The classi cation of possible errors in object intents is given and some possibilities of exploring them are discussed. Two techniques for nding some types of errors in new object intents are introduced. After comparing the better technique is developed further in order to guarantee the absence of certain errors given enough information. Based on this technique an approach for debugging source code is presented and discussed. It is shown that the new approach yields bug hypothesis in a strict logical form. Using the new approach it is possible to come closer to debugging programs on a logical level not checking executions line by line. An example of applying the new approach is presented.
In this work we present a new approach to debug programs. This work was
inspired by the Delta Debugger project [
In this paper we rst introduce a new technique for nding errors in new object intents. This technique was rst introduced in our previous work; we partly repeat the results and refer to our previous work for more details. In this paper we recall two di erent approaches for revealing errors in new object intents: one based on computing the implication system of the context and another one based on computing the closures of the subsets of the new object intent. Since computing closures may be performed much faster we improve and generalize this approach and nally obtain a procedure for nding all possible errors of the considered types. We also provide experimental results to compare two approaches. After that we present a new approach of debugging based on the discussed above technique of nding errors in data. An example of debugging is provided.
All sets and contexts we consider in this paper are assumed to be nite. 2
Let G and M be sets. Let I G M be a binary relation between G and M . Triple K := (G; M; I) is called a (formal) context.
The set G is called a set of objects. The set M is called a set of attributes.
Consider mappings ' : 2G ! 2M and : 2M ! 2G: '(X) := fm 2 M j gIm for all g 2 Xg; (A) := fg 2 G j gIm for all m 2 Ag: For any X1; X2 G, A1; A2 M one has Mappings ' and de ne a Galois connection between (2G; ) and (2M ; ), i.e. '(X) A , (A) X. Usually, instead of ' and a single notation ( )0 is used. ( )0 is sometimes called a derivation operator. For X G the set X0 is called the intent of X and is denoted int(X). Similarly, for A M the set A0 is called the extent of A and is denoted ext(A).
Let Z M or Z G. (Z)00 is called the closure of Z in K. Applying Properties 1 and 2 consequently one gets the monotonicity property: for any Z1; Z2 G or Z1; Z2 M one has Z1 Z2 ) Z100 Z200.
Let m 2 M; X G, then m is called a negated attribute. m 2 X0 whenever no x 2 X satis es xIm. Let A M ; A X0 i all m 2 A satisfy m 2 X0.
An implication of K := (G; M; I) is de ned as a pair (A; B), written A ! B, where A; B M . A is called the premise, B is called the conclusion of the implication A ! B. The implication A ! B is respected by a set of attributes N if A * N or B N . The implication A ! B holds (is valid) in K if it is respected by all g0, g 2 G, i.e. every object, that has all the attributes from A, also has all the attributes from B. Implications satisfy Armstrong rules : A ! A ;
A ! B A [ C ! B ;
A ! B; B [ C ! D
A [ C ! D A support of an implication in context K is the set of all objects of K, whose intents contain the premise and the conclusion of the implication. A unit implications is de ned as an implication with only one attribute in the conclusion, i.e. A ! b, where A M; b 2 M . Every implication A ! B can be regarded as the set of unit implications fA ! b j b 2 Bg. One can always observe only unit implications without loss of generality.
An implication basis of a context K is de ned as a set L of implications of K, from which any valid implication for K can be deduced by the Armstrong rules and none of the proper subsets of L has this property.
A minimal implication basis is an implication basis minimal in the number of
implications. A minimal implication basis was de ned in [
We say that an object g is reducible in a context K := (G; M; I) i 9X G : g0 = T j0.
j2X 3
In this section we use the idea of data domain dependency. Usually objects and attributes of a context represent entities. Dependencies may hold on attributes of such entities. However, such dependencies may not be implications of a context as a result of an error in object intents. Thereby, data domain dependencies are such rules that hold on data represented by objects in a context, but may erroneously be not valid implications of a context.
