Interference freedom

Edsger W. Dijkstra introduced the principle of non-interference in EWD 117,

"Programming Considered as a Human Activity", written about 1965.{{cite web

|website=E. W. Dijkstra Archive

|publisher=University of Texas

|url=https://www.cs.utexas.edu/~EWD/ewd01xx/EWD117.PDF

|title=Programming Considered as a Human Activity}}

This principle states that:

The correctness of the whole can be established by taking into

account only the exterior specifications (abbreviated specs throughout) of the parts, and not their

interior construction. Dijkstra outlined the general steps in using this principle:

1. Give a complete spec of each individual part.
2. Check that the total problem is solved when program parts meeting their specs are available.
3. Construct the individual parts to satisfy their specs, but independent of one another and the context in which they will be used.

He gave several examples of this principle outside of programming. But

its use in programming is a main concern. For example, a programmer using a method (subroutine, function, etc.) should rely only on its spec to determine what it does and how to call it, and never on its implementation.

Program specs are written in Hoare logic, introduced by Sir Tony Hoare, as exemplified in the specs of processes {{math|S1}} and {{math|S2}}:

{{spaces|2|em}}{{math|{pre-S1}}}{{spaces|5.5|em}}{{math|{pre-S2}}}
{{spaces|2|em}}{{math|S1}}{{spaces|15|em}}{{math|S2}}

{{spaces|2|em}}{{math|{post-S1}}}{{spaces|3.6|em}}{{math|{pre-S2}}}

Meaning: If execution of {{math|Si}} in a state in which precondition {{math|pre-Si}} is true terminates, then upon termination, postcondition

{{math|post-Si}} is true.

Now consider concurrent programming with shared variables. The specs of two (or

more) processes {{math|S1}} and {{math|S2}} are given in terms of their pre- and post-conditions,

and we assume that implementations of {{math|S1}} and {{math|S2}} are given that satisfy their

specs. But when executing their implementations in parallel, since they share variables,

a race condition can occur; one process changes a shared variable to a value

that is not anticipated in the proof of the other process, so the other

process does not work as intended.

Thus, Dijkstra's Principle of non-interference is violated.

In her PhD thesis of 1975 in Computer Science, Cornell University, written under

advisor David Gries, Susan Owicki developed the notion of interference freedom.

If processes {{math|S1}} and {{math|S2}} satisfy interference freedom, then their parallel execution will work as planned.

Dijkstra called this work the first significant step toward applying Hoare logic to concurrent

processes.{{Cite book

|title = Selected Writings on Computing: A Personal Perspective

|last = Dijkstra

|first = Edsger W.

|publisher = Springer-Verlag

|year = 1982

|doi =

|series = Monographs in Computer Science

|isbn = 0387906525

|chapter = EWD 554: A personal summary of the Gries-Owicki Theory

|pages = 188–199

}}

To simplify discussions, we restrict attention to only two concurrent processes,

although Owicki-Gries allows more.

Owicki-Gries introduced the proof outline

for a Hoare triple {{math|{P}S{Q}}}. It contains all details needed for a proof of correctness of

{{math|{P}S{Q}}} using the axioms and inference rules of Hoare logic. (This work uses the assignment statement {{math|x:}}{{math|{{=}} e}}, {{math|if-then}} and {{math|if-then-else}} statements, and the {{math|while}} loop.) Hoare alluded to proof outlines in his early work; for interference freedom, it had to be formalized.

A proof outline for {{math|{P}S{Q}}} begins with precondition {{math|P}} and ends with postcondition {{math|Q}}.

Two assertions within braces { and } appearing next to each other indicates that the first must imply the second.

Example: A proof outline for {{math|{P} S {Q}}}

where {{math|S}} is:

{{math|x}}{{math|:}}{{math|{{=}} a; if e then S1 else S2}}

{{spaces|2|em}}{{math|{P}}}

{{spaces|2|em}}{{math|{P1[x/a]}}}

{{spaces|2|em}}{{math|x}}{{math|:}}{{math|{{=}} a;}}

{{spaces|2|em}}{{math|{P1}}}

{{spaces|2|em}}{{math|if e then {P1 ∧ e}}}

{{spaces|17|em}}{{math|S1}}

{{spaces|17|em}}{{math|{Q1}}}

{{spaces|8|em}}{{math|else {P1 ∧ ¬ e}}}

{{spaces|17|em}}{{math|S2}}

{{spaces|17|em}}{{math|{Q1}}}

{{spaces|2|em}}{{math|{Q1}}}

{{spaces|2|em}}{{math|{Q}}}

{{math|P ⇒ P1[x/a]}} must hold,

where {{math|P1[x/a]}} stands for {{math|P1}} with every

occurrence of {{math|x}} replaced by {{math|a}}. (In this example, {{math|S1}} and {{math|S2}} are basic statements, like an assignment statement, skip, or an await statement.)

Each statement {{math|T}} in the proof outline is preceded by a precondition {{math|pre-T}} and followed by a postcondition {{math|post-T}}, and

{{math|{pre-T}T{post-T}}} must be provable using some axiom or inference rule of Hoare logic. Thus, the proof outline contains all the information necessary to prove that {{math|{P}S{Q}}} is correct.

Now consider two processes {{math|S1}} and {{math|S2}} executing in parallel, and their specs:

{{spaces|2|em}}{{math|{pre-S1}}}{{spaces|5.5|em}}{{math|{pre-S2}}}
{{spaces|2|em}}{{math|S1}}{{spaces|15|em}}{{math|S2}}

{{spaces|2|em}}{{math|{post-S1}}}{{spaces|3.6|em}}{{math|{post-S2}}}

Proving that they work suitably in parallel will require restricting them as follows.

