Abstract:
A method and apparatus are provided for analyzing software programs. The invention combines data flow analysis and symbolic execution with a new constraint solver to create a more efficient and accurate static software analysis tool. The disclosed constraint solver combines rewrite rules with arithmetic constraint solving to provide a constraint solver that is efficient, flexible and capable of satisfactorily expressing semantics and handling arithmetic constraints. The disclosed constraint solver comprises a number of data structures to remember existing range, equivalence and inequality constraints and incrementally add new constraints. The constraint solver returns an inconsistent indication only if the range constraints, equivalence constraints, and inequality constraints are mutually inconsistent.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention relates generally to writing, debugging, or maintaining software programs, and more specifically, the invention relates to the use of static analysis techniques to write, debug, or maintain software programs (or a combination of the foregoing).  
         BACKGROUND OF THE INVENTION  
         [0002]    There are two general techniques for debugging software programs. Dynamic debugging methods form a set of test-cases and the expected result(s) for each test-case. The program is then executed on the set of test cases and the result(s) of the execution are compared with the expected result(s). A mismatch is a symptom of an error in the program. Static debugging methods, on the other hand, form a set of properties that the program should satisfy. For example, a static debugging technique may require that a program should not crash; should satisfy given rules of accessing data; and should have outputs with a given relation to its inputs.  
           [0003]    Static methods analyze the input source code without executing it. They search for a path violating one of the properties that is to be reported as an error. In this search, static methods tradeoff efficiency for accuracy. A key issue is the determination of whether the path is feasible, i.e., are there input values that would cause the path to be executed. In general, static debugging techniques excel at discovering rare bugs whereas dynamic debugging techniques excel at finding common bugs and testing multiple modules. Thus, the two test methods are complementary.  
           [0004]    “Lint” software testing and debugging tools place a high degree of importance on efficiency and do not determine the feasibility of paths. Commercial implementations of Lint tools include Parasoft, Flexlint and Reasoning. Lint tools do not try to avoid “false errors.” “Formal verifiers,” on the other hand, are software debugging tools that determine feasibility. For that purpose, formal verifiers collect the constraints for a path to be feasible, and pass those constraints to a constraint solver. If the constraint solver determines the constraints to be consistent, then an error can be reported.  
           [0005]    Static analysis tools parse the source programs to produce a parse tree. A parse tree is a representation of the structure of the given input source programs. Parsing is performed using standard compiler techniques. In addition, static analysis tools perform semantic analysis to produce a flow graph from the given parse tree, using standard compiler techniques (where in place of emitting code, flow-graph nodes are generated). The nodes represent data flow operations, such as “+,” as well as control flow operations, such as variable assignments. There are also nodes representing conditional branching that record the condition(s) of the test. Thereafter, an analysis of the flow graph is performed. The actual form of flow graph analysis differs for different tools, but in general involves traversing the flow graph and doing some operations for each node traversed. Tools that determine feasibility of paths have to take into account the nodes representing conditional branches. From these conditional branch nodes, the tools collect the constraints for following each path. These constraints involve operations and predicates from various domains: arithmetic, pointers, arrays, and other data structures.  
           [0006]    The constraint solvers need to understand these domains, and they use several approaches for that purpose. For example, arithmetic is in general undecidable, but there is a decidable subset, referred to as Presburger arithmetic, that is adequate for the purposes of software analysis. For a detailed discussion of Presburger arithmetic, see, for example, Presburger, On the Completeness of a Certain System of Arithmetic of Whole Numbers in Which Addition Occurs as the only Operation, Hist. Philos. Logic, 12(2):225-233, 1991, Translated from German and with commentaries by Dale Jacquette, incorporated by reference herein. However, as the decision procedure for Presburger arithmetic has a super exponential lower bound, Presburger arithmetic is too expensive for the purposes of software analysis. Therefore, only subsets of Presburger arithmetic are being used.  
           [0007]    Solvers employing Presburger arithmetic, or derivatives thereof, such as linear integer programing, however, are inefficient. Such solvers are complete even for types of constraints unnecessary in software analysis, making them less efficient. At the same time, such solvers are inflexible, i.e., it is not possible to add operators outside of their theory. Another general approach to constraint solving relies on rewrite rules. For a detailed discussion of rewrite rules, see, for example, N. Dershowitz &amp; J. P. Jouannaud, Rewrite Systems, Handbook of Theoretical Computer Science, Volume B, Chapter 15, North-Holland, 1989, incorporated by reference herein. Generally, rewrite rules modify the constraints (or the flow graph) in order to arrive at an answer. While solvers employing rewrite rules express the semantics well, they are inefficient with arithmetic constraints.  
           [0008]    The static techniques (referred to as lint above) that do not evaluate the feasibility of paths tend to issue too many complaints that, in fact, do not represent any error in the program. As a result, programmers tend to ignore all complaints issued by such tools. Formal verifiers check a given implementation against a user-supplied specification. Verifiers spend more time than other source code analysis tools, achieving the highest degree of accuracy. However, there is still uncertainty. First, the verification tool may not know which input combinations are considered legal and, secondly, the problem may be too large for the verifier to handle. Both of these kinds of uncertainties are resolved by placing the burden of proof on the user. Specifically, an error is reported if the user-provided information does not allow the verifier to prove the absence of error.  
           [0009]    Static techniques that evaluate the feasibility of paths rely on a constraint solver. A constraint solver should be efficient; sound (i.e., what percentage of constraints declared inconsistent are indeed inconsistent); complete (i.e., what percentage of constraints declared consistent are indeed consistent); and flexible (i.e., how easy is it to extend the solver). As it is impossible to satisfy all four properties perfectly, traded-offs must be made. The main tradeoff is between efficiency and completeness. Ideally, a solver should be only as complete as required by the application of software analysis; being less complete would result in incorrect error reports, being more complete would result in reduced efficiency (although more program errors would be discovered).  