In this work we consider only dependencies that do not have negations of attributes in premises. As mentioned above there is no need to specially observe non-unit implications. Consider possible types of such dependencies (A M; b; c 2 M ): The types 1 and 2 are most simple and common dependencies. In this work we try to nd the algorithm to reveal these two types of dependencies and nd corresponding errors. 4
Below we assume that we are given a context (possibly empty) with correct data and a number of new object intents that may contain errors. This data is taken from some data domain and we may ask an expert whose answers are always 1. A ! b 2. A ! b 3. A ! b _ c 4. A ! , where = a _ (b ^ c)
is any logical formula not considered above, for example, correct. However, we should ask as few questions as possible.
We introduce two di erent approaches to nding errors. The rst one is based on inspecting the canonical basis of a context. When adding a new object to the context one may nd all implications from the canonical basis of the context such that the implications are not respected by the intent of the new object. These implications are then output as questions to an expert in form of unit implications. If at least one of these implications is accepted, the object intent is erroneous. Since the canonical basis is the most compact (in the number of implications) representation of all valid implications of a context, it is guaranteed that the minimal number of questions is asked and no valid dependencies of Type 1 are left out.
Although this approach allows one to reveal all dependencies of Type 1, there
are several issues. The problem of producing the canonical basis with known
algorithms is intractable. Recent theoretical results suggest that the canonical
base can hardly be computed with better worst-case complexity than that of the
existing approaches ([
However, since we are only interested in implications corresponding to an object, it may be not necessary to compute a whole implication basis. Here is the second approach. Let A M be the intent of the new object not yet added to the context. m 2 A00 i 8g 2 G : A g0 ) m 2 g0, in other words, A00 contains the attributes common to all object intents containing A. The set of unit implications fA ! b j b 2 A00 n Ag can then be shown to the expert. If all implications are rejected, no attributes are forgotten in the new object intent. Otherwise, the object is erroneous. This approach allows one to nd errors of Type 1. 5
Obviously, applying the derivation operator two times is a much easier task than computing the canonical basis, and can be performed in polynomial time. However, the following case is possible. Let A M be the intent of the new object such that @g 2 G : A g0. In this case A00 = M and the implication A ! A00 n A has empty support. This may indicate an error of Type 2, because the object intent contains a combination of attributes impossible in the data domain, but the object may be correct as well. An expert could be asked if the combination of attributes in the object intent is consistent in the data domain. For such a question the information already input in the context is not used. More than that, this question is not su cient to reveal an error of Type 1. Proposition 1. Let K = (G; M; I); A
M . The set
IA = fB ! d j B 2 MCA; d 2 B00 n A [ A n Bg; where MCA = fB 2 CAj@C 2 CA : B Cg and CA = fA \ g0 j g 2 Gg, is the set of all unit implications (or their non-trivial consequences with some attributes added in the premise) of Types 1 and 2 such that implications are valid in K, not respected by A, and have not empty support.
Proposition 1 allows one to nd an algorithm for computing the set of questions to an expert revealing possible errors of Types 1 and 2. The pseudocode is pretty straightforward and is not shown here for the sake of compactness. Since computing the closure of a subset of attributes takes O(jGj jM j) time in the worst case, and we need to compute respective closures for every object in the context, the time complexity of the whole algorithm is O(jGj2 jM j). We may now conclude that we are able to nd possibly broken dependencies of two most common types in new objects. However, this does not always indicate broken real dependency, as we not always have enough information already input in our context. That is why we may only develop a hypothesis and ask an expert if it holds.
For more details, example, and proof of Proposition 1, please, refer to [
Normally debugging starts with a failure report. Such a report contains the input on which the program failed. By this we mean that our program was not able to output the expected result or did not nish at all. This implicitly de nes \goal" function which is capable of determining either a program run was successful or not. We could imagine a case where we do not have any successful inputs, i.e. those inputs which were processed successfully by the program. However, it does not seem reasonable. In a such a case the best option seems to rewrite the code or look for obvious mistakes. Modern techniques of software development suggests running tests even before writing code itself; unless the tests are passed code is not considered nished. Therefore, successful inputs are at least those contained in the test suites.
