed Sets

A set S is weakly commitment-closed if all nodes not in S are S-weakly committing. This S is not weakly commitment-closed.

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Definition: Weakly Commitment-closed Sets

A set S is weakly commitment-closed if all nodes not in S are S-weakly committing. This S is not weakly commitment-closed. Now it is!

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Weakly Commitment-closed Sets in Reactive Systems

So let’s see how it works for reactive systems.

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Weakly Commitment-closed Sets in Reactive Systems

Which nodes are S-weakly committing?

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Weakly Commitment-closed Sets in Reactive Systems

Which nodes are S-weakly committing? v1, q and v4.

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Weakly Commitment-closed Sets in Reactive Systems

Which nodes are S-weakly committing? v1, q and v4. But not p. So S is not weak commitment-closed.

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Weakly Commitment-closed Sets in Reactive Systems

Which nodes are S-weakly committing? v1, q and v4. But not p. So S is not weak commitment-closed. So the induced graph is bad.

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Weakly Commitment-closed Sets in Reactive Systems

Now S is weakly commitment-closed!

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Weakly Commitment-closed Sets in Reactive Systems

Now S is weak commitment-closed! So the induced graph is good.

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Theorem 1: Soundness and Completeness of WCC

For each weak form of control dependence c in the literature, a set S is closed under c if and only if S is weakly commitment-closed.

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Generality of WCC

The beauty of weak commitment-closedness is that there is no need to consider special cases considered by previous authors. It works for them all.

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Generality of WCC

Using Weak Commitment-Closedness, things like end reachability are

  • irrelevant. It ‘works’ for all directed graphs.

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Algorithm for WCC

We have an algorithm O(n3)log(n) which given any node set V , computes the minimal weakly commitment closed set containing V .

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Using WCC

Because of Theorem 1, this algorithm can be used in all cases instead of the weak forms of control dependence in the literature.

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Traditional Slicing using Weakly Commitment-closed Sets

So in traditional slicing, given a slicing criterion V ′ we must find the minimal weakly commitment closed set containing V ′.

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Traditional Slicing using Weakly Commitment-closed Sets

So in traditional slicing, given a slicing criterion V ′ we must find the minimal weakly commitment closed set containing V ′. We have an O(n3)log(n) algorithm for this. This is the same as for Amtoft’s WOD − − − →.

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Traditional Slicing using Weakly Commitment-closed Sets

So in traditional slicing, given a slicing criterion V ′ we must find the minimal weakly commitment closed set containing V ′. We have an O(n3)log(n) algorithm for this. This is the same as for Amtoft’s WOD − − − →. We believe it can be improved to O(n3).

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Another Example

This set is weakly commitment-closed.

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Another Example

This set is weakly commitment-closed. What is the induced graph?

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Another Example

This set is weakly commitment-closed. This is the induced graph.

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Another Example

Any comments regarding non-termination?

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WCC does not preserve non-termination

It certainly does not preserve non-termination.

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SLIDE 90

WCC does not preserve non-termination

It certainly does not preserve non-termination. But that’s not surprising because this is weak commitment-closedness.

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We need Strong Commitment Closedness for that.

To preserve non-termination we need strong commitment closedness.

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S-avoiding Nodes

A node is S-avoiding if no paths from it reach S.

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S-avoiding Nodes

A node is S-avoiding if no paths from it reach S. q, v3, v4 are S-avoiding .

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S-Strongly Committing Nodes

A node is S-strongly committing if it is S-weakly committing and all paths from it eventually reach S.

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S-Strongly Committing Nodes

A node is S-strongly committing if it is S-weakly committing and all paths from it eventually reach S. i.e. all paths from it reach the same element of S first.

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S-Strongly Committing Nodes

A node is S-strongly committing if it is S-weakly committing and all paths from it eventually reach S. i.e. all paths from it reach the same element of S first. v1 is S-strongly committing.

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SLIDE 97

Strong Commitment Closedness

S is strongly commitment-closed if all elements not in S are either S-avoiding or S-strongly committing.

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SLIDE 98

Strong Commitment Closedness

S is strongly commitment-closed if all elements not in S are either S-avoiding or S-strongly committing. p is neither S-avoiding nor S-strongly committing.

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SLIDE 99

Strong Commitment Closedness

S is strongly commitment-closed if all elements not in S are either S-avoiding or S-strongly committing. p is neither S-avoiding nor S-strongly committing. So S is not strongly commitment-closed.

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Strong Commitment Closedness

S is strongly commitment-closed if all elements not in S are either S-avoiding or S-strongly committing. p is neither S-avoiding nor S-strongly committing. So S is not strongly commitment-closed. Now it is!

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SLIDE 101

Graphs Induced from Strongly Commitment Closed Sets

So let’s look at this induced graph.

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SLIDE 102

Graphs Induced from Strongly Commitment Closed Sets

So let’s look at this induced graph.

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SLIDE 103

Incomplete Predicates

So let’s look at this induced graph. p is an ‘incomplete’ predicate.