Each expression {{math|E}} in {{math|S1}} or {{math|S2}} may refer to at most one variable {{math|y}} that can be

changed by the other process while {{math|E}} is being evaluated, and {{math|E}} may refer to {{math|y}} at most once.

A similar restriction holds for assignment statements {{math|x:}}{{math|{{=}} E}}.

With this convention, the only indivisible action need be the memory reference. For example, suppose process {{math|S1}} references variable {{math|y}} while {{math|S2}} changes {{math|y}}. The value {{math|S1}} receives for {{math|y}}

must be the value before or after {{math|S2}} changes {{math|y}}, and not some spurious in-between value.

Definition of Interference-free

The important innovation of Owicki-Gries was to define what it means for a statement {{math|T}} not to interfere with the proof of {{math|{P}S{Q}}}. If execution of {{math|T}} cannot falsify any assertion given in the proof

outline of {{math|{P}S{Q}}}, then that proof still holds even in the face of concurrent execution of {{math|S}} and {{math|T}}.

Definition. Statement {{math|T}} with precondition {{math|pre-T}}

does not interfere with the proof of {{math|{P}S{Q}}} if two conditions hold:

(1) {{math|{Q ∧ pre-T} T {Q}}}

(2) Let {{math|S'}} be any statement within {{math|S}} but not within an {{math|await}} statement (see later section).

Then {{math|{pre-S' ∧ pre-T} T {pre-S'}}}.

Read the last Hoare triple like this: If the state is such that both {{math|T}} and {{math|S'}} can

be executed, then execution of {{math|T}} is not going to falsify {{math|pre-S'}}.

Definition. Proof outlines for {{math|{P1}S1{Q1}}} and {{math|{P2}S2{Q2}}} are interference-free if the following holds.

Let {{math|T}} be an {{math|await}} or assignment statement (that does not appear in an {{math|await}}) of process {{math|S1}}.

Then {{math|T}} does not interfere with the proof of {{math|{P2}S2{Q2}}}.

Similarly for {{math|T}} of process {{math|S2}} and {{math|{P1}S1{Q1}}}.

Two statements were introduced to deal with concurrency. Execution of the statement {{math|cobegin S1 // S2 coend}} executes {{math|S1}} and {{math|S2}} in parallel. It terminates when both {{math|S1}} and {{math|S2}} have terminated.

Execution of the {{math|await}} statement {{math|await B then S}} is delayed until condition {{math|B}} is true.

Then, statement {{math|S}} is executed as an indivisible action—evaluation of {{math|B}} is part of that indivisible action. If two processes are waiting for the same condition {{math|B}}, when it becomes true, one of them continues waiting while the other proceeds.

The {{math|await}} statement cannot be implemented efficiently and is not proposed to be inserted into the programming language. Rather it provides a means of representing several standard primitives such as semaphores—first express the semaphore operations as {{math|awaits}}, then apply the techniques described here.

Inference rules for {{math|await}} and {{math|cobegin}} are:

{{math|await}}

{{math|{{sfrac|{P ∧ B} S {Q}|{P} await B then S {Q} }}}}

{{math|cobegin}}

{{math|{{sfrac|{P1} S1 {Q1}, {P2} S2 {Q2}
interference-free|{P1∧P2} cobegin S1//S2 coend {Q1∧Q2} }}}}

An auxiliary variable does not occur in the program but is introduced in the proof of correctness to make reasoning simpler —or even possible. Auxiliary variables are used only in assignments to auxiliary variables, so their introduction neither alters the

program for any input nor affects the values of program variables. Typically,

they are used either as program counters or to record histories of a computation.

Definition. Let {{math|AV}} be a set of variables that appear in {{math|S'}} only in assignments

{{math|x:}}{{math|{{=}} E}}, where {{math|x}} is in {{math|AV}}. Then {{math|AV}} is an auxiliary variable set for {{math|S'}}.

Since a set {{math|AV}} of auxiliary variables are used only in assignments to variables in {{math|AV}}, deleting all assignments to them doesn't change the program's correctness, and we have the inference rule {{math|AV}} elimination:

{{spaces|2|em}}{{math|{{sfrac|{P} S' {Q}|{P} S {Q} }}}}

{{math|AV}} is an auxiliary variable set for {{math|S'}}. The variables in {{math|AV}} do not occur in {{math|P}} or {{math|Q}}. {{math|S}} is obtained from {{math|S'}} by deleting all assignments to the variables in {{math|AV}}.

Instead of using auxiliary variables, one can introduce a program counter into the proof system,

but that adds complexity to the proof system.

Note:

Apt {{Cite journal

|last1=Apt

|first1=Krzysztof R.

|title=Recursive assertions and parallel programs

|journal=Acta Informatica

|volume=15

|pages=219–232

|date=June 1981

|issue=3

|url=https://doi.org/10.1007/BF00289262

|doi=10.1007/BF00289262

|s2cid=42470032

}} discusses the Owicki-Gries logic in the context of recursive assertions, that is,

effectively computable assertions. He proves that all the assertions in proof outlines

can be recursive, but that this is no longer the case if auxiliary variables are used only as program counters and not to record histories of computation.

Lamport, in his similar work, uses assertions about token positions instead of auxiliary variables, where a token on an edge of a flow chart is akin to a program counter. There is no notion of a history variable.

This indicates that Owicki-Gries and Lamport's approach are not equivalent

when restricted to recursive assertions.

Owicki-Gries deals mainly with partial correctness:

{{math|{P} S {Q}}} means: If {{math|S}} executed in

a state in which {{math|P}} is true terminates, then {{math|Q}} is true of the state upon termination.