           [0010]    A constraint solver is needed that remembers former constraints and adds new constraints incrementally. The solver should be efficient, flexible and capable of satisfactorily expressing semantics and handling arithmetic constraints.  
         SUMMARY OF THE INVENTION  
         [0011]    Generally, a method and apparatus are provided for analyzing software programs. The invention combines data flow analysis and symbolic execution with a new constraint solver to create a more efficient and accurate static software analysis tool. The disclosed constraint solver combines rewrite rules with arithmetic constraint solving to provide a constraint solver that is efficient, flexible and capable of satisfactorily expressing semantics and handling arithmetic constraints.  
           [0012]    From the process point of view, the disclosed constraint solver analyzes a path in a software program. Initially, input constraints are received for a path in the software program to be feasible. Thereafter, one or more rewrite rules are applied to a flow graph of the software program, where the one or more rewrite rules define how the flow graph can change. At least one new node or new edge is added to the flow graph based on the rewrite rules. Finally, new constraints are derived by arithmetic constraint solving from the input constraints, flow graph and one or more existing constraints and added to the existing constraints.  
           [0013]    The disclosed constraint solver comprises a number of data structures to remember existing constraints and incrementally add new constraints. In particular, the constraint solver includes a range-constraint data structure having at least one node record corresponding to a range constraint, each node record having a node identifier identifying a node that is an operation in a flow graph of the software program and zero or more intervals associated with the respective node, the intervals including all of the possible values that the node can have during the execution of the software program; an equivalence data structure having at least one record that identifies zero or more sets of equivalent nodes that have an equivalence constraint, the equivalence constraint indicating that each of the nodes in one of the sets of equivalent nodes have the same value during a time in an execution of the software program; an inequality data structure having at least one inequality record, each defining an inequality constraint, the inequality constraint being that the product of a first value and a first node added to the product of a second value and a second node is within an inequality range; and a processor that returns an inconsistent indication only if the three data structures are mutually inconsistent.  
           [0014]    A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0015]    [0015]FIG. 1 is a block diagram of one embodiment of the present invention;  
         [0016]    [0016]FIGS. 2 a  and  2   b  are examples of software programs processed by the present invention;  
         [0017]    [0017]FIG. 3 is a flow chart describing an exemplary software analysis system using the constraint solver of the present invention;  
         [0018]    [0018]FIG. 4 is an example of a flow graph of FIG. 3;  
         [0019]    [0019]FIG. 5 is a sample table from an exemplary range-constraint data structure;  
         [0020]    [0020]FIG. 6 is a sample table from an exemplary equivalence class data structure;  
         [0021]    [0021]FIG. 7 is a sample table from an exemplary inequality data structure;  
         [0022]    [0022]FIG. 8 is a block diagram of one embodiment of a constraint solver architecture of FIG. 3;  
         [0023]    [0023]FIG. 9 is a flow chart of an exemplary check consistency process of FIG. 8;  
         [0024]    [0024]FIG. 10 is an example of a rewrite rule;  
         [0025]    [0025]FIG. 11 is a flow chart of an exemplary add range process of FIG. 8;  
         [0026]    [0026]FIG. 12 is a flow chart of an exemplary add equivalence process of FIG. 8;  
         [0027]    [0027]FIG. 13 is a flow chart of an exemplary add inequality process of FIG. 8;  
         [0028]    [0028]FIG. 14 is a flow chart of an exemplary inequality range process; and  
         [0029]    [0029]FIG. 15 is a block diagram of an example error report produced by the present invention. 
     
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS  
       [0030]    The present invention combines data flow analysis and symbolic execution with a new constraint solver to create a more efficient and accurate static software analysis tool. The tool overcomes the main weakness of symbolic execution, namely, path explosion, by using data-flow analysis to find potential faults. The goal is to discover the faults while requiring minimal effort from the programmer. The tool reports a fault symptom only if it is associated with a feasible path, i.e., a path that can possibly execute. The constraint solver of the present invention combines rewrite rules with arithmetic constraint solving. As previously indicated, conventional rewrite rule solvers express the semantics well, but are very inefficient with arithmetic constraints. Thus, integrating the two methods results in an efficient and accurate constraint solver.  
         [0031]    [0031]FIG. 1 is a block diagram of one preferred embodiment of the present invention. As shown in FIG. 1, a static analysis is performed by a static analysis system  150  on source code  110  to generate an error report  180 . The source code  110  is a set of one or more input files (programs), as input into a standard compiler. The error report  180  is a list of error symptoms in the source code  110 , as will be described in more detail below in conjunction with FIG. 15. The static analysis system  150  interacts with the constraint solver  390  to perform the static analysis of the source code  110 . In one exemplary embodiment, the static analysis system  150  offers a list of symptoms that can be detected and a user selects (from this list) the symptoms to be reported. The selection is based on the fact that none of the selected symptoms should occur in the intended program.  
         [0032]    [0032]FIGS. 2 a  and  2   b  show an example of programs  210 ,  220  that may be processed by the present invention. The program  210  in FIG. 2 a  contains an error in statement  215  {if(!I)}. A corrected version of the program is shown in FIG. 2 b , where the statement  225  is corrected {if(I)}. As a result, the function in statement  215  may return an uninitialized value of X, which is a symptom that can be detected.  