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SLIDE 104

Interpreting Incomplete Predicates

So let’s look at this induced graph. p is an ‘incomplete’ predicate. How do we interpret this?

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SLIDE 105

Interpreting Incomplete Predicates

So let’s look at this induced graph. p is an ‘incomplete’ predicate. How do we interpret this? If p evaluates to T then we get silent non-termination.

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SLIDE 106

The Advantage of Incomplete Predicates

Using incomplete predicates for silent non-termination means that we don’t have to include ‘ghost’ control sinks that may introduce further unnecessary dependences.

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SLIDE 107

Theorem 2: Soundness and Completeness of SCC

For each form c of the strong forms of control dependence in the literature, S is closed under c if and only if S is strongly commitment-closed.

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SLIDE 108

Non-termination Sensitive Slicing using Strongly Commitment-closed Sets

So in Non-termination Sensitive Slicing slicing, given a slicing criterion V ′ we must find the minimal strongly commitment-closed set containing V ′.

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SLIDE 109

Non-termination Sensitive Slicing using Strongly Commitment-closed Sets

So in Non-termination Sensitive Slicing slicing, given a slicing criterion V ′ we must find the minimal strongly commitment-closed set containing V ′. Again, we have an O(n3)log(n) algorithm for this.

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SLIDE 110

Non-termination Sensitive Slicing using Strongly Commitment-closed Sets

So in Non-termination Sensitive Slicing slicing, given a slicing criterion V ′ we must find the minimal strongly commitment-closed set containing V ′. Again, we have an O(n3)log(n) algorithm for this. Again, we believe it can be improved to O(n3).

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SLIDE 111

Semantics?

What is the semantic relationship between a graph and the graphs induced by a weakly and strongly commitment-closed sets?

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SLIDE 112

Semantics Induced by Weakly Commitment-closed Sets

What is the semantic relationship between a graph and a graph induced by a weakly commitment-closed set?

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SLIDE 113

Walks

Walks are like paths where we also record whether the T or F branches were taken at the predicates.

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SLIDE 114

Examples of Walks

start, v1, v2

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Examples of Walks

start, v1, v2, (p, T)

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SLIDE 116

Examples of Walks

start, v1, v2, (p, T), (q, F), v4, v3, (q, T), v3

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SLIDE 117

Examples of Walks

start, v1, v2, (p, F), v5, end

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SLIDE 118

Walks of the Induced Graph

G1 G2 Let’s compare walks of the original graph with the walks of a graph induced by a weakly commitment closed set.

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SLIDE 119

Walks of the Induced Graph

G1 start, v1, v2 G2 start, v2

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SLIDE 120

Walks of the Induced Graph

G1 start, v1, v2, (p, T) G2 start, v2, (p, T)

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SLIDE 121

Walks of the Induced Graph

G1 start, v1, v2, (p, T), (q, F), v4, v3, (q, T), v3 G2 start, v2, (p, T), v3, v3

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SLIDE 122

Walks of the Induced Graph

G1 start, v1, v2, (p, F), v5, end G2 start, v2, (p, F), v5, end

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SLIDE 123

Walks of Graphs Induced from WCC Sets

G1 G2 What is the relationship between the walks of G1 and the walks of G2?

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SLIDE 124

Weak Projections

G1 G2 Every walk of G1 when restricted to G2 is a walk of G2.

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SLIDE 125

Weak Projections

G1 G2 Every walk of G1 when restricted to G2 is a walk of G2. We say G2 is a weak projection of G1.

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SLIDE 126

Theorem 3: Semantics of WCC

G1 G2 The graph induced from V is a weak projection if and only if V is weakly commitment-closed.

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SLIDE 127

Result

G1 G2 Theorems 1 and 3 imply that sets closed under all weak forms of control dependence in the literature induce weak projections.

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SLIDE 128

Weak Projections

G1 G2 So weak projection captures semantically what all previous authors of definitions of weak control dependence wanted to achieve!

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SLIDE 129

Weak Control Dependence

G1 G2 So authors of future definitions should also prove their definitions satisfy this property!

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SLIDE 130

Strong Control Dependence

G1 G2 What about graphs induced from strongly commitment-closed sets?

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SLIDE 131

Maximal Walks

Maximal walks are those which are not a prefix of any other walk.

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SLIDE 132

Maximal Walks corresponding to termination

The only maximal walks that correspond to termination are those whose final element is end.

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SLIDE 133

Maximal Walks corresponding to non-termination

All other finite maximal walks are considered non-terminating.

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SLIDE 134

Walks of Graphs Induced from SCC sets

start, v1, v2, (p, T), (q, T), v3, (q, F), v4, v3 . . .

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SLIDE 135

Walks of Graphs Induced from SCC sets

start, v1, v2, (p, T), (q, T), v3, (q, F), v4, v3 . . . start, v2, (p, T)

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SLIDE 136

Walks of Graphs Induced from SCC sets

What is the relationship between the two?