However, Owicki-Gries also gives some practical techniques that use information obtained

from a partial correctness proof to derive other correctness properties,

including freedom from deadlock, program termination, and mutual exclusion.

A program is in deadlock if all processes that have not terminated are executing

{{math|await}} statements

and none can proceed because their {{math|await}} conditions are false.

Owicki-Gries provides conditions under which deadlock cannot occur.

Owicki-Gries presents an inference rule for total correctness of the while loop. It uses

a bound function that decreases with each iteration and is positive as long as the

loop condition is true. Apt et al {{Cite book

|last1=Apt

|first1=Krzysztof R.

|author-link=

|last2=de Boer

|first2=Frank S.

|author-link2=

|last3=Olderog

|first3=Ernst-Rüdiger

|author-link3=

|chapter=Proving termination of parallel programs

|editor1-last=Gries

|editor1-first=D.

|editor2-last=Feijen

|editor2-first=W.H.J.

|editor3-last=van Gasteren

|editor3-first=A.J.M.

|editor4-last=Misra

|editor4-first=J.

|date=1990

|pages=0–6

|title=Beauty is Our Business

|series=Monographs in Computer Science

|publisher=Springer Verlag

|url=https://link.springer.com/book/10.1007/978-1-4612-4476-9

|location=New York|doi=10.1007/978-1-4612-4476-9

|isbn=978-1-4612-8792-6

|s2cid=24379938

}} show that this new inference rule does not satisfy interference freedom. The fact that the bound function is positive as long as the loop condition is true was not

included in an interference test. They show two ways to rectify this mistake.

Consider the statement:

{{indent|7}}{{math|{x{{=}}0} }}

{{indent|7}}{{math|cobegin await true then x:}}{{math|{{=}} x+1}}

{{indent|18}}// {{math|await true then x:}}{{math|{{=}} x+2}}

{{indent|7}}{{math|coend}}

{{indent|7}}{{math|{x{{=}}3} }}

The proof outline for it:

{{spaces|0|em}}{{math|{x{{=}}0} }}

{{spaces|0|em}}{{math|S: cobegin}}

{{spaces|7|em}}{{math|{x{{=}}0} }}

{{spaces|7|em}}{{math|{x{{=}}0 ∨ x{{=}}2} }}

{{spaces|7|em}}{{math|S1: await true then }}{{math|x}}{{math|:}}{{math|{{=}}}} {{math|x+1}}

{{spaces|7|em}}{{math|{Q1: x{{=}}1 ∨ x{{=}}3} }}

{{spaces|5|em}}//

{{spaces|7|em}}{{math|{x{{=}}0} }}

{{spaces|7|em}}{{math|{x{{=}}0 ∨ x{{=}}1} }}

{{spaces|7|em}}{{math|S2: await true then }}{{math|x}}{{math|:}}{{math|{{=}}}} {{math|x+2}}

{{spaces|7|em}}{{math|{Q2: x{{=}}2 ∨ x{{=}}3} }}

{{spaces|5|em}}{{math|coend}}

{{spaces|0|em}}{{math|{(x{{=}}1 ∨ x{{=}}3) ∧ (x{{=}}2 ∨ x{{=}}3)} }}

{{spaces|0|em}}{{math|{x{{=}}3} }}

Proving that {{math|S1}} does not interfere with the proof of {{math|S2}} requires proving two Hoare triples:

(1) {{math|{(x{{=}}0 ∨ x{{=}}2) ∧ (x{{=}}0 ∨ x{{=}}1} S1 {x{{=}}0 ∨ x{{=}}1} }}

{{indent|0}}(2) {{math|{(x{{=}}0 ∨ x{{=}}2) ∧ (x{{=}}2 ∨ x{{=}}3} S1 {x{{=}}2 ∨ x{{=}}3} }}

The precondition of (1) reduces to {{math|x{{=}}0}} and the precondition of (2) reduces to {{math|x{{=}}2}}. From this, it is easy to see that these Hoare triples hold. Two similar Hoare triples are required to show that {{math|S2}} does not interfere with the proof of {{math|S1}}.

Suppose {{math|S1}} is changed from the {{math|await}} statement to the assignment

{{math|x:}}{{math|{{=}} x+1}}. Then the proof outline does not satisfy the requirements, because the assignment contains two occurrences of shared variable {{math|x}}. Indeed, the value of {{math|x}} after execution of the {{math|cobegin}} statement could be 2 or 3.

Suppose {{math|S1}} is changed to the {{math|await}} statement

{{math|await true then x:}}{{math|{{=}} x+2}}, so it is the same as {{math|S2}}.

After execution of {{math|S}}, {{math|x}} should be 4. To prove this, because the two assignments are the same, two auxiliary variables are needed, one to indicate whether {{math|S1}} has been executed; the other, whether {{math|S2}} has been executed.

We leave the change in the proof outline to the reader.

A. Findpos. Write a program that finds the first positive element of an array (if there is one). One process checks all array elements at even positions of the array and terminates when it finds a positive value or when none is found. Similarly, the other process checks array elements at odd positions of the array. Thus, this example deals with while loops. It also has no {{math|await}} statements.

This example comes from Barry K. Rosen.{{cite journal

|last1= Rosen

|first1= Barry K

|date= 1976

|title= Correctness of parallel programs: The Church-Rosser Approach.

|journal= Theoretical Computer Science

|volume= 2

|issue= 2

|pages= 183–207

|doi= 10.1016/0304-3975(76)90032-3|doi-access= free

}}

The solution in Owicki-Gries, complete with program, proof outline, and discussion of interference freedom, takes less than two pages. Interference freedom is quite easy to check, since there is only one shared variable. In contrast, Rosen's

article

uses {{math|Findpos}} as the single, running example in this 24 page paper.

