In many cases it might be convenient to use a concurrency model in which threads don’t share state (Erlang style). I haven’t yet done that and I guess it could be a bit painful to do it in Java. But you’d get rid of the dangerous side-effects.

That is a great thing to do. In fact, all of those possible interactions are irrelevant when you do exactly that. That’s why we avoid instance variables in Servlets and pass everything on the stack.

In general, any time you can reduce the scope of information (e.g. not sharing data between threads), you reduce the scope of possible side-effects.

In some cases, however, you’re trying to get multiple threads working on the same problem, especially when you’re working with I/O bound problems. In that case, you have common data. But you want to minimize the scope of possible interaction both to reduce the chance of problems creeping up and to reduce contention.

Could someone explain how we can derive the formula Total = (N*T)! / ( T!) ^(N) ?

This is from an email Uncle Bob Sent me (so give the credit to him):

With N steps and T threads there are T x N total steps. Prior to each step there is a context switch that chooses between the T threads. Each path can thus be represented as a string of digits denoting the context switches. Given steps A and B and threads 1 and 2 the six possible paths are 1122, 1212, 1221, 2112, 2121, and 2211. Or in terms of steps it is: A1B1A2B2, A1A2B1B2, A1A2B2B1, A2A1B1B2, A2A1B2B1, and A2B2A1B1.

For three threads the sequence is 112233, 112323, 113223, 113232, 112233, 121233, 121323, 121332, 123132. 123123, ...

One characteristic of these strings is that there must always be N instances of each T. So the string 111111 is invalid because it has 6 instances of 1 and zero instances of 2 and 3.

So we want the permutations of N 1’s, N 2’s, ... and N T’s. This is really just the permutations of N x T things taken N x T at a time (which is (N x T)!), but with all the duplicates removed. So the trick is to count the duplicates and subtract that from (N x T)!.

Given two steps and two threads, how many duplicates are there? Each four digit string has two 1’s and two 2’s. Each of those pairs could be swapped without changing the sense of the string. You could swap the 1s, or the 2s, both, or neither. So there are four isomorphs for each string, which means that there are three duplicates. So three out of four are duplicates. Or only 1/4 of the permutations are NOT duplicates. 4! * .25 = 6. So this reasoning seems to work.

How many duplicates are there? In the case where N=2 and T=2 I could swap the 1’s the 2’s or both. In the case where N=2 and T=3 I could swap the 1’s, the 2’s, the 3’s, 1’s and 2’s, 1’s and 3’s, or 2’s and 3’s. Swapping is just the permutations of N. Let’s say there are P permutations of N. The number of different ways to arrange those permutations are PT.

So the number of possible isomorphs is N!T. And so the number of paths is (T x N)!/(N! ^ T). Again in our T=2, N=2 case we get 6. (24/4).

For N=2 and T=3 we get 720/8 = 90. (Does that seem right?)

For N=3 and T=3 we get 9!/6^3 = 1680.

I am not convinced by the proof. It is not a question of duplicates, but of removing invalid combinations among the (N*T)! possible ways to arrange them. Invalid combinations are those where step 2 comes first before step 1 is done by a particular thread.

The proof seems more like a trial and error effort than a rigorous one.

Just curious about the proof. The formula seems right though.

I’m not sure where you got the impression that anything I was doing was a proof. Ultimately, when Bob gave me the formula, I was satisfied. That the estimate I derived numerically matched his formula so well told me that I probably correctly wrote the program. However, the program was simply a tool to figure out the number of possibilities.

As to your concerns, here’s the method in a little more detail.

1. I used TDD to verify that I was producing all combinations correctly. I ended up working through various algorithms until I eventually wrote a recursive algorithm that looks like how you’d approach it using lisp. While I only verified that the I got all permutations correctly for up to 4, I did verify that for several larger numbers of nodes I got the correct number.
2. I next used TDD to develop a class that used the permutation generator and then removed invalid paths. By invalid path, I reviewed the entire chain of orderings and looked to see if any transition was invalid. To do so, I used an O(N^2) algorithm. Starting at the first element, I asked (given my dependencies) if any of the following nodes were invalid. I then did that for the next node, the next, etc. I believe I generated the transitive closure correctly so that I did not forget to filter out any paths.