         [0033]    The symptoms or errors detected by the static analysis system  150  can generally be divided into three levels of difficulty. For a more detailed discussion of an exemplary set of difficulty levels, see D. Brand, A Software Falsifier, Int&#39;l Symposium on Software Reliability Engineering, IEEE Computer Society Press, 174-85 (October, 2000), incorporated by reference herein. These symptoms are both generic (i.e., violations of the programming language semantics), as well as project specific (mainly violations of constraints on data base accesses). Symptoms of difficulty 0 are not associated with a path. They are typically associated with just one statement, and they could cause a failure no matter which path is followed in reaching that statement. These kinds of symptoms are not the focus of the present invention, because many of them are covered by existing commercial tools. Only those difficulty 0 symptoms that have been explicitly requested by a user are detected.  
         [0034]    Difficulty 1 symptoms are violations of a finite state property along some execution path. Difficulty 2 symptoms include “index out of array range,” “dereferencing a null pointer,” or “failed assertion.” In general, all symptoms of difficulty 2 are expressible as assertions, which is what the static analysis system  150  relies on for project specific symptoms of difficulty 2. Such assertions are inserted automatically into the code  110  during parsing, and therefore do not require special consideration. Both types of symptom share the problem of identifying a feasible path whose execution will cause the symptom.  
         [0035]    In contrast to a verifier or lint, the goal of the static analysis system  150  is to ensure that any reported error can actually cause a failure during execution. In this sense, the static analysis system  150  is related to a compiler with the exception that the static analysis system  150  is allowed more time so as to detect more difficult errors. For example, the static analysis system  150  can run overnight to check code  110  written during the day.  
         [0036]    [0036]FIG. 3 is a schematic block diagram of an exemplary static analysis system  150  in accordance with the present invention. As shown in FIG. 3, the exemplary static analysis system  150  interacts with a constraint solver  390  incorporating features of the present invention. For a detailed discussion of a suitable static analysis system  150 , see, for example, D. Brand, Error Detection by Data Flow Analysis Restricted to Executable Paths, RC 21484, IBM T.J. Watson Research Center, (May, 1999), incorporated by reference herein. While the static analysis system  150  is not within the scope of the present invention, those portions of the static analysis system  150  that interact with the constraint solver  390  of the present invention are briefly discussed hereinafter.  
         [0037]    As shown in FIG. 3, the static analysis system  150  includes a parser  305  that analyzes the syntactic structure of the source code  110  (as is done in standard compilers) and produces a parse tree  310 , a representation of that syntactic structure. Thereafter, a semantic analyzer  320  extracts the semantic meaning of the parse tree  310  to produce a flow graph  330 . An exemplary flow graph  330  is shown in FIG. 4. The flow graph  330  is a graphical representation of the semantics of the source code  110  and consists of nodes, such as nodes  431 , and edges, such as edges  432 . The nodes  431  represent executable statements and are connected by edges  432  if control can flow from one node  431  to another node  431 . If the flow is only conditional, then the condition  434  is attached to the arc (edge)  433  connecting the two nodes  431 .  
         [0038]    The semantic analyzer  320  may optionally utilize the constraint solver  390  to eliminate paths  439  through the source code  110  that will not be traversed during execution of the program  110 . Thus, the overall software analysis is more efficient, since the semantic analyzer  320  will not have to be performed on the entire source code  110 . When the semantic analyzer  320  encounters a conditional node  435  in the parse tree  310 , the semantic analyzer  320  will form a constraint  350  for following one of the branches  433  emanating from the conditional node  435  and pass the constraint  350  to the constraint solver  390 .  
         [0039]    As shown in FIG. 3 and discussed further below in conjunction with FIGS. 5 through 7, respectively, the constraint solver  390  includes one or more range constraint data structures  500 , equivalence class data structures  600  and inequality data structures  700 . Generally, the constraint solver  390  determines whether a particular set of conditions is consistent. The data structures  500 ,  600 ,  700  record the constraints, equivalences and inequivalences, respectively, embodied in a given program  110 . The data structures  500 ,  600 ,  700  are generated and maintained by the constraint solver  390  in accordance with the present invention.  
         [0040]    The constraint solver  390  will process a new constraint  350  received from the semantic analyzer  320  to determine if the new constraint is inconsistent with the existing constraints  500 ,  600 ,  700 . Thereafter, the constraint solver  390  will inform the semantic analyzer  320  of the result  355 . If the new constraint is inconsistent, the semantic analyzer  320  can skip over the corresponding portion  437  of the parse tree  310 .  
         [0041]    The static analysis system  150  simplifies the graph representation  330 . The simplification has two goals: reducing the graph  330  in size for efficiency and, more importantly, making the graph  330  canonical, where possible. To reduce the size of the graph  330 , some standard compiler optimizations are performed, as described in A. V. Aho &amp; J. D. Ullman, Compilers: Principles, Techniques and Tools, Addison-Wesley (1989), such as constant propagation or value numbering. Code motion is generally not performed because that would make it harder to report a fault symptom to the user in terms of his program  110 .  
         [0042]    For ease of deduction, it is important to make the graph  330  as canonical as possible. In other words, expressions need to be rewritten to allow maximum sharing of subexpressions. For example, suppose that the two expressions 2*A−2*B&lt;12 and B−A+6&lt;1 appear in the program. They will be brought into the formats A−B&lt;6 and A−B&gt;5, sharing the common subexpression A−B so that, if their consistency ever needs to be established, the result will be immediate.  
         [0043]    Once the flow graph  330  has been simplified, the static analysis system  150  performs a data flow analysis  340 , in a known manner. For a detailed discussion of a suitable data flow analysis  340  technique, see, for example, D. Brand, Error Detection by Data Flow Analysis Restricted to Executable Paths, RC 21484, IBM T.J. Watson Research Center, (May, 1999), incorporated by reference herein. Generally, the data flow analyzer  340  traverses paths  439  in the flow graph  330  and produces a list of potential errors  345 . A potential error in the list  345  is a set of paths  439  through the flow graph  330 . If any of the paths  439  could be executed, then the software  110  would fail at the last node  438  in the path  439 . The potential errors  345  identified by the data flow analyzer  340  are then processed by a symbolic execution stage  360 , in a known manner, to generate the final error report  180 . For a detailed discussion of a suitable symbolic execution stage  360 , see, for example, D. Brand, Error Detection by Data Flow Analysis Restricted to Executable Paths, RC 21484, IBM T.J. Watson Research Center, § 5.3, at 17 (May, 1999).  