An outline of both processes in a general environment:

{{spaces|0|em}}{{math|cobegin producer: ...}}

{{spaces|5|em}}{{math|await in-out < N then skip;}}

{{spaces|5|em}}{{math|add: b[in mod N]:{{=}} next value;}}

{{spaces|5|em}}{{math|markin:}} {{math|in:}}{{math|{{=}}}} {{math|in+1;}}

{{spaces|5|em}}{{math|...}}

{{spaces|2|em}}//

{{spaces|5|em}}{{math|consumer: ...}}

{{spaces|5|em}}{{math|await in-out > 0 then skip;}}

{{spaces|5|em}}{{math|remove:}} {{math|this value:{{=}}

{{spaces|7|em}} b[out mod N];}}

{{spaces|5|em}}{{math|markout:}} {{math|out:}}{{math|{{=}}}} {{math|out+1;}}

{{spaces|0|em}}{{math|coend}}

{{spaces|0|em}}{{math|...}}

B. Bounded buffer consumer/producer problem.

A producer process generates values and puts them into bounded buffer {{math|b}} of size {{math|N}};

a consumer process removes them.

They proceed at variable rates.

The producer must wait if buffer {{math|b}} is full; the consumer must wait if buffer {{math|b}} is empty.

In Owicki-Gries, a solution in a general environment is shown; it is then embedded in a program that copies an array {{math|c[1..M]}} into an array {{math|d[1..M]}}.

This example exhibits a principle to reduce interference checks to a minimum: Place as much as possible in an assertion that is invariantly true everywhere in both processes. In this case the assertion is the definition of the bounded buffer and bounds on variables that indicate how many values have been added to and removed from the buffer. Besides buffer {{math|b}} itself, two shared variables record the number {{math|in}} of values added to the buffer and the number {{math|out}} removed from the buffer.

C. Implementing semaphores.

In his article on the THE multiprogramming system, Dijkstra introduces the semaphore {{math|sem}} as a synchronization primitive:

{{math|sem}} is an integer variable that can be referenced in only two ways, shown below; each is an indivisible operation:

1. {{math|P(sem)}}: Decrease {{math|sem}} by 1. If now {{math|sem < 0}}, suspend the process and put it on a list of suspended processes associated with {{math|sem}}.

2. {{math|V(sem)}}: Increase {{math|sem}} by 1. If now {{math|sem ≤ 0}}, remove one of the processes from the list of suspended processes associated with {{math|sem}}, so its dynamic progress is again permissible.

The implementation of {{math|P}} and {{math|V}} using {{math|await}} statements is:

{{spaces|0|em}}{{math|P(sem):}}

{{spaces|6|em}}{{math|await true then}}

{{spaces|10|em}}{{math|begin sem:}}{{math|{{=}}}}{{math| sem-1;}}

{{spaces|14|em}}{{math|if sem < 0 then}}

{{spaces|16|em}}{{math|w[this process]:}}{{math|{{=}}}}{{math| true}}

{{spaces|10|em}}{{math|end;}}

{{spaces|10|em}}{{math|await ¬w[this process]}}

{{spaces|16|em}}{{math|then skip}}

{{spaces|0|em}}

{{spaces|0|em}}{{math|V(sem):}}

{{spaces|6|em}}{{math|await true then}}

{{spaces|10|em}}{{math|begin sem:}}{{math|{{=}}}}{{math| sem+1;}}

{{spaces|14|em}}{{math|if sem ≤ 0 then}}

{{spaces|17|em}}{{math|begin choose p such}}

{{spaces|35|em}}{{math|that w[p];}}

{{spaces|28|em}}{{math|w[p]:}}{{math|{{=}}}}{{math| false}}

{{spaces|17|em}}{{math|end}}

{{spaces|10|em}}{{math|end}}

Here, {{math|w}} is an array of processes that are waiting because they have been suspended;

initially, {{math|w[p]}}{{math|{{=}}}}{{math|false}} for every process {{math|p}}.

One could change the implementation to always waken the longest suspended process.

D. On-the-fly garbage collection. At the 1975 Summer School Marktoberdorf, Dijkstra discussed an on-the-fly garbage collector as an exercise in understanding parallelism. The data structure used in a conventional implementation of LISP is a directed graph in which each node has at most two outgoing edges, either of which may be missing: an outgoing left edge and an outgoing right edge. All nodes of the graph must be reachable from a known root. Changing a node may result in unreachable nodes, which can no longer be used and are called garbage. An on-the-fly garbage collector has two processes: the program itself and a garbage collector, whose task is to identify garbage nodes and put them on a free list so that they can be used again.

Gries felt that interference freedom could be used to prove the on-the-fly garbage collector correct.

With help from Dijkstra and Hoare, he was able to give a presentation at the end of the Summer School,

which resulted in an article in CACM.{{Cite journal

|last1=Gries

|first1=David

|authorlink1=David Gries

|title=An exercise in proving parallel programs correct

|doi=10.1145/359897.359903

|journal=Communications of the ACM

|volume=20

|issue=12

|pages=921–930

|date=December 1977

|s2cid=3202388

|doi-access=free

}}

E. Verification of readers/writers solution with semaphores. Courtois et al{{Cite journal

|last1=Courtois

|first1=P.J.

|last2=Heymans

|first2=F.

|last3=Parnas

|first3=D.L.

|authorlink3=David Parnas

|title=Concurrent control with "readers" and "writers"

|doi=10.1145/362759.362813

|journal=Communications of the ACM

|volume=14

|issue=10

|pages=667–668

|date=October 1971

|s2cid=7540747

|doi-access=free

}} use semaphores to give two versions of the readers/writers problem, without proof. Write operations block both reads and writes, but read operations can occur in parallel. Owicki{{cite tech report

|last=Owicki

|first=Susan

|authorlink=Susan Owicki

|title=Verifying concurrent programs with shared data classes

|number=147

|publisher=Digital Systems Laboratory, Stanford University

|date=August 1977

|url=http://i.stanford.edu/pub/cstr/reports/csl/tr/77/147/CSL-TR-77-147.pdf

|access-date=2022-07-08}} provides a proof.