Here’s one of the unit tests I used for the permutation generator:

    @Test
    public void fourGives24() {
        PermutationGenerator generator = new PermutationGenerator(4);
        List<String> allPermutations = generator.validPermutations();
        assertEquals(24, allPermutations.size());
        assertTrue(allPermutations.contains("abcd"));
        assertTrue(allPermutations.contains("abdc"));
        assertTrue(allPermutations.contains("acbd"));
        assertTrue(allPermutations.contains("acdb"));
        assertTrue(allPermutations.contains("adbc"));
        assertTrue(allPermutations.contains("adcb"));

        assertTrue(allPermutations.contains("bacd"));
        assertTrue(allPermutations.contains("badc"));
        assertTrue(allPermutations.contains("bcad"));
        assertTrue(allPermutations.contains("bcda"));
        assertTrue(allPermutations.contains("bdac"));
        assertTrue(allPermutations.contains("bdca"));

        assertTrue(allPermutations.contains("cabd"));
        assertTrue(allPermutations.contains("cadb"));
        assertTrue(allPermutations.contains("cbad"));
        assertTrue(allPermutations.contains("cbda"));
        assertTrue(allPermutations.contains("cdab"));
        assertTrue(allPermutations.contains("cdba"));

        assertTrue(allPermutations.contains("dabc"));
        assertTrue(allPermutations.contains("dacb"));
        assertTrue(allPermutations.contains("dbac"));
        assertTrue(allPermutations.contains("dbca"));
        assertTrue(allPermutations.contains("dcab"));
        assertTrue(allPermutations.contains("dcba"));
    }

Here’s a sampling of some of the tests I used to verify that only the paths that should be generated, were:

   @Test
public void noNodesNoPaths() {
    calculator = new PathCalculator(0);
    assertEquals(0, calculator.totalPaths());
}

@Test
public void oneNodeResultsInOnePossiblePath() {
    calculator = new PathCalculator(1);
    int totalPaths = calculator.totalPaths();
    assertEquals(1, totalPaths);

}

@Test
public void threeNodesNoDependencies6Paths() {
    calculator = new PathCalculator(3);
    int totalPaths = calculator.totalPaths();
    assertEquals(6, totalPaths);
}

@Test
public void fourNodesNoDependencies24Paths() {
    calculator = new PathCalculator(4);
    int totalPaths = calculator.totalPaths();
    assertEquals(24, totalPaths);
}

@Test
public void twoNodesOneDependencyOnePath() {
    calculator = new PathCalculator(2);
    createLine('a', 'b');
    int totalPaths = calculator.totalPaths();
    assertEquals(1, totalPaths);
}

@Test
public void threeNodesInALineOnePath() {
    calculator = new PathCalculator(3);
    createLine('a', 'b');
    createLine('b', 'c');
    int totalPaths = calculator.totalPaths();
    assertEquals(1, totalPaths);
}

@Test
public void fourNodesWithTwoIndependentDependneciesShouldResultInSixPahts() {
    calculator = new PathCalculator(4);
    createLine('a', 'b');
    createLine('c', 'd');
    int totalPaths = calculator.totalPaths();
    assertEquals(6, totalPaths);
}

@Test
public void twoNodesWithFlowInBothDirectionsShouldResultInNoPaths() {
    calculator = new PathCalculator(2);
    createLine('a', 'b');
    createLine('b', 'a');
    int totalPaths = calculator.totalPaths();
    assertEquals(0, totalPaths);
}

But regardless of that, if you are convinced of the formula, then that’s really all that matters. The primary point is that there are a lot of different ways in which a very small amount of code can execute.

I did the quickest of sanity checks, and the expression (N*T)! / (T!^N) doesn’t collapse to 1 when T=1, so it can’t be right.