         [0044]    General symbolic execution  360  considers all feasible paths  439 , the number of which could grow exponentially with the size of a program  110 . (In the presence of loops, the number of paths  439  would be infinite, but loops are replaced by recursive procedures.) In contrast, dataflow analysis  340  combines information calculated for two reconverging paths  439 , resulting in behavior that is linear with the size of the program  110 . Dataflow analysis  340  has the advantage of efficiency, but its results cannot generally be used to report a fault symptom. Any time information is merged, some details are lost, and it is uncertain whether there is actually a feasible path  439  to the symptom. The results of symbolic execution  360  can be used to report a fault symptom, but symbolic execution  360  suffers from an exponential explosion of paths  439 .  
         [0045]    Therefore, the static analysis system  150  combines the advantages of the two types of analyses. Dataflow analysis  340  is used as a filter to screen out areas definitely containing no fault symptom. If dataflow analysis  340  finds the possibility of a fault, a “bundle” of paths is calculated that can lead to the symptom of the fault. Symbolic execution  360  is then restricted to this bundle of paths  439 , which is normally small enough to be efficient. Symbolic execution  360  selects one path  439  from the bundle to be reported to the user; however, the bundle information is also given to the user because knowing which other paths  439  lead to the symptoms helps to determine what is relevant to the fault.  
         [0046]    In addition, the data flow analyzer  340  may optionally use the constraint solver  390  to eliminate paths in the flow graph  330  that will not be traversed during execution of the program  110 . When the data flow analyzer  340  encounters a conditional node  435  in the flow graph  330 , the data flow analyzer  340  will form a constraint  350  for following one of the branches  433  emanating from the conditional node  435  and pass the constraint  350  to the constraint solver  390 . The constraint solver  390  will process the new constraint  350  to determine if the new constraint is inconsistent with the existing constraints  500 ,  600 ,  700  and will inform the data flow analyzer  340  of the result  355 . If the new constraint is inconsistent, the data flow analyzer  340  can skip over the corresponding portion  437  of the flow graph  330 .  
         [0047]    The symbolic executor  360  traverses paths  439  in the flow graph  430  that are identified by the potential errors  345 . When the symbolic executor  360  encounters a conditional node  435  in the flow graph  330 , the symbolic executor  360  will form a constraint  350  for following one of the branches emanating from the conditional node  435  and pass the constraint  350  to the constraint solver  390 . The constraint solver  390  will process the new constraint  350  to determine if it is inconsistent with the existing constraints  500 ,  600 ,  700  and will inform the symbolic executor  360  of the result  355 . If the new constraint is inconsistent, the symbolic executor  360  can skip over the corresponding portion  437  of the flow graph  330 . If the symbolic executor  360  reaches the end of the path  438  before the constraint solver  390  reports an inconsistency  355 , then there is an error in the source code  110 . For a description of the preferred embodiment of the symbolic executor  260 , see D. Brand, Error Detection by Data Flow Analysis Restricted to Executable Paths, RC 21484, IBM T.J. Watson Research Center, (May, 1999).  
       Constraint Solver  
       [0048]    As previously indicated, the constraint solver  390  determines whether a particular set of conditions is consistent. In a falsifier, such as the static analysis system  150 , the static analysis system  150  must show that a path  439  containing a fault is feasible. Therefore, if a particular path  439  is too difficult to decide, it is acceptable to give up and report nothing. In a verifier, however, the user must show the absence of error. If the verifier is unable to prove that the evidence provided by the user is sufficient, the user must be able to provide some additional information or evidence. The evidence tends to be in the form of assertions describing the state of the program  110 , or in the form of properties of some procedures, on which the program  110  relies. Such assertions need to describe the results of iteration; to describe the results of iteration requires quantifiers or some other forms of iteration. Therefore, the constraint solver  390  of a verifier must deal with quantifiers or some form of induction. But for a falsifier, no quantifiers are needed, or more exactly, all variables have an implicit existential quantifier.  
         [0049]    In general, the constraint solver  390  contains data structures  500 ,  600 ,  700  which at any time contain a set of predicates and has the following operations:  
         [0050]    1) a solver  390  can be initialized to any set of predicates;  
         [0051]    2) a solver  390  can be queried as to whether its set of predicates is satisfiable (consistent);  
         [0052]    3) a solver  390  implies a predicate p if any parameter values satisfying the predicates of the solver  390  also satisfy p;  
         [0053]    4) the solver  390  in union with p is a new solver  390  obtained from the original solver  390  by adding the predicate p;  
         [0054]    5) the intersection of a first constraint solver  390  and a second constraint solver  390  is a new solver  390  containing those predicates implied by both the first constraint solver  390  and the second constraint solver  390 , and  
         [0055]    6) the “simplification” of a predicate p under the conditions of a constraint solver  390  is another predicate that is equivalent to p whenever all the constraints of the solver are true.  
         [0056]    The input of the constraint solver  390  is a set of conditions  350  and the possible outputs  355  of the constraint solver  390  include: “the conditions are satisfiable;” “the conditions are not satisfiable;” or “cannot decide within given time limit.” Satisfiable conditions normally imply that a feasible path  439  containing a fault is found. If the solver  390  cannot decide whether the conditions in the set of conditions  350  are satisfiable, then no error is generated for the user.  