F. Peterson's algorithm, a solution to the 2-process mutual exclusion

problem, was published by Peterson in a 2-page article.{{Cite journal

|last1=Peterson

|first1=Gary L.

|title=Myths about the mutual exclusion problem

|journal=IPL

|volume=12

|issue=3

|pages=115–116

|date=June 1981

|url=https://dx.doi.org/10.1016/0020-0190%2881%2990106-X

|doi=10.1016/0020-0190(81)90106-X

|url-access=subscription

}} Schneider and Andrews provide a correctness proof.{{Cite conference

|last1=Schneider

|first1=Fred B.

|authorlink1=Fred B. Schneider

|last2=Andrews

|first2=Gregory R.

|title=Concepts for concurrent programming

|pages=669–716

|date=1986

|book-title=Current Trends in Concurrency

|series=Lecture Notes in Computer Science

|editor1=J.W. Bakker

|editor2=W.P. de Roever

|editor3=G. Rozenberg

|volume=224

|location=Noordwijkerhout, the Netherlands

|publisher=Springer Verlag

|doi=10.1007/BFb0027049

|isbn=978-3-540-16488-3

|url=https://doi.org/10.1007/BFb0027049

|url-access=subscription

}}

The image below, by Ilya Sergey, depicts the flow of ideas that have been implemented in logics that deal with concurrency. At the root is interference freedom. The file {{Citation

| url = https://ilyasergey.net/assets/other/CSL-Family-Tree.pdf

| title = CSL-Family-Tree

| ref = none

}} contains references. Below, we summarize the major advances.

File:Program-logics-graph.png

• Rely-Guarantee. 1981. Interference freedom is not compositional. Cliff Jones{{Cite thesis

|last=Jones

|first=C.B.

|title=Development Methods for Computer Programs including a Notion of Interference

|date=June 1981

|degree=DPhil

|publisher=Oxford University

|url=http://www.cs.ox.ac.uk/files/9025/PRG-25.pdf

|doi=}}{{cite conference

|title= Specification and design of (parallel) programs

|first1=Cliff B.

|last1=Jones

|author-link1= Cliff Jones (computer scientist)

|year=1983

|conference=9th IFIP World Computer Congress (Information Processing 83)

|editor= R.E.A. Mason

|publisher=North-Holland/IFIP

|pages=321–332

|isbn=0444867295

}} recovers compositionality by abstracting interference into two new predicates in a spec: a rely-condition records what interference a thread must be able to tolerate and a guarantee-condition sets an upper bound on the interference that the thread can inflict on its sibling threads. Xu et al {{Cite journal

|last1=Xu

|first1=Qiwen

|last2=de Roever

|first2=Willem-Paul

|last3=He

|first3=Jifeng

|title=The Rely-Guarantee method for verifying shared variable concurrent programs

|doi=10.1007/BF01211617

|journal=Formal Aspects of Computing

|volume=9

|pages=149–174

|date=1997

|issue=2

|s2cid=12148448

|url=https://doi.org/10.1007/BF01211617

|doi-access=free

}} observe that Rely-Guarantee is a reformulation of interference freedom; revealing the connection between these two methods, they say, offers a deep understanding about verification of shared variable programs.

• CSL. 2004. Separation logic supports local reasoning, whereby specifications and proofs of a program component mention only the portion of memory used by the component. Concurrent separation logic (CSL) was originally proposed by Peter O'Hearn,{{cite conference

| url = https://link.springer.com/book/10.1007/b100113

| title = Resources, Concurrency and Local Reasoning

| first = Peter W.

| last = O'Hearn

| author-link = Peter O'Hearn

| date = 2004-09-03

| conference = CONCUR 2004

| editor1 = P. Gardner

| editor2 = N. Yoshida

| volume =

| edition =

| book-title = CONCUR 2004 -- Concurrency Theory

| publisher = Springer Verlag Berlin, Heidelberg

| location = London, UK

| pages = 49–67

| isbn = 978-3-540-28644-8

| doi = 10.1007/b100113

| access-date = 2022-07-06| url-access = subscription

}}{{cite journal|last1=O'Hearn|first1=Peter|title=Resources, Concurrency and Local Reasoning|journal=Theoretical Computer Science|date=2007|volume=375|issue=1–3|pages=271–307|doi=10.1016/j.tcs.2006.12.035|url=http://www.cs.ucl.ac.uk/staff/p.ohearn/papers/concurrency.pdf|doi-access=free}} We quote from: "the Owicki-Gries method involves explicit checking of non-interference between program components, while our system rules out interference in an implicit way, by the nature of the way that proofs are constructed."