Having read what Uncle Bob wrote, his reasoning seems correct, you’ve just mistranscribed his result: he got (N*T)! / (N!^T)

Ah! You are correct. Thanks for finding my error.

I’ve corrected the original post.

Ok. With the newer version, it is easy to prove the formula.

It is interesting how a little knowledge of Mathematics can help eliminate a lot of code (and tests.)

In this case, once we know the formula for the total number of valid paths, not a single line of code is required. After all, the intent was to show the possible paths and how synchronize can reduce them. With the formula, we know the number of paths; there is no need to write any code or tests.

On another topic, the test code shown has horrendous duplication. Why could not one write a single test method, pass it the value of “T” and “N” and the expected result and see whether it works? That way, it is just a matter of passing a collection and seeing which tests succeed and which fail.

Adding more tests is as simple as adding more rows (elements) to the collection. If the test input data is stored in a data source (database, text file, ...) then one never has to change the test code! And isn’t that what “soft”ware is about, the ability to change things easily?

Code duplication is why I don’t like the current way of writing one method for each test case. The test code in the comment by the original author proves my point.

Sorry for being harsh, but as a contributor to a somewhat popular blog, you should hold yourself to high standards. There is the chance that others may take what they read here as the gospel truth without thinking critically for themselves.

In this particular case, a little knowledge of mathematics does indeed obviate all need to write any code.

Your comments about code duplication are well taken and in this case you’re correct. It is possible. In fact, below is just such an example.

Before I get to that, I do want to ask about this:

Code duplication is why I don’t like the current way of writing one method for each test case. The test code in the comment by the original author proves my point.

I don’t see any comments in any of the code in the entire post. What am I missing?

Should I infer from this that you don’t, in general, like writing test methods but prefer instead to write things closer to the below:

This is a a test that verifies a filter on class packages works as expected. The method annotated with @Parameters represents a collection of tests. Add a new Object[] array of values to the collection, get a new test:

package schuchert.contest;

import static junit.framework.Assert.assertEquals;

import java.util.ArrayList;
import java.util.Collection;

import org.junit.Before;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.junit.runners.Parameterized;
import org.junit.runners.Parameterized.Parameters;

@RunWith(Parameterized.class)
public class DynamicInstrumentorFilterTest {
    DynamicInstrumentor transformer;
    private final String filter;
    private final boolean comShouldBeInstrumented;
    private final boolean javaShouldBeInstrumented;
    private final boolean schuchertShouldBeInstrumented;
    private final boolean fooShouldBeInstrumented;

    @Before
    public void setup() {
        transformer = new DynamicInstrumentor();
    }

    @Parameters
    public static Collection<Object[]> data() {
        ArrayList<Object[]> values = new ArrayList<Object[]>();
        values.add(new Object[] { "", true, true, true, true });
        values.add(new Object[] { "-java.*", true, false, true, true });
        values.add(new Object[] { "+java.*", false, true, false, false });
        values.add(new Object[] { "-java.*::-com.*", false, false, true, true });
        values.add(new Object[] { "+java.*::-schuchert.*::-com.*", false, true, false, false });
        values.add(new Object[] { "+[js].*", false, true, true, false });
        values.add(new Object[] { "+.*m$", true, false, false, false });
        return values;
    }

    public DynamicInstrumentorFilterTest(String filter,
            boolean matchComIbm, boolean matchJavaIo, boolean matchSchuchert, boolean matchFoo) {
        this.filter = filter;
        this.comShouldBeInstrumented = matchComIbm;
        this.javaShouldBeInstrumented = matchJavaIo;
        this.schuchertShouldBeInstrumented = matchSchuchert;
        this.fooShouldBeInstrumented = matchFoo;
    }

    @Test
    public void checkPattern() {
        transformer.setFilter(filter);
        assertFilterMatchedCorrectly("com/ibm", comShouldBeInstrumented);
        assertFilterMatchedCorrectly("java/io", javaShouldBeInstrumented);
        assertFilterMatchedCorrectly("schuchert/contest/", schuchertShouldBeInstrumented);
        assertFilterMatchedCorrectly("foo", fooShouldBeInstrumented);
    }

    private void assertFilterMatchedCorrectly(String string, boolean expected) {
        boolean result = transformer.isClassWeShouldProcess(string);
        assertEquals(String.format("Filter: %s, pattern: %s", filter, string), expected, result);

    }
}

Here’s a more detailed description of Parameterized Tests

By “comment” I meant the “response(s)” to the original post, not comments in the code. Sorry for not being clear about it.