         [0057]    In one exemplary embodiment, the present invention contemplates two levels of constraint solvers  390 , namely, state-sensitive and state-insensitive constraint solvers  390 . A state-insensitive solver  390  is used by a state-insensitive dataflow analysis  340 ; while the more accurate state-sensitive solver  390  is used by a state-sensitive data-flow analysis  340  and symbolic execution  360 . The difference lies in the treatment of the variables appearing in the list of conditions. While the state-insensitive solver  390  assumes that all the variables are independent of each other, the state-sensitive solver  390  takes into consideration the structure of the flow graph  330  defining the values of the variables. This then provides the difference between the state-sensitive and state-insensitive analysis as was explained earlier.  
         [0058]    The state-insensitive solver  390  works by ‘anding’ all the conditions, while ignoring the graph  330 . If that results in an inconsistency, then the conditions are inconsistent even in the more accurate state-sensitive sense and the candidate fault is not feasible. Conditions that are consistent in the state-insensitive sense, however, might not be consistent in the state-sensitive sense.  
         [0059]    The state-sensitive solver  390  works by building a set of equalities and inequalities  600 ,  700 , respectively, concerning edges  432  in the graph  330 . The equalities and inequalities  600 ,  700  are obtained by propagating information about inputs of a node  431  to its outputs or vice versa. For example, from 
         aε(0, 3), bε(1, 4) it can be deduced that a+bε(1, 7). 
         [0060]    Such information is typically propagated using rewrite rules  820 , discussed further below in conjunction with FIGS. 8 through 10, in a known manner. The rewrite rules  820  for propagating information correspond to the laws of arithmetic, or any other data domain. More rules  820  make the solver  390  more powerful, but also slower. Only those rules  820  that are actually found needed in an application domain, such as design automation software, are given to the solver  390 . In the case of design automation software, it was sufficient to have only the rules  820  concerning the operations and relations of arithmetic and bit-wise operators. In addition to the small needs for arithmetic, very little propositional calculus sophistication was needed. Resolution with unit clauses proved sufficient.  
         [0061]    On the other hand, it is important for the solver  390  to understand the interaction between the data flow and control flow. The issue concerns variables appearing in the conditions  350  input to the solver  390 . These variables refer to values they had been assigned in the program  110 , and the state-sensitive solver  390  needs to use these values. One approach to the issue considers one path  439  at a time; then all the variables can be replaced unambiguously by their values. Considering one path  439  at a time, however, is too inefficient. It is necessary to consider a bundle of paths  439  at a time. When a bundle of paths  439  are considered, a variable could be assigned different values along different paths  439  in the bundle. Additional approaches to this problem, give the solver  390  a complete description of the values each variable is assigned, and the corresponding conditions, but the description is usually unacceptably large.  
         [0062]    The solver  390  can simplify the given flow graph  330  under the conditions  350  given to the solver  390 , which may resolve the values of variables. In any case, the simplification will identify two variables as having the same contents if that is implied by the given conditions.  
         [0063]    Both the state-sensitive and state-insensitive solvers  390  are incremental in the sense that adding a new condition does not require recalculating what was derived for the prior conditions. This is important because intra-procedural analysis traverses paths  439  and constantly asks the solver  390  whether the partial path  439  traversed so far is feasible.  
         [0064]    [0064]FIG. 5 is a diagram of an exemplary range constraint data structure  500 . As previously indicated, the range constraint data structure  500  records each of the range constraints for a given program  110 . As shown in FIG. 5, the range constraint data structure  500  consists of a plurality of records, each associated with a different range constraint  550 . For each range constraint  550 , the range constraint data structure  500  identifies the associated node in field  505 , and indicates one or more corresponding ranges in a range field  510 . Each range in the range field  510  consists of zero or more intervals. Each interval contains a pair of integers. The range constraint  550  is therefore a (node, range) pair constraining all possible executions to those which will cause the given node to have an integer value in the given range.  
         [0065]    [0065]FIG. 6 is a diagram of an exemplary equivalence class data structure  600 . As previously indicated, the equivalence class data structure  600  records each of the equivalence classes for a given program  110 . As shown in FIG. 6, the equivalence class data structure  600  contains zero or more records, each corresponding to an equivalence class  650 . Each equivalence class  650  consists of one or more nodes  431  that have been determined to be equivalent. One of the equivalent nodes is selected to be the representative of the equivalence class and is identified in field  610 . The remaining nodes in the equivalence class  650  are identified in field  620 .  
         [0066]    [0066]FIG. 7 is a diagram of an exemplary inequality data structure  700 . As previously indicated, the inequality class data structure  700  records each of the inequality constraints for a given program  110 . As shown in FIG. 7, the inequality data structure  700  contains zero or more records each associated with a different inequality constraint  750 . Each inequality constraint  750  is a quintuple (a, A, b, B, R) consisting of two coefficients a, b, identified in fields  710  and  730 , respectively, two nodes A, B, identified in fields  720  and  740 , respectively, and a range R x  identified in field  745 . An inequality constraint  750  constrains all possible executions to those which will cause the given nodes A, B to have integer values where aA+bB is in the range R. The coefficients a, b are any integers.  
         [0067]    In general an inequality constraint  750  is the most general form of a linear constraint involving two nodes. A linear constraint involving just one node would be of the form aA in R, which is equivalent to a range constraint  550 . A linear constraint involving three nodes would be of the form aA+bB+cC in R, and similarly for linear constraints involving more nodes. Linear constraints involving more than two nodes are not necessary, as they do not occur often, and when they do occur they can be handled using the rewrite rules. On the other hand, linear constraints involving two nodes occur very often and for efficiency reasons special handling of the form described here is provided by the present invention. Examples of linear constraints between two nodes include 
         A&lt;B represented as 1*A+(−1)*B in (−infinity, 0) 
         A !=B represented as 1*A+(−1)*B in (−infinity, 0)v (0, infinity) 
         2*A&lt;3*B+5 represented as 2*A+(−3)*B in (−infinity, 5) 
         A=B represented as 1*A+(−1)*B in {0} 
         [0068]    The last example is an equality, which is more efficiently represented by the equivalence classes  600 . Therefore, the linear constraints between two nodes are used to represent just inequalities.  