• Deriving concurrent programs. 2005-2007. Feijen and van Gasteren show how to use Owicki-Gries to design concurrent programs, but the lack of a theory of progress means that designs are driven only by safety requirements. Dongol, Goldson, Mooij, and Hayes have extended this work to include a "logic of progress" based on Chandy and Misra's language Unity, molded to fit a sequential programming model. Dongel and Goldson{{cite journal

| url = https://doi.org/10.2168%2Flmcs-2%281%3A6%292006

| title = Extending the theory of Owicki and Gries with a logic of progress

| last1 = Dongol

| first1 = Brijesh

| last2 = Goldson

| first2 = Doug

| journal = Logical Methods in Computer Science

| year = 2006

| volume = 2

| orig-date = January 12, 2005

| publisher = Centre pour la Communication Scientifique Directe (CCSD)

| arxiv = cs/0512012v3

| doi = 10.2168/lmcs-2(1:6)2006

| s2cid = 302420

| access-date =

| url-status =}} describe their logic of progress. Goldson and Dongol{{Cite conference

|url=https://dl.acm.org/doi/pdf/10.5555/1082260.1082265

|last1=Goldson

|first1=Doug

|last2=Dongol

|first2=Brijesh

|title=Concurrent Program Design in the Extended Theory of Owicki and Gries

|pages=41–50

|date=January 2005

|book-title=CATS '05: Proc 2005 Australasian Symp on Theory of Computing

|editor1=Mike Atkinson

|editor2=Frank Dehne

|volume=41

|publisher=Australian Computer Society, Inc.

|doi=

|isbn=

}} show how this logic is used to improve the process of designing programs, using Dekker's algorithm for two processes as an example. Dongol and Mooij{{Cite conference

|url=https://link.springer.com/content/pdf/10.1007/11783596_11

|last2=Mooij

|first2=Arjan J

|last1=Dongol

|first1=Brijesh

|title=Progress in deriving concurrent programs: emphasizing the role of stable guards

|pages=14–161

|date=July 2006

|book-title=MPC'06: Proc. 8th Int. Conf. on Mathematics of Program Construction

|editor1=Tarmo Uustalu

|volume=41

|publisher=Springer Verlag, Berlin, Heidelberg

|location=Kuressaare, Estonia

|doi=10.1007/11783596_11

|isbn=

|url-access=subscription

}} present more techniques for deriving programs, using Peterson's mutual exclusion algorithm as one example. Dongol and Mooij{{cite journal

|last1=Dongol |first1=Brijesh

|last2= Mooij|first2=Arjan J

|date=2008

|title=Streamlining progress-based derivations of concurrent program

|url=https://dl.acm.org/doi/pdf/10.1007/s00165-007-0037-4

|journal=Formal Aspects of Computing

|volume=20

|issue= 2

|pages= 141–160

|doi=10.1007/s00165-007-0037-4

|s2cid=7024064

|access-date=|doi-access=free

}} show how to reduce the calculational overhead in formal proofs and derivations and derive Dekker's algorithm again, leading to some new and simpler variants of the algorithm. Mooij{{Cite conference

|url=

|last1=Mooij

|first1=Arjan J.

|title=Calculating and composing progress properties in terms of the leads-to relation

|pages=366–386

|date=November 2007

|book-title=ICFEM'07: Proc. Formal Engineering Methods 9th Int. Conf. on Formal Methods and Software Engineering

|editor1=Michael Butler

|editor2=Michael G. Hinchey

|editor3=María M. Larrondo-Petrie

|location=Boca Raton, Florida

|volume=

|publisher=Springer Verlag, Berlin, Heidelberg

|doi=

|isbn=978-3540766483

}} studies calculational rules for Unity's leads-to relation. Finally, Dongol and Hayes{{cite tech report

|last1=Dongol

|first1=Brijesh

|last2=Hayes

|first2=Ian

|date=April 2007

|url=https://core.ac.uk/download/pdf/14986002.pdf

|title=Trace semantics for the Owicki-Gries theory integrated with the progress logic from UNITY

|institution=University of Queensland

|number=SSE-2007-02}} provide a theoretical basis for and prove soundness of the process logic.

• OGRA. 2015. Lahav and Vafeiadis strengthen the interference freedom check to produce (we quote from the abstract) "OGRA, a program logic that is sound for reasoning about programs in the release-acquire fragment of the C11 memory model." They provide several examples of its use, including an implementation of the RCU synchronization primitives.{{cite conference

|url=https://link.springer.com/chapter/10.1007/978-3-662-47666-6_25

|title= Owicki-Gries reasoning for weak memory models

|first1=Ori

|last1=Lahav

|first2=Viktor

|last2=Vafeiadis

|year=2015

|conference=ICALP 2015

|editor1=Halldórsson, M.

|editor2=Iwama, K.

|editor3=Kobayashi, N.

|editor4=Speckmann, B.

|volume=9135

|book-title=Automata, Languages, and Programming. ICALP 2015

|series=Lecture Notes in Computer Science

|publisher=Springer

|location=Berlin, Heidelberg

|pages=311–323

|doi=10.1007/978-3-662-47666-6_25|url-access=subscription

}}

• Quantum programming. 2018. Ying et al {{cite arXiv

|last1=Ying

|first1=Mingsheng

|last2=Zhou

|first2=Li

|last3=Li

|first3=Yangjia

|title=Reasoning about Parallel Quantum Programs

|date=2018

|class=cs.LO

|eprint=1810.11334}} extend interference freedom to quantum programming. Difficulties they face include intertwined nondeterminism: nondeterminism involving quantum measurements and nondeterminism introduced by parallelism occurring at the same time. The authors formally verify Bravyi-Gosset-König's parallel quantum algorithm solving a linear algebra problem, giving, they say, for the first time an unconditional proof of a computational quantum advantage.