Yes, something along those lines. But I would go even further by changing the code in

@Test
    public void checkPattern() {
        transformer.setFilter(filter);
        assertFilterMatchedCorrectly("com/ibm", comShouldBeInstrumented);
        assertFilterMatchedCorrectly("java/io", javaShouldBeInstrumented);
        assertFilterMatchedCorrectly("schuchert/contest/", schuchertShouldBeInstrumented);
        assertFilterMatchedCorrectly("foo", fooShouldBeInstrumented);
    }

to

@Test
    public void checkPattern() {
        transformer.setFilter(filter);

        for (Object[] testCase : allTestCases) { 
           assertFilterMatchedCorrectly ( 
               (String) testCase[0], // input value
               (Boolean) tesCase[1].boolValue()); // value to compare with
       }
    }

Of course. this handles one specific type of test. But even here the reduction in code is obvious.

By the way, from the example code I could not understand where the ArrayList is being used, or how.

That’s an example of how JUnit supports so-called parameterized tests. It’s a bit obtuse. The ArrayList has a collection of Object[]. The types of objects in various object arrays in the array list are used to match a constructor. So the basic algorithm is:

1. Get all tests by calling method with @Parameters annotation
2. For each element in the collection
   1. call matching ctor
   2. call @Test methods

The test is annotated with: @RunWith(Parameterized.class), for further details, check out the Parameterized class that ships with JUnit 4.

JUnit is using reflection to select and call the ctor once for each “test” in the array list. In that example, there are 7 items in the array list so there are 7 tests executed.

It is interesting how a little knowledge of Mathematics can help eliminate a lot of code (and tests.)

In this case, once we know the formula for the total number of valid paths, not a single line of code is required. After all, the intent was to show the possible paths and how synchronize can reduce them. With the formula, we know the number of paths; there is no need to write any code or tests.

I disagree. Constructing a mathematical proof is similar to writing a program. There can be subtle bugs. It’s always nice to see empirical results that confirm the conclusions of your proofs.

When I constructed the line of reasoning that led the the formula, I was far from convinced it was correct. Even now I worry that I overlooked something. The fact that Brett’s empirical model seems to agree with the formula decreases that worry somewhat.

Constructing a mathematical proof is similar to writing a program. There can be subtle bugs.

By definition, a “proof” is correct and without bugs. Once we’ve got a proof, airtight, there is no need for empirical validation.

More to the point, a valid mathematical proof is true for all possible values of the parameter, while an empirical validation can only be assumed to work for the specific cases tested. There could be a counterexample that invalidates the empirical proof.

A mathematical proof is true for all possible values of the parameters (within the domain of possible values.) There is no chance of a counterexample to a valid proof.

I had absolutely no doubts about the proof once the correct formula was posted. Hence, to me the empirical method was simply a waste of time.

Proof != Code

When I took a discrete mathematics course some 20+ years ago, I found a lot of value in the course as a whole, e.g. complexity analysis, proof by induction, and others. HOWEVER, program correctness proofs were, to my mind, completely useless for several reasons.

1. Practically speaking, proving a trivial for loop was too much work. Proving anything of any size was little more than a way to get a paper published that nobody would/should read.
2. Proving code is correct != proving that code does what someone wants. More on this below…

While your comments about proving something are true with respect to math, they have no relationship to software development. WHY?

Problems are stated in natural language. Yes, there have been efforts to write formal requirements specifications, and tools in support of such efforts, all of which amount to mental masturbation, IMO.