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SLIDE 137

Remember Weak Projections

G2 is a weak projection of G1 means every walk of G1 when restricted to G2 is a walk of G2.

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SLIDE 138

Strong Projections

G2 is a strong projection of G1 means every maximal walk of G1 when restricted to G2 is a maximal walk of G2.

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SLIDE 139

Theorem 4: Semantics of SCC

The graph induced from V is a strong projection if and only if V is strongly commitment-closed.

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SLIDE 140

Result

Theorems 2 and 4 imply that sets closed under all strong forms of control dependence in the literature induce strong projections.

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SLIDE 141

Strong Projection

So, again, strong projection captures semantically what all previous authors of definitions of strong control dependence wanted to achieve.

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SLIDE 142

Strong Projections are Non-Termination Preserving

It follows that strong projections are non-termination preserving.

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SLIDE 143

Strong Projections are Non-Termination Preserving

From this it follows that strong projections are non-termination preserving (as required!).

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SLIDE 144

Strong Projections with end preserve both

Also notice, strong projections are weak projections.

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SLIDE 145

Strong Projections with end preserve both

Also notice, strong projections are weak projections. But not vice-versa.

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SLIDE 146

Strong Projections with end preserve both

If end is in the weak projection, then the weak projection preserves termination.

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SLIDE 147

Strong Projections with end preserve both

So if end is in the strong projection, then the strong projection preserves both termination and non-termination.

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SLIDE 148

Conclusion

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007)

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SLIDE 149

Conclusion

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way?

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SLIDE 150

Conclusion

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way? yes!

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SLIDE 151

Conclusion

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way? yes! Are there underlying semantic properties captured by all these different forms of control dependence?

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SLIDE 152

Conclusion

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

WOD

− − − → (Amtoft 2007) Can they be generalised in a nice high-level way? yes! Are there underlying semantic properties captured by all these different forms of control dependence? yes!

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SLIDE 153

Conclusion

Weak (Non-termination sensitive):

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

WOD

− − − → (Amtoft 2007)

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SLIDE 154

Conclusion

Weak (Non-termination sensitive):

W-controls

− − − − − − → (Weiser 1979)

F-controls

− − − − − − → (Ferrante and Ottenstein 1987)

WOD

− − − → (Amtoft 2007) Strong (Non-termination sensitive):

PC-weak

− − − − − → (Podgurski and Clarke 1990)

NTSCD

− − − − − → and DOD − − − → (Ranganath et al 2006)

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SLIDE 155

Conclusions

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SLIDE 156

Conclusions

We have generalised non-termination insensitive and sensitive control dependence by defining and giving algorithms for weak and strong commitment-closedness.

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SLIDE 157

Conclusions

We have generalised non-termination insensitive and sensitive control dependence by defining and giving algorithms for weak and strong commitment-closedness. Theorem 1: A set is closed under each of the weak forms of control dependence in the literature if and only if it is weakly commitment-closed.

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SLIDE 158

Conclusions

We have generalised non-termination insensitive and sensitive control dependence by defining and giving algorithms for weak and strong commitment-closedness. Theorem 1: A set is closed under each of the weak forms of control dependence in the literature if and only if it is weakly commitment-closed. Theorem 2: A set is closed under each of the strong forms of control dependence in the literature if and only if it is strongly commitment-closed.

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SLIDE 159

Conclusions

We have generalised non-termination insensitive and sensitive control dependence by defining and giving algorithms for weak and strong commitment-closedness. Theorem 1: A set is closed under each of the weak forms of control dependence in the literature if and only if it is weakly commitment-closed. Theorem 2: A set is closed under each of the strong forms of control dependence in the literature if and only if it is strongly commitment-closed. We defined semantic relations: weak and strong projections between graphs in terms of walks.

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SLIDE 160

Conclusions

We have generalised non-termination insensitive and sensitive control dependence by defining and giving algorithms for weak and strong commitment-closedness. Theorem 1: A set is closed under each of the weak forms of control dependence in the literature if and only if it is weakly commitment-closed. Theorem 2: A set is closed under each of the strong forms of control dependence in the literature if and only if it is strongly commitment-closed. We defined semantic relations: weak and strong projections between graphs in terms of walks. Theorem 3: The graph induced from V is a weak projection if and

  • nly if V is weakly commitment-closed.

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SLIDE 161

Conclusions

We have generalised non-termination insensitive and sensitive control dependence by defining and giving algorithms for weak and strong commitment-closedness. Theorem 1: A set is closed under each of the weak forms of control dependence in the literature if and only if it is weakly commitment-closed. Theorem 2: A set is closed under each of the strong forms of control dependence in the literature if and only if it is strongly commitment-closed. We defined semantic relations: weak and strong projections between graphs in terms of walks. Theorem 3: The graph induced from V is a weak projection if and

  • nly if V is weakly commitment-closed.

Theorem 4: The graph induced from V is a strong projection if and

  • nly if V is strongly commitment-closed.

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SLIDE 162

The End

Thanks for listening?

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SLIDE 163

The End

Thanks for listening? Any questions?

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