         [0069]    [0069]FIG. 8 is a block diagram of a constraint solver  390  incorporating features of the present invention. As shown in FIG. 8, the constraint solver  390  is accessed through a check consistency process  810  or an add range process  830 . As previously indicated, the constraint solver  390  can be called by the procedures  320 ,  340 ,  360  in the static analysis system  150 . If a procedure  320 ,  340 ,  360  issues a call  351  to the add range process  830 , the constraint solver  390  will execute the add range process  830 , as well as an add equivalence process  840 , as appropriate, each discussed below in conjunction with FIGS. 11 and 12. If a procedure  320 ,  340 ,  360  issues a call  351  to the check consistency process  810 , the constraint solver  390  will execute a rewrite rules process  820 , discussed below in conjunction with FIG. 9. The rewrite rules may call the add range process  830 , add equivalence process  840  or the add inequality process  850 , as appropriate. Since the rewrite rules process  820  is a computationally expensive process, it is typically only called by the symbolic execution process  360 . As shown in FIG. 3, each procedure call  351  includes a new constraint  350  to be evaluated by the constraint solver  390  for consistency.  
         [0070]    [0070]FIG. 9 is a flow chart describing an exemplary check consistency process  810 . If a procedure call  351  is made to the check consistency process  810 , the process  810  will determine whether the set of contraints  350  in the data structures  500 ,  600 ,  700  are consistent with the flow graph The rewrite rules invoked by the check consistency process  810  may, if appropriate, add new nodes  431  and edges  432  to the flow graph  330  and they may call the add range process  830  (FIG. 11), the add equivalence process  840  (FIG. 12) and/or the add inequality process  850  (FIG. 13) which will add new constraints  550 ,  650 ,  750  to data structures  500 ,  600 ,  700 , respectively.  
         [0071]    For example, the add range process  830  takes as input a range constraint  550  from the rewrite rules  820  and adds the range constraint  550  to the existing constraints contained in the range data structure  500 . In addition, the add range process  830  may derive other constraints and, if appropriate, call the add equivalence process  840  with a new equivalence constraint  650  to add new nodes  431  to the existing equivalence class(es)  650  and/or add new equivalence classes  650  to the equivalence data structure  600 . As noted earlier, each equivalence constraint  650  is a pair of nodes which constrains the set of executions to those where the two nodes  431  have identical values.  
         [0072]    Similarly, the add inequality process  850  takes as input an inequality constraint  750  and adds the inequality constraint  750  to the inequality data structure  700 . In addition, the add inequality process  650  may derive other constraints from the inequality constraints  750  and may call the add equivalence process  840  to add equivalence constraints  650  to the equivalence data structure  600 .  
         [0073]    Upon entry to the check consistency process  810 , a timer is initialized to TIME_LIMIT and the timer is started. As shown in FIG. 9, a test is performed during step  910  to determine if any node  431  has been marked such that rewrite rules  820  would apply. If it is determined during step  910  that no node  431  has been marked, then CONSISTENT is returned to the calling process  320 ,  340 ,  360  during step  915 . Otherwise, if it is determined during step  910  that a node  431  has been marked, then a further test is performed during step  920  to determine if the check consistency process  810  has been running longer than a preset time limit, TIME_LIMIT.  
         [0074]    If it is determined during step  920  that the timer has expired, then OUT_OF_TIME is returned during step  925  to the calling process  320 ,  340 ,  360 . Otherwise, if it is determined during step  920  that the timer has not expired, then the rewrite rules  820  are applied during step  930 .  
         [0075]    Initially, all the nodes are ordered during step  930  according to the topology and then each node  431  is visited from the first node to the last node in the ordered list. If a node  431  is a representative of an equivalence class  650 , any rule applicable to the node  431  is applied. A test is performed during step  935  to determine if any rewrite rule discovers an inconsistency in the constraints  500 ,  600 ,  700 . If it is determined during step  935  that a rewrite rule has discovered an inconsistency in the constraints  500 ,  600 ,  700 , then INCONSISTENT is returned to the calling process  320 ,  340 ,  360  during step  938 .  
         [0076]    Rules are then applied in reverse topological order during step  940 . Each node  431  is visited from the last node to the first node in the ordered list. If a node  431  is a representative of an equivalence class  650 , then any rule applicable to the node  431  is applied. A test is performed during step  945  to determine if any rule has discovered an inconsistency in the constraints  500 ,  600 ,  700 . If it is determined during step  945  that a rewrite rule has discovered an inconsistency in the constraints  500 ,  600 ,  700 , then INCONSISTENT is returned to the calling process  320 ,  340 ,  360  during step  948 . If, however, it is determined during step  945  that a rewrite rule has not discovered an inconsistency in the constraints  500 ,  600 ,  700 , then program control returns to step  910  and continues in the manner described above.  
         [0077]    [0077]FIG. 10 illustrates an example of a rewrite rule  820 . A flow graph  1010  is a portion of the larger flow graph  330  prior to the application of a rewrite rule  820 . As shown in FIG. 10, three nodes A, B, C are applied to an addition node “+,” which is labeled D. A range constraint  1015  associated with node D is present in the range data structure  500 . The range associated with node D is equal to zero. The result of applying the rewrite rules  820  to the flow graph  1010  is the flow graph  1020 , whereas node E has been added to the original flow graph  1010  to represent the expression −(A+B). More specifically, the range constraint  1015  associated with node D (i.e., that D is equal to 0), implies that the expression defined by flow graph  1010  (i.e., (A+B+C)=D) must equal 0 and may be expressed as follows: 
           A+B=−C  or  C=− ( A+B ). 