Most people speak in natural language. Natural language is ambiguous – in fact, much of its expressiveness comes from ambiguity. EVEN IF we could get people to speak in formal languages, there’s still the issues that:

1. People still don’t know what they want until they see it.
2. When they do see something, what they want changes
3. Reality kicks in and what they want/need changes

I immediately rejected formal specification of requirements as well as program correctness proofs, both of which I saw as, simply put, a big wast of time. I still believe that.

Furthermore, a formal specification does not change the reality that what we want is a moving target, so even if formal specification were possible, it’s nearly irrelevant because the existence of change makes it so.

There was/are efforts to get formal “sign off” on requirements. I used to do that when I was working as an architect or technical lead years ago – but those were always ways to “stop the insanity”.

When I started practicing incremental & iterative development, that helped. I worked on getting sign-off for a short period. Then a few weeks later, I asked for a little more.

Trying to get “sign-off” is attempting to fix a symptom that is an indication of a good thing… When requirements change it comes from either a perceived new need, or feedback from seeing something. I don’t want to inhibit change, I want to embrace and encourage it. It means I’ve got something that live and worth doing. No changes coming down the pike? Bad…

Trying to put controls on the work we do and equate it to mathematics, is, in general, a bad idea. There are places where it is relevant. If your domain involves optimal layout, optimal/minimal cost analysis, all of those problems that collapse in the traveling salesmen problem (NP-Complete problems), heuristics and mathematical models can help. But even so, that’s generally part of a system and not THE SYSTEM.

So there might be parts of problems that collapse into provable subsystems, but general software development does not fit that mold.

When it does – and I don’t think it ever will – software development will be simply clicking buttons in a wizard and entering formulas. Remember the CASE tool debacle? I’m not worried about losing my job to a case tool anytime soon. And if you look at software development in general, it is becoming more fragmented, not less. There are some trends:

1. Virtual Machines are mainstream and getting much better all the time.
2. Dynamic languages are on the cusp of becoming mainstream (FINALLY – I still love smalltalk)
3. People are accepting multi-paradigm solutions, another good trend
4. Java Script is a language to watch, with fast VM’s and better browser support

There’s no “next big” thing on the horizon. There’s a suite of tools and techniques, with enabling technologies such as dynamic typing and virtual machines, dynamic instrumentation or meta object protocols, mixed-mode development, etc.

All of this makes the notion of proving something correct more and more quaint.

I’m looking forward to the ensuing chaos !

Well, it looks like we’ve digressed quite a bit from the original topic, which was to show the large number of possible paths that can occur when multiple threads simultaneously access even small code fragments ( a small number of low-level instructions.) It started by attempting to find out how many possible valid paths exist for a program with N steps and T threads accessing it.

Two ways were given to find the number of valid paths: a small program that could calculate this value for a given N and T; and a formula that would do the same.

My argument was that once you verified the formula using logic and first principles, there would be no reason to write the code to find the answer for specific cases. Simply plugging in the numbers into a scientific calculator would suffice.

I did not mention anything about the “correctness” of a software program, or a “proof of correctness.” These topics, interesting though they are in their own right, are not what the original discussion was about.

Just like everything else, mathematics is a tool that we can use when the occasion arises. We do not have to use it for every single software problem, using it only where it is appropriate.

By definition, a “proof” is correct and without bugs.

         a = b
       a^2 = ab
 a^2 - b^2 = ab - b^2
(a+b)(a-b) = b(a-b)
       a+b = b
        2b = b
         2 = 1
Q.E.D.

There is a bug in that proof; but many people puzzle over it until it is explained to them. The point is that proofs can appear valid while containing subtle flaws.

A widely accepted proof is correct and without bugs. The example shown above merely tests our knowledge of mathematics principles, and, since the bug is known to the person providing the proof, may not be relevant to this case.

There is a chapter by Jon Bentley in “Beautiful Code” which states that the most beautiful code is the one he never wrote because he found a mathematical validation for it. And that is beauty, indeed!

Anyway, I believe that a good understanding of the fundamentals of several disciplines makes one a better programmer.

Anyway, I believe that a good understanding of the fundamentals of several disciplines makes one a better programmer.

No argument there.

I would have thought that the ‘proof’ was overkill anyway – all that we really need to know is that the number of routes through a piece of code increases exponentially – in relation to the number of steps/threads – if not managed with the synchronised keyword.

Interesting discussion, in any case.

“If you’ve made it this far, wow! Congratulations!”

I really enjoyed your article, because it doesn’t contradict any of my ideas about software development.

There are 2 problem, IMHO, with this article though:

1. Has a very long and difficult explanation of why multithreaded programming is difficult. You don’t need such complex demonstrations, instead it is enough to realize that most software is riddled with bugs, so many that some bugs are considered features, other bugs are kept for compatability purposes and other bugs hide other bugs.

2. Ending an article with a comment saying “I expect that whoever is reading me is an idiot” is not considered good manners. I agree that most developers seem to think most of the people are idiots, and they are probably right, if your definiton of idiot is “he doesn’t know what I know”. Going by that definition only can mean that being idiotic is something relative (it depends of who you are talking to), and since “everyone knows where his shoes are too tight”, we live in the land or mordon.

Replacing a clever algorithm with a formula is always faster.

Testing a formula with a test is a sure way to detet bugs in formulas.

Either way, the EJB experiment has shown you don’t need multithreaded programs nor programmers who can understand multithreaded programs. All you need is programmers who can devise single threaded algorithms and a deployment environment that can take the single threaded algorithm and run it in parallel.

Has a very long and difficult explanation of why multi-threaded programming is difficult.

I’ve seen lots of books/articles that describe the difficulty with multi-threaded coding but never one that shows things at the level of the JVM. I think many (most) people who develop in Java would understand it much better with just a little background in:

1. How the JVM structures bytecodes
2. Frames – or what the term I learned, which was activation records
3. A mental model of the stack, heap, where things live and how threads interact with all of that

I don’t think everybody needs to understand everything at a deep level, but it seems that many people (even some “senior consultants” I’ve worked with), really don’t have a fundamental understanding of what’s going on. That leads to a lot of “hunt-and-peck” programming – keep typing still the s*&t works.

And besides, it’s a freaking blog.

2. Ending an article with a comment saying “I expect that whoever is reading me is an idiot” is not considered good manners. I agree that most developers seem to think most of the

Since you ”” quoted me, I can categorically deny what you wrote. I did a search and I did not see the word idiot before your first mention of it. I certainly did not call anybody an idiot. If you got that out of what I wrote you might want to ask yourself why you did. Also, from your perspective, my attempt at a self effacing comment failed.

Either way, the EJB experiment has shown you don’t need multithreaded programs nor programmers who can understand multithreaded programs. All you need is programmers who ca

Also disagree. I’ve done a fair bit of EJB development in large team environments and it is not for the light-hearted. It only appears to take away the need to avoid threading. There are still issues:

1. Reentrant code
2. Shared objects in the web tier
3. Shared objects in the ejb tier
4. Difficulty of defining good ejbs
5. Stateful versus stateless – I’m not in the came of “everything must be stateless”, though I strive to keep my state as focused as possible
6. Object allocation causing terrible garbage collection – not a threading issue but an issue anyway

The list goes on. EJB, before the 3.0 spec, was a solution looking for a problem. Using just a web container + spring + hibernate is more than enough for many (most?) enterprise-level applications that scale well.

I really like the direction taken in EJB 3 and JPA. I think it’s too little, too late. IBM is only now coming to the market with the technology (JBoss and was it Oracle got there some time ago).

Frankly, I think the EJB “experiment” was an excellent demonstration of a disaster, which is only being made worse by adding a bus, web services and calling it SOA – that’s a way to sell licenses, but solutions? Not so much.

So now that I’ve disagreed with your ideas about software development (and article writing apparently), I’m guessing you may no longer like the article – or at least my reply to your comments.

oh ,my god ,the last one

Very pleased that the article is what I want! Thank you

I’ve done a fair bit of EJB development in large team environments and it is not for the light-hearted. It only appears to take away the need to avoid threading. virtuemart templates

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