As discussed in the beginning of this paper the problem of nding appropriate inputs was considered by di erent authors. This problem is indeed of essential importance for debugging. However, we do not aim at solving it. Instead we assume that inputs are already found (using user reports, random generator, or something else), processed (it is better if inputs are minimized, however, not necessary), and are at hands. We focus on processing the program runs on given inputs.
Our approach consists in the following. We construct two contexts: rst with successful runs as objects, second with failed runs. In both cases attributes are the lines of the code (conveniently presented via line numbers). We put a cross if during processing of the input the program has covered the corresponding line. So in both cases we record the information about covered lines during processing of the inputs. After contexts are ready we treat all the objects from the context with failed runs as new objects and try to nd errors as described in the previous sections. Expected output is an implication A ! B. The interpretation is as follows; in successful runs whenever lines numbers A are covered, lines numbers B are covered as well. For some reason in the inspected failed run this is not the case. Debugging consists now in nding this reason. This is not absolutely automatic debugging, however, we receive some more clues and may nd a bug without checking the written code line by line. More than that, this approach is strict, that is we say that it always happens, not with any probability. And it corresponds to the real situation: the bug is there, not with any probability. 6.2
Consider the following function written in Python (example taken from [
Listing 1.1: remove html markup [
The goal of the function, as follows from its name, is to remove html markup from the input, no matter if it occurs inside or outside quotes. Therefore, we may formulate our goal as: no < in output. Such a formulation does not allow us to catch all the bugs (check input "foo" in contexts below), but it su ces for our purposes.
The function works as follows. After initialisation we have four \if" cases. The rst and the second one checks if we have encountered a tag symbol outside of quotes. If so, the value of \tag" is changed. The third one checks if we have encountered a quote symbol inside tag. This is important for not closing a tag if the closing symbol happens to be in one of the parameters (see inputs). If so, the value of \quote" is changed. The last \if" adds the current character to the output if we are outside the tag.
We consider the following set of inputs: foo, <b>foo</b>, "<b>foo</b>", "<b>a</b>", "<b></b>", "<>", "foo", 'foo', <em>foo</em>, "", <"">, <p>, <a href=">">foo</a>
Using the given outputs we obtain two contexts: Context with successful inputs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 foo <b>foo</b> "foo" 'foo' <em>foo</em> <a href=">">foo</a> "" <""> <p> "<b>foo</b>" "<b>a</b>" "<b></b>" "<>" Context with failed inputs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
It is easy to notice that the only di erence between failed inputs as context objects is their names.
Adding any of failed inputs to the rst context yields the following implication: 7; 13; 15 ! 8; 11 What is essentially said is if we happened to cover lines 7, 13, 15, we should have also had been inside tag (lines 8 and 11). Given some thought and attention we realize that this is absolutely true, because it is not clear how we could reach line 15 without having \tag"= true, as this condition is checked in line 14. In Python as well as in many other languages logical operation \and" has a higher priority as \or", so condition of the third \if" (c == '"' or c == "'" and tag) is implicitly transformed in (c == '"' or (c == "'" and tag)), that is why on lines 12 and 13 brackets are forgotten. After debugging the condition should look as follows: ((c == '"' or c == "'") and tag) and the program runs correctly.
A technique for nding errors of two types in new object intents is presented. As opposed to nding the canonical basis of the context the proposed algorithm terminates much faster. Based on this technique an approach for debugging source code is presented. This approach is capable of nding strict dependencies between lines of source code covered in successful and failed runs. The output is a logical expression which allows to debug the source code using the logic of the program. This may get us one step closer to automated debugging.