         [0078]    Thus, the node E is added to the flow graph  1020  to express the following: 
           E=− ( A+B ), 
         [0079]    and the equivalence structure  1030  is added to the data structure  600  to indicate that the nodes C and E are equivalent.  
         [0080]    For example, the rewrite rules  820  may optionally include one or more of the following exemplary rules:  
         [0081]    Rule 1: Consider  
         [0082]    signed int S;  
         [0083]    unsigned int U;  
         [0084]    U=(unsigned int) S; or S=(signed int) U  
         [0085]    Then  
         [0086]    Add_Range(U, RangeOf(S) &amp; (−1, 2{circumflex over ( )}31) | RangeOf(S) &amp; (−2{circumflex over ( )}31−1,0)+2{circumflex over ( )}32)  
         [0087]    Add_Range(S, RangeOf(U) &amp; (−1, 2{circumflex over ( )}31) | RangeOf(U) &amp; (−2{circumflex over ( )}31−1,0)−2{circumflex over ( )}32)  
         [0088]    Rule 2: Consider  
         [0089]    X=A ? B:C;  
         [0090]    if RangeOf(A) &amp; {0}==empty then Add_Equivalence(X, B)  
         [0091]    if RangeOf(A) is a subset of {0} then Add_Equivalence(X, C)  
         [0092]    if B=C then Add_Equivalence(X, C)  
         [0093]    Rule 3: Consider  
         [0094]    X=A op B, where op is an operator +, −, *, /, %, and po is the inverse of op  
         [0095]    then  
         [0096]    Add_Range(X, RangeOf(A) op RangeOf(B))  
         [0097]    Add_Range(A, RangeOf(X) po RangeOf(B))  
         [0098]    Rule 4: Suppose  
         [0099]    X=a*A+b*B+C, where a and b are integer constants  
         [0100]    then  
         [0101]    Add_Inequality(a, A, b, B, RangeOf(X)−RangeOf(C))  
         [0102]    Rule 5: Suppose  
         [0103]    0=A+B+C  
         [0104]    then  
         [0105]    Add_equivalence(C, −A−B)  
         [0106]    Rule 5 is applied only to the term C that is latest in topological order.  
         [0107]    Rule 6:  
         [0108]    A(I)=u; x=A(J);  
         [0109]    becomes  
         [0110]    A(I)=u; x=u; provides I=J  
         [0111]    alternatively it becomes  
         [0112]    x=A(J); A(I)=u; provided RangeOf(I) &amp; Range(J) is empty  
         [0113]    Rule 7:  
         [0114]    if (a) {S} else {T} x=u;  
         [0115]    becomes  
         [0116]    − if (a) {S} else {T; x=u;} provided a=0  
         [0117]    or  
         [0118]    − if (a) {S; x=u;} else {T} provided RangeOf(a) does not contain 0  
         [0119]    Rule 8: For any operation  
         [0120]    X=A op B  
         [0121]    if A=A′ and B=B′ and X′=A′ op B′ 
         [0122]    then Add_Equivalence(X, X′)  
         [0123]    [0123]FIG. 11 is a flow chart describing an exemplary add range process  830 . As previously indicated, the add range process  830  will add a range constraint  550  to the range data structure  500 . For example, the data structure  500  can initially contain a range constraint indicating that a given node, A, has a corresponding Range 0 . A new range constraint is input using the add range process  830  indicating that the given node, A, also has a corresponding Range. Initially, the add range process  830  forms the intersection of Range 0  and Range during step  1120  to test the contents of the intersection. If it is determined during step  1120  that the intersection is empty, then the add range process  830  returns INCONSISTENT to the calling routine during step  1124 . If it is determined during step  1120  that the new Range is equal to the original Range 0 , then there are no changes to the constraint(s) and NOTHING 13  NEW is returned to the calling process during step  1126 . If it is determined during step  1120  that the intersection consists of a single number k, then the add equivalence process  640  is called during step  1128  to add the equivalence node pair (A, k) to the data structure  600 . Here, k is a node  431  representing the integer k.  
         [0124]    In step  1130 , the data structure  500  is updated by changing the range constraint  550  of each node  431  sharing the equivalence class of A (as indicated by data structure  600 ) to contain the intersection of Range 0  and Range.  
         [0125]    In step  1140 , the inequality data structure  700  is updated using the process  1400  on each inequality record present in  700 . If the process  1400  returns INCONSISTENT for any of the inequalities, then so does process  830 . Otherwise, CONSISTENT is returned to the calling process during step  1150 .  
         [0126]    [0126]FIG. 12 is a flow chart of an exemplary add equivalence process  840 . As previously indicated, the add equivalence process  840  will add an equivalence constraint  650  to the equivalence data structure  600 . For example, assume that the range data structure  500  contains two range constraints indicating that a node A has a range, Range A , and a node B has a range, Range B . In addition, assume that the equivalence data structure  600  contains two equivalence classes indicating that a representative node, A 0 , has equivalent nodes including a node A, and a representative node, B 0 , has equivalent nodes including a node B.  