• POG. 2020. Raad et al present POG (Persistent Owicki-Gries), the first program logic for reasoning about non-volatile memory technologies, specifically the Intel-x86.{{Cite conference

|last1=Raad

|first1=Azalea

|last2=Lahav

|first2=Ori

|last3=Vafeiadis

|first3=Viktor

|title=Persistent Owicki-Gries reasoning: a program logic for reasoning about persistent programs on Intel-x86

|date=13 November 2020

|book-title=Proceedings of the ACM on Programming Languages

|volume=4

|issue=OOPSLA

|pages=1–28

|publisher=ACM

|doi=10.1145/3428219

|doi-access=free

|hdl=10044/1/97398

|hdl-access=free

}}

• On A Method of Multiprogramming, 1999.{{cite book

|last1= Van Gasteren

|first1= A.J.M.

|last2= Feijen

|first2= W.H.J.

|editor-last1=Gries

|editor-first1=David

|editor-link1= David Gries

|editor-last2=Schneider

|editor-first2=Fred B.

|editor-link2=Fred B. Schneider

|year=1999

|title=On A Method Of Multiprogramming

|series=Monographs in Computer Science

|publisher=Springer-Verlag New York Inc.

|pages=370

|doi=10.1007/978-1-4757-3126-2

|isbn=978-1-4757-3126-2|s2cid= 13607884

}} Van Gasteren and Feijen discuss the formal development of concurrent programs entirely on the idea of interference freedom.

• On Current Programming, 1997.{{cite book

|last1= Schneider

|first1= Fred B.

|author-link1= Fred B. Schneider

|editor-last1=Gries

|editor-first1=David

|editor-link1= David Gries

|editor-last2=Schneider

|editor-first2=Fred B.

|editor-link2=Fred B. Schneider

|year=1997

|title=On Concurrent Programming

|series=Graduate Texts in Computer Science

|publisher=Springer-Verlag New York Inc.

|doi=10.1007/978-1-4612-1830-2

|isbn=978-1-4612-1830-2|s2cid= 9980317

}} Schneider uses interference freedom as the main tool in developing and proving concurrent programs. A connection to temporal logic is given, so arbitrary safety and liveness properties can be proven. Control predicates obviate the need for auxiliary variables for reasoning about program counters.

• Verification of Sequential and Concurrent Programs, 1991,{{cite book

|last1= Apt

|first1= Krzysztof R.

|author-link1=

|last2= Olderog

|first2= Ernst-Rüdiger

|editor-last1=Gries

|editor-first1=David

|editor-link1= David Gries

|year=1991

|title=Verification of Sequential and Concurrent Programs

|series=Texts in Computer Science

|publisher=Springer-Verlag Germany

}} 2009.{{cite book

|last1= Apt

|first1= Krzysztof R.

|author-link1=

|last3= Olderog

|first3= Ernst-Rüdiger

|author-link3=

|last2=Boer

|first2=Frank S.

|author-link2=

|editor-last1=Gries

|editor-first1=David

|editor-link1= David Gries

|editor-last2=Schneider

|editor-first2=Fred B.

|editor-link2=Fred B. Schneider

|year=2009

|edition=3rd

|title=Verification of Sequential and Concurrent Programs

|series=Texts in Computer Science

|publisher=Springer-Verlag London

|pages=502

|url=http://link.springer.com/10.1007/978-1-84882-745-5

|doi=10.1007/978-1-84882-745-5

|bibcode= 2009vscp.book.....A

|isbn=978-1-84882-744-8}} This first text to cover verification of structured concurrent programs, by Apt et al, has gone through several editions over several decades.

• Concurrency Verification: Introduction to Compositional and Non-Compositional Methods, 2112.{{cite book

|last1= de Roever

|first1= Willem-Paul

|last2= de Boer

|first2= Willem-Paul

|last3= Hanneman

|first3= Ulrich

|last4= Hooman

|first4= Jozef

|last5= Lakhnech

|first5= Yassine

|last6= Poel

|first6= Mannes

|last7= Zwiers

|first7= Job

|editor-last1=Abramsky

|editor-first1=S.

|year=2012

|pages=800

|title=Concurrency Verification: Introduction to Compositional and Non-Compositional Methods

|series=Cambridge Tracts in Theoretical Computer Science

|publisher=Cambridge University Press USA

|isbn=978-0521169325}} De Roever et al provide a systematic and comprehensive introduction to compositional and non-compositional proof methods for the state-based verification of concurrent programs

• 1999: Nipkow and Nieto present the first formalization of interference freedom and its compositional version, the rely-guarantee method, in a theorem prover: Isabelle/HOL.{{Cite thesis

|last=Nieto

|first=Leonor Prensa

|title=Verification of Parallel Programs with the Owicki-Gries and Rely-Guarantee Methods in Isabelle/HOL

|date=2002-01-31

|degree=Dr. rer. nat.

|publisher=Technischen Universitaet Muenchen

|type= PhD thesis

|url=http://tumb1.biblio.tu-muenchen.de/publ/diss/allgemein.html

|pages=198

|access-date=2022-07-05

}}{{Cite conference

|last1=Nipkow

|first1=Tobias

|last2=Nieto

|first2=Leonor Prensa

|title=Owicki/Gries in Isabelle/HOL

|volume=1577

|conference=FASE 1999

|book-title=Fundamental Approaches to Software Engineering

|editor=J.P. Finance

|publisher=Springer Verlag

|location=Berlin Heidelberg

|series=Lecture Notes in Computer Science

|date=1999-03-22

|pages=188–203

|isbn=978-3-540-49020-3

|doi=10.1007/978-3-540-49020-3_13|doi-access=free

}}

• 2005: Ábrahám's PhD thesis provides a way to prove multithreaded Java programs correct in three steps: (1) Annotate the program to produce a proof outline, (2) Use their tool Verger to automatically create verification conditions, and (3) Use the theorem prover PVS to prove the verification conditions interactively.{{Cite thesis

|last=Ábrahám

|first=Erika

|title=An Assertional Proof System for Multithreaded Java - Theory and Tool Support

|date=2005-01-20

|degree=PhD

|publisher=Universiteit Leiden

|type= PhD thesis

|url=https://hdl.handle.net/1887/584

|pages=220

|hdl=1887/584

|access-date=2022-07-05

|isbn=9090189084}}{{cite journal

|last1=Ábrahám

|first1=Erika

|last2=Boer

|first2=Frank, S., de

|last3=Roever

|first3=Willem-Paul, de

|last4=Martin

|first4=Steffen

|title=An assertion-based proof system for multithreaded Java

|journal=Theoretical Computer Science

|publisher=Elsevier

|volume=331

|date=2005-02-25

|issue=2–3

|pages=251–290

|doi=10.1016/j.tcs.2004.09.019|doi-access=free

}}

• 2017: Denissen{{Cite thesis

|last=Denissen

|first=P.E.J.G

|title=Extending Dafny to Concurrency: Owicki-Gries style program verification for the Dafny program verifier

|date=November 2017

|degree=Masters

|publisher=Eindhoven University of Technology

|url=https://research.tue.nl/en/studentTheses/extending-dafny-to-concurrency}} reports on an implementation of Owicki-Gries in the "verification ready" programming language Dafny.{{cite web

|url=https://dafny.org

|title=Dafny Programming Language

|access-date=2022-07-20}} Denissen remarks on the ease of use of Dafny and his extension to it, making it extremely suitable when teaching students about interference freedom. Its simplicity and intuitiveness outweighs the drawback of being non-compositional. He lists some twenty institutions that teach interference freedom.

• 2017: Amani et al combine the approaches of Hoare-Parallel, a formalisation of Owicki-Gries in Isabelle/HOL for a simple while-language, and SIMPL, a generic language embedded in Isabelle/HOL, to allow formal reasoning on C programs.{{cite conference

| url = http://dx.doi.org/10.1145/3018610.3018627

| title = COMPLX: A verification framework for concurrent imperative programs

| first1 = S.

| last1 = Amani

| first2 = J.

| last2 = Andronick

| first3 = M.

| last3 = Bortin

| first4 = C.

| last4 = Lewis

| first5 = C.

| last5 = Rizkallah

| first6 = J.

| last6 = Tuong

| date = 16 January 2017

| conference = CPP 2017: Proc 6th ACM SIGPLAN Conference on Certified Programs and Proof

| editor1 = Yves Bertot

| editor2 = Viktor Vafeiadid

| publisher = ACM

| location = Paris, France

| pages = 138–150

| isbn = 978-1-4503-4705-1

| doi = 10.1145/3018610.3018627| url-access = subscription

}}

• 2022: Dalvandi et al introduce the first deductive verification environment in Isabelle/HOL for C11-like weak memory programs, building on Nipkow and Nieto's encoding of Owicki–Gries in the Isabelle theorem prover.{{Cite journal

|last1=Dalvandi

|first1=Sadegh

|last2=Dongol

|first2=Brijesh

|last3=Doherty

|first3=Simon

|last4=Wehrheim

|first4=Heike

|title=Integrating Owicki–Gries for C11-Style memory models into Isabelle/HOL

|doi=10.1007/s10817-021-09610-2

|journal=Journal of Automated Reasoning

|volume=66

|pages=141–171

|date=February 2022

|s2cid=215238874

|doi-access=free

|arxiv=2004.02983

}}

• 2022: This webpage {{cite web |url=https://civl-verifier.github.io

|title=Civl: A verifier for concurrent programs

|access-date=2022-07-22}} describes the Civl verifier for concurrent programs and gives instructions for installing it on your computer. It is built on top of Boogie, a verifier for sequential programs. Kragl et al {{Cite conference

|last1=Kragl

|first1=Bernhard

|last2=Qadeer

|first2=Shaz

|last3=Henzinger

|first3=Thomas A.

|title=Refinement for structured concurrent programs

|date=2020

|book-title=CAV 2020: Computer Aided Verification

|series=Lecture Notes in Computer Science

|editor1=S. Lahiri

|editor2=C. Wang

|volume=12224

|publisher=Springer Verlag

|doi=10.1007/978-3-030-53288-8_14

|isbn=978-3-030-53288-8

|doi-access=free

}} describe how interference freedom is achieved in Civl using their new specification idiom, yield invariants. One can also use specs in the rely-guarantee style. Civl offers a combination of linear typing and logic that allows economical and local reasoning about disjointness (like separation logic). Civl is the first system that offers refinement reasoning on structured concurrent programs.

• 2022. Esen and Rümmer developed TRICERA,{{Cite conference

|last1=Esen

|first1=Zafer

|last2=Rümmer

|first2=Philipp

|title=TRICERA Verifying C Programs Using the Theory of Heaps

|date=October 2022

|book-title=Proc. 22nd Conf. on Formal Methods in Computer-Aided Design – FMCAD 2022

|editor1=A. Griggio

|editor2=N. Rungta

|volume=

|issue=

|pages=360–391

|publisher=TU Wien Academic Press

|doi=10.34727/2022/isbn.978-3-85448-053-2_45

|doi-access=free

}} an automated open-source verification tool for C programs. It is based on the concept of constrained Horn clauses, and it handles programs operating on the heap using a theory of heaps. A web interface to try it online is available. To handle concurrency, TRICERA uses a variant of the Owicki-Gries proof rules, with explicit variables to added to represent time and clocks.

{{Reflist}}

Category:Formal methods

Category:Program logic

Category:Logic in computer science