         [0127]    As shown in FIG. 12, a new equivalence constraint  1250  containing nodes A and B is input to the add equivalence process  840 . In step  1260 , if nodes A and B are in the same equivalence class  650 , then there are no changes to the existing constraints and NOTHING 13  NEW is returned to the calling process during step  1265 . If A and B are not in the same equivalence class  650 , then a variable, NEW 13  RANGE, is set equal to the intersection of Range A , and Range B . A test is performed during step  1280  to determine if the variable, NEW_RANGE, is empty. If it is determined during step  1280  that the variable is empty, then the constraints  500 ,  600  are inconsistent and INCONSISTENT is returned during step  1285  to the calling process. If it is determined during step  1280  that the variable is not empty, the constraints  500 ,  600  are consistent and the two equivalence classes indicating that a representative node, A 0 , has equivalent nodes including a node A, and a representative node, B 0 , has equivalent nodes including a node B will be merged during step  1290  with their union. In step  1295 , the inequality data structure  700  is updated using the process  1400 , which may return INCONSISTENT, in which case the process  1200  also returns INCONSISTENT during step  1297 . Otherwise, CONSISTENT is returned to the calling program during step  1298 .  
         [0128]    [0128]FIG. 13 is a flow chart of an exemplary add inequality process  850 . As previously indicated, the add inequality process  850  will add an inequality constraint  750  to the inequality data structure  700 . As shown in FIG. 13, an inequality constraint {a, A, b, B, Range}  1310  representing the relation a*A+b*BεRange is applied to the add inequality process  850 . A test is initially performed during step  1320  (using the inequality range process  1400  of FIG. 14) to determine whether the given inequality constraint  1310  is inconsistent with the data structures  500 ,  600 ,  700 . If the inequality constraint  1310  is inconsistent with the data structures  500 ,  600 ,  700 , then INCONSISTENT is returned to the calling procedure during step  1324 .  
         [0129]    If the inequality constraint  1310  is implied by the constraints of data structures  500 ,  600 ,  700 , then there is no new information and NOTHING_NEW is returned to the calling procedure  320 ,  340 ,  360  during step  1328 . Otherwise, step  1330  is performed, where the input inequality constraint  1310  is normalized. The normalization of the input inequality constraint  1310  will, e.g., remove any common divisors, and ensure that the inequality expressions are always written in the same canonical way.  
         [0130]    For example, in step  1330 , if node A  431  does not come before node B  431  in topological order, then nodes A and B are swapped. In addition, if a is less than zero, then the values a, b, and Range are multiplied by −1. Likewise, if the values a, b are not relatively prime, then the values a, b and Range are divided by their greatest common divisor.  
         [0131]    In step  1340 , a subsumption test is performed. If the inequality data structure  700  contains an inequality a, A, b, B, Range′ and Range′ is a subset of Range, then no new information is obtained and NOTHING_NEW is returned to the calling procedure  320 ,  340 ,  360  during step  1345 . If the inequality data structure  700  contains an inequality a, A, b, B, Range′ and Range′ is a superset of Range, then the inequality constraint a, A, b, B, Range′ is deleted from the inequality data structure  700  and Range is set equal to Range{circumflex over ( )}Range′.  
         [0132]    In step  1350 , if an inequality a, A, b, B, Range contradicts an existing inequality a′, A, b′, B, Range′, then INCONSISTENT is returned to the calling procedure during step  1355 . Otherwise, the equality is evaluated during step  1360 . In step  1360 , if Range consists only of the number 0 and a*b equals −1, then the add equality process  840  is called during step  1365  to add the equivalent node pair (A,B)  431  to the equivalence data structure  600 . Otherwise, the inequality data structure  700  is updated during step  1370 . In step  1370 , the inequality data structure  700  is updated by adding the inequality a, A, b, B, Range to the list of inequalities  700 . Each inequality constraint (b′, B, c, C, Range′) in data structure  700  is tested and, if the expression Add_Inequality(a*b′, A, −b*c, C, b′*Range−b*Range′) equals inconsistent, then INCONSISTENT is returned in step  1374 . Each inequality constraint (a′, A, c, C, Range′) in data structure  700  is tested and, if the expression Add_Inequality(a′*b, B, −a*c, C, a′*Range−a*Range′) equals inconsistent, then INCONSISTENT is returned in step  1374 . Otherwise, CONSISTENT is returned in step  1380 .  
         [0133]    [0133]FIG. 14 is a flow chart for an exemplary inequality range process  1400 . Generally, the add inequality range process  1400  derives information about the ranges of nodes A and B from the inequality a*A+b*B in Range. The process  1400  updates the Range and returns the updated range to the calling process. In step  1410 , the process  1400  is passed the inequality constraint (a, A, b, B, Range). In step  1420 , a*RangeOf (A)+b*RangeOf (B) is evaluated to determine if the result is a subset of Range. If a*RangeOf (A)+b*RangeOf (B) is a subset of Range, then NOTHING_NEW is returned in step  1428 . Otherwise, Range is set equal to Range &amp; (a*RangeOf (A)+b*RangeOf (B)) in step  1430 .  
         [0134]    In step  1440 , Range is tested to determine if it is empty. If Range is empty, then INCONSISTENT is returned in step  1424 . Otherwise, the range constraint (A, (Range−b*RangeOf (B))/a) is added to the range constraint data structure  500  by calling add range in step  1450 . The result of the add range call is then tested in step  1460 . If the add range result is inconsistent, then INCONSISTENT is returned in step  1465 . Otherwise, the range constraint (B, (Range−a*RangeOf (A))/b) is added to the range constraint data structure  500  by calling add range in step  1470 . The result of the add range call is then tested in step  1480 . If the add range result is inconsistent, then INCONSISTENT is returned in step  1485 ; otherwise, CONSISTENT is returned in step  1490 .  
         [0135]    [0135]FIG. 15 is an example error report  180  produced by the invention. As shown in FIG. 15, the error report  180  includes a list of branch points  1581  in the source code  110 , which must be followed for the error to occur. For example, the exemplary error report  180  is an uninitialized variable error and indicates at least one possible path leading to the error.  
         [0136]    It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention.