Patent Publication Number: US-8984343-B2

Title: Error propagation in a system model

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority to U.S. Provisional Patent Application Ser. No. 61/442,648 entitled “Method to Propagate Error Associated with Type, Range, and Signal Value Data through a Behavioral Model” filed on Feb. 14, 2011, the disclosure of which is hereby incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     This invention was made with Government support under Contract No. NNA10DE73C awarded by NASA. The Government may have certain rights in the invention. 
    
    
     BACKGROUND 
     Model-based design can be used for hardware and software systems (e.g., cyber-physical systems (CPSs)). Data flow semantics can be used to specify control algorithms. One area in which model-based design is increasingly applied is for the design and certification of flight-critical software. In this area, MATLAB Simulink and Esterel Technologies SCADE, in particular, are widely used in the aerospace industry for modeling and simulation-based evaluation of avionics CPSs. Both Simulink and SCADE use data flow models for model-based design. 
     Verification tools exist to analyze type and range data in the context of data flow models, according to the DO-178B software certification process. Such tools can automate a number of previously manual tasks, including code reviews, model analysis, and object code testing. 
     SUMMARY 
     One exemplary embodiment is directed to a method providing an input signal range corresponding to a range of expected values for an input signal to a functional block. A minimum value error range corresponding to a range of error for a minimum value endpoint of the input signal range and a maximum value error range corresponding to a range of error for a maximum value endpoint of the input signal range is also provided. The method maps the input signal range to one or more output signal ranges as a function of a range mapping function corresponding to the functional block. The method also calculates a set of error extended input signal ranges by: adding a min endpoint of the minimum value error range to the minimum value of the input signal range; adding a max endpoint of the minimum value error range to the minimum value of the input signal range; adding the min endpoint of the maximum value error range to the maximum value of the input signal; and adding the max endpoint of the maximum value error range to the maximum value of the input signal range. The set of error extended input signal ranges are mapped to a set of error extended output signal ranges as a function of the range mapping function. Finally, a minimum output error range and a maximum output error range are calculated as a function of a difference between the set of error extended output signal ranges and the output signal ranges. 
    
    
     
       DRAWINGS 
       Understanding that the drawings depict only exemplary embodiments and are not therefore to be considered limiting in scope, the exemplary embodiments will be described with additional specificity and detail through the use of the accompanying drawings. 
         FIG. 1  illustrates a computer for execution of a software verification tool in accordance with one embodiment. 
         FIG. 2  illustrates an example of a data flow model for a system under test in accordance with one embodiment. 
         FIG. 3  illustrates an example of an interval and error ranges associated with endpoints of the interval. 
         FIG. 4  illustrates a method for propagating signal value error through a functional block in a model in accordance with one embodiment. 
         FIG. 5  illustrates a data flow model for a system under test in which signal value error is propagated through the model in accordance with one embodiment. 
         FIG. 6  illustrates a data flow model for a system under test having both continuous and discrete signals in which signal value error is propagated through the model in accordance with one embodiment. 
         FIG. 7  illustrates a model having a feedback signal in accordance with one embodiment. 
         FIG. 8  illustrates a model in which the pattern of a feedback loop from  FIG. 6  has been replaced with the functional block implementing a feedback counter function. 
     
    
    
     In accordance with common practice, the various described features are not drawn to scale but are drawn to emphasize specific features relevant to the exemplary embodiments. 
     DETAILED DESCRIPTION 
     In the following detailed description, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific illustrative embodiments. However, it is to be understood that other embodiments may be utilized and that logical, mechanical, and electrical changes may be made. Furthermore, the method presented in the drawing figures and the specification is not to be construed as limiting the order in which the individual steps may be performed. The following detailed description is, therefore, not to be taken in a limiting sense. 
     Current software verification tools do not directly handle errors associated with signal values (also referred to herein as “signal value errors”). A signal value error can occur when a representation of a signal value, over which computation is performed, is different than its corresponding ground truth values. That is, a signal value error can include a difference between an actual output signal from a component in a system and the ideal output signal from the component. 
     This challenge is commonly faced when abstractions, assumptions, and restrictions are utilized with the goal of increasing the scalability of analysis methods. For example, synchronous languages can rely on synchrony and zero-time execution assumptions that are typically not valid in a physical implementation. 
     One example of an abstraction includes a floating-point representation of certain numeric values. This floating-point representation error is generally proportional to signal magnitude. In other words a bound for a signal value of one million will tend to be greater than a bound for a signal value of ten or one. This is because correct rounding may be performed only to a limited number of decimal places (e.g., 7 for 32-bit floats, 16 for 64-bit floats, and 34 for 128 bit floats). If a decision within a system is dependent upon an expression using floating-point rounding, the resulting behavior may not be deterministically predictable. Additionally, this source of error can be exacerbated for accumulated error. For example, for a loop in which a 32-bit float variable is incremented a thousand times, the effect of the error can be present in values near one hundred thousand rather than one hundred million. 
     A second source of signal value error can be due to hardware floating-point units that may produce signal value error. So, for example, even if two floating-point values that have no error are multiplied together, the result can still have signal value error. 
     A third source of signal value error can be due to mixed continuous and discrete computation. For example, a continuous sensor signal can be periodically buffered and reported. Clock skew, however, can result in the continuous signal being captured too early to too late, which may result in a value different from the ground truth value. Also, when computation uses periodic data that is sampled between periods, interpolation can be used. Since interpolation is an estimate, signal value error can be produced from interpolation. These error sources tend to be bounded. For example, the error arising from clock skew can be bounded by the maximum rate of signal change corresponding to the maximum clock skew. 
     Yet another source of signal value error is sensor accuracy. This source of error can be constant (e.g., plus or minus 0.001) across a range of operational values and undefined outside of this range. 
     Signal value errors can lead to differences between the semantics of system models and the actual behavior of the system observed on an execution platform. By quantifying the differences between the “ideal” case and the actual behavior, the effects of errors can be systematically analyzed, and the correctness of the implementation can be shown in the presence of errors. 
     Not only can output values be different than ground truth (e.g., ideal) values, but signal value error may also result in timing jitter as errors push condition values across discrete decision points in the code. Essentially, signal value errors can non-deterministically affect the behavior and performance of the system controlled by the software. 
     Conventional systems have been certified by relying on the argument that the error is bounded. In these previous approaches, a static and conservative error threshold (e.g., a tolerance) has been applied to an output signal of the model. That is, an output signal value for a model is compared to an error interval for that output signal. If the output signal value is outside of the interval, the output signal is considered to have failed. Each output signal can have a different error threshold. However, the sizes of the intervals typically depend upon the underlying data type. Analysis is typically not performed to determine whether the particular tolerance is appropriate for the given model. Accordingly, the error for a given signal is not propagated through the system. 
     For example, if a tolerance is plus or minus 0.0001 and an output signal value of 2.13 is expected, but a value of 2.12999999 is measured, the test is considered passed. This is because 2.13-0.0001&lt;2.12999999&lt;2.13+0.0001. 
     As system complexity has increased, a signal static tolerance factor can be overly conservative in most cases, while not being conservative enough in rare cases. Furthermore, applying a tolerance to output signals does not provide a mechanism to analyze timing jitter and potential for non-deterministic behavior that may originate from signal value error. 
     Embodiments of the present subject matter can enable the analysis of signal value errors for system models. In an example, signal value errors can be propagated through the functional blocks of a system model to analyze possible effects as the signal value errors impact incident functional blocks. This propagation of the errors can be applicable to many models of computation including avionics models, synchronous data flow and Kahn process networks. 
     Signal value errors can have a number of sources. One potential source includes a floating-point representation of certain numeric values. This can be one of the most widespread sources of error in the modeling of practical flight-critical systems. 
     Accordingly, the characteristics of each source of signal value error can be different. Embodiments of the subject matter described herein can represent these different error characteristics and can enable translation between these error characteristics. 
     In an example, the subject matter herein can be used to determine a magnitude of error on an output of a system given a characterization of signal value error for input signal(s) to the system. In an example, the subject matter herein can be used to determine if signal value error can non-deterministically change the behavior of the system. For example, the subject matter herein can be used to identify if and where signal value error can potentially cause a mode change to occur too early or too late. 
     In an example, the subject matter herein can be used to detect error-induced underflow. Underflow can include the condition when the result of a floating-point operation is smaller than the smallest representable value. For example, if a feedback loop increments a signal by one each period and the signal is stored as a 32-bit float, after the signal increments to near 100,000,000 the rounding error may negate the increment operation. Thus, the signal might not ever reach a particular value that is greater than 100,000,000. Error-induced underflow is similar, but can be due to any source of signal value error. Error-induced underflow can be detected by quantifying the signal value error for an output signal from a functional block and determining if the signal value error can negate the operation performed by the functional block. 
     In an example, the subject matter herein can be used to determine if signal value error can move ranges of an output signal from a functional block outside of accepted range. An output signal outside of the accepted range can cause overflows or exceptions. Analyzing the range of output signals with signal value error propagation can be used to determine whether anomalous behavior is possible for the system. 
       FIG. 1  illustrates a computer  100  for execution of a software verification tool. The computer  100  can include one or more processing devices  102  (e.g., a central processing unit (CPU), microcontroller, microprocessor, etc.) coupled to one or more memory devices  104  (e.g., random access memory (RAM), a hard drive, an optical medium (CD), etc.). The one or more memory devices  104  can include instructions which, when executed by the one or more processing devices  102 , can cause the one or more processing devices  102  to perform the functions of a software verification tool as described herein. 
     Separate from or in addition to the one or more memory devices  104 , the instructions can be stored on any appropriate computer readable medium used for storage of computer readable instructions or data structures. The computer readable medium can be implemented as any available media that can be accessed by a general purpose or special purpose computer or processor, or any programmable logic device. Suitable processor-readable media can include tangible media such as magnetic or optical media. For example, tangible media can include conventional hard disks, Compact Disk-Read Only Memory (CD-ROM), volatile or non-volatile media such as Random Access Memory (RAM) (including, but not limited to, Synchronous Dynamic Random Access Memory (SDRAM), Double Data Rate (DDR) RAM, RAMBUS Dynamic RAM (RDRAM), Static RAM (SRAM), etc.), Read Only Memory (ROM), Electrically Erasable Programmable ROM (EEPROM), and flash memory, etc. Suitable processor-readable media can also include transmission media such as electrical, electromagnetic, or digital signals, conveyed via a communication medium such as a network and/or a wireless link. 
     In an example, the computer  100  can include one or more input devices  106  (e.g., a mouse, keyboard, touchscreen, microphone, etc.) for receiving inputs from a user. The computer  100  can also include one or more output devices  108  (e.g., a monitor, speaker, light, etc.) for providing output to a user. The computer  100  can comprise a desktop computer, workstation, laptop, tablet, mobile phone, or other computing device. In some examples, the computer  100  can be distributed in nature. 
     As mentioned above, the instructions on the one or more memory device  104  can cause the one or more processing device  102  to perform the functions of a software verification tool as described herein. In an example, the software verification tool can utilize an extension of interval arithmetic to represent ranges of feasible signal values in order to support the computation of range propagation and analysis. 
     An interval may be represented by the tuple I=(min, max, includeMin, includeMax) where min represents the lower bound endpoint of the interval, max represents the upper bound endpoint of the interval, includeMin indicates if the min value is included in the interval, includeMax indicates if the max value is included in the interval. As used herein, the common interval notation of the min and max values enclosed in parentheses or brackets depending on whether or not the associated endpoint is included in the internal. For example, the interval (3, 7] indicates the interval I=(3, 7, false, true). 
     A range may be comprised of a set of intervals. In one embodiment, the set of intervals may be a non-overlapping set of intervals. 
     A range can include data type information. Data type information may be used for checking for value overflow and underflow and to guide certain type-specific range operations. The data type information can indicate the included set of values between the min and max endpoints. If data type information in unavailable, a default data type can be assigned. 
     In one embodiment a range can include a property “type” from the set T, where T={Boolean, Integer8, Integer16, Integer32, Integer64, Unsigned8, Unsigned16, Unsigned32, Unsigned64, Float32, Float64, Undefined}. Other sets of types are also possible. 
     A range can be specified as the tuple R=(min, max, includeMin, includeMax, C, T), where: min represents the lowest bound endpoint of all min values of all included intervals, max represents the upper bound endpoint of all max values of all included intervals in the range, includeMin indicates if the min value is included in the range, includeMax indicates if the max value is included in the range, C is the ordered set of intervals. For example, the range [1, 5] may include the intervals [1, 2) and [2, 5]. In one embodiment, a range can be hierarchically composed of multiple non-overlapping child ranges. 
     In an example, the software verification tool can use a model of a system under test having one or more functional blocks connected according to a graph structure. In an example, the graph structure can include a directed graph. The software verification tool can use the model to compute one or more ranges for one or more output signals from the model. In an example, the software verification tool can use the model along with specifications for one or more ranges of one or more input signals to the model to compute the one or more ranges for the one or more output signals. To compute the range(s) for the output signal(s), the software verification tool can propagate the range(s) for the input signal(s) topologically through the functional blocks until the output signal(s) are reached. A functional block v of the set V can have a set of inputs (X) and a set of outputs (Y), corresponding to edge sources and targets. A functional block can map one or more ranges for one or more input signals for the functional block to one or more ranges for one or more output signals for the functional block. The mapping of an output signal for a functional block from one or more input signals for the functional block can be defined by a range mapping function. Different output signals for a functional block can have different range mapping functions corresponding thereto. The range mapping function(s) for a functional block can be used to determine the range(s) for the output signal(s) for the functional block as a function of the range(s) of the input signal(s) for the functional block. 
       FIG. 2  illustrates an example of how range propagation can be performed for a model  200  for a system under test as described in U.S. patent application Ser. No. 13/006,750 entitled “Type and Range Propagation through Data-Flow Models” filed on Jan. 14, 2011, the disclosure of which is hereby incorporated herein by reference in its entirety. The model  200  can include a first input signal block  202  and a second input signal block  204 . An input signal block represents an input signal to the model. In an example, the first input signal can have a range of expected values of −100.0 to 200.0 (i.e., [−100.0, 200.0]). The expected values for the input signal can include the one or more values in which the input signal can have not including signal value error for the input signal. In this example, the second input signal can have a range of expected value of [−50.0, 50.0]. For simplicity, in this example, the floating-point type of all ranges is (T=Float32), however, it should be understood that other floating-point types can be used. 
     In model  200 , the first input signal  202  is the input signal to the numerator of two functional blocks  206 ,  208  implementing a divide function. The second input signal  204  is the input signal to a function block  210  implementing a range limit function. Functional block  210  can limit the input signal to values within defined bounds (e.g., minimum and maximum bounds). Values outside the bounds can be constrained to the minimum or maximum bound as appropriate. In this example, functional block  210  can limit an input signal to the range [−10.0, 10.0]. Accordingly, for the range [−50.0, 50.0] of the second input signal  204 , the range of the output signal for functional block  210 , according to the range mapping function for functional block  210 , is [−10.0, 10.0]. 
     In model  200 , the output signal from functional block  210  is input to two functional blocks  212 ,  214  implementing sum functions. The ranges for the other input signals to functional blocks  212 ,  214  are constants. Accordingly, the range for the output signal from functional block  212  is [−5.0, 15.0] and the range for the output signal from functional block  214  is [2.0, 22.0]. These output signals are provided to the functional blocks  206 ,  208 . The range of the output signal from functional block  208  is [−50.0, 100.0], and the range of the output signal from functional block  206  is (−infinity, infinity) since the range of the input signal for the denominator of functional block  206  includes zero. The model  200  can include output signal blocks  216 ,  218  that are connected to the output signals of functional blocks  206 ,  208  respectively. An output signal block represents the output signal of the model. As shown, the value being input to output block  216  can spike to positive or negative infinity which may cause a run-time exception in the system under test. 
     To support detection of such potential exceptions, the range propagation routine for the divide block can include a check to ensure there is a safe region around zero of the denominators signal range. If this safe region is entered, a warning can be produced. This is an example of a block-centric range analysis that is built on top of range propagation. Examples of other checks includes: (i) determining if the ranges of the inputs signals of functional blocks implementing square root functions are greater than or equal to zero, (ii) determining if switch controller ranges are greater than or equal to one and less than or equal to the number of switched signals, and (iii) determining if signal ranges are contained within the minimum and maximum values of their associated language-independent types for integer data types (e.g., between 32,768 and 32,767 for a 16-bit integer). 
     Range propagation can also be used to prune the search space associated with requirements-based test generation as described in U.S. Pat. No. 7,644,334, entitled “Requirements-Based Test Generation” filed on Nov. 26, 2007, the disclosure of which is hereby incorporated herein by reference in its entirety. 
     In an example, the software verification tool does not rely on continuous dynamics or notion of time, as used in hybrid automata or time automata to capture and analyze a mix of continuous dynamics and discrete state transitions. This is not a limitation in practice, as exhaustive analysis is typically infeasible on complex practical models of a system under test. In fact, many large avionics control systems are currently designed using discrete time notation, from a practical perspective, since the flight code implementation uses periodic execution and scheduling. The advantage of the discrete approach for time representation is scalability on complex models, which has many practical benefits. 
     In order to analyze the effect of the structure of a model of a system under test as well as the effect of error on potential behavioral non-determinism, range propagation can be extended to also propagate signal value error and analyze its potential effect. 
     In order to capture signal value errors, the endpoints of all intervals I in range R have an associated signal value error. One or more intermediate values from a signal range may have associated signal value error represented. The more values for which signal value error is represented, the more accurate the error analysis can be. However, the computation required to perform the error propagation can also increase. 
     In one embodiment, an intermediate value in the range may have associated signal value error represented. Accordingly, the signal value error for the intermediate value can be propagated through a functional block to determine an intermediate value error range. The signal value error for the intermediate value can be sent through the function block by using the range mapping function in the same manner as discussed with respect to the signal value error for the endpoints of a range. In one example, the intermediate value for which signal value error is sent through the functional block can be a value near zero. In another embodiment, the intermediate value may be a midpoint in the range. 
     In one embodiment, the signal value error is represented as an interval associated with a particular value of the signal range. In an example, intermediate signal error values can be determined by interpolation of endpoint signal error values. 
     In another embodiment, the signal value error is represented as a single value, k, that is relative to all feasible values of a range. In an example, endpoint signal error values and/or intermediate signal value errors can be determined by multiplying the particular value of the range by the value k. 
     In yet another embodiment, the signal value error is represented as an error function of a value in the range. In this example, endpoint signal error values and/or intermediate signal value errors can be determined by applying the error function to the particular value of the range. 
       FIG. 3  graphically illustrates how an error interval can be associated with both endpoints of an interval. It shows interval [2, 7) with minimum value error [−0.5, 0.5] and maximum value error [−0.5, 1.0]. In this example, the actual values observed may range from [1.5, 8). 
       FIG. 4  illustrates an example of a method  300  for propagating signal value error through a functional block in a model. In this example, method  300  is described with respect to a functional block b with n input signals having ranges that have been determined. 
     At block  302 , one or more ranges for the one or more output signals of the functional block b can be determined as a function of the ranges corresponding to the expected values (e.g., without signal value error) for the n input signals as described with respect to  FIG. 2 . That is, range(s) for the output signal(s) can be determined using the range mapping function for the functional block b based on the range(s) of the expected values of the input signal(s) not including signal value error for the input signal(s). A range for an input signal can include a maximum value and a minimum value and one or more intervals. 
     At block  304 , a set of error extended ranges for an input signal can be calculated. The set of error extended ranges for an input signal can include all permutations of the ranges of the signal value errors for the input signal added to the respective maximum and minimum values of the range of the input signal. 
     A set of error extended ranges for an input signal is calculated by adding the min and max of the signal value error intervals to each range interval endpoint. This results in m pairs of error extended ranges where m is the total number of end-points (e.g., unique min and max values in all intervals of a range. 
     At block  306 , the set of ranges for the error extended output signals can be determined. The set of ranges for the error extended output signal can be determined by propagating each range of the set of error extended ranges for the input signals through the functional block. That is, each range of the set of error extended ranges for the first input signal is matched with each error extended range for the second and remaining input signals. All permutations of error extended ranges can be mapped to an error extended output signal range by using the range mapping function of the functional block b. 
     At block  308 , the ranges of the minimum and maximum values for the set of ranges of error extended output values can be determined. Each range in the set of ranges of error extended output values has a maximum and a minimum value. Accordingly, the maximum values of the set of ranges occupy a certain range and the minimum values of the set of ranges occupy a certain range. These ranges are determined accordingly. 
     At block  310 , the ranges for the minimum and maximum values for the set of ranges of error extended output values can be subtracted from the minimum and maximum value from the range of output signal determined at block  302  to determine the ranges of signal value error for the output signal of the functional block b. That is, a range of signal value error corresponding to the maximum value in the range of the output signal determined at block  302  can be determined as the range of the maximum values for the set of error extended output values subtracted from the maximum value of the range of output signal determined in block  302 . Similarly, a range of signal value error corresponding to the minimum value in the range of the output signal determined at block  302  can be determined as the range of the minimum values for the set of error extended output values subtracted from the minimum value of the range of output signal determined in block  302 . 
     Once determined the range of the output signal (without error) as determined in block  302  and the ranges for the signal value error corresponding to the output signal can be provided to a functional block downstream from the functional block b. In examples where the functional block b has multiple output signals, the method  300  can be performed for each output signal. 
     Example pseudocode for method  300  is as follows: 
     
       
         
           
               
               
             
               
                   
               
             
            
               
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                 propagate the set of ranges through the incident block, b, as normal to 
               
               
                   
                 result in output signal ranges without error 
               
               
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                 for all input signal ranges, r, of block b do 
               
            
           
           
               
               
            
               
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                 for all endpoints v of r do 
               
               
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                 create two error-extended endpoint values by adding to v the min 
               
               
                   
                 and max values of its associated error 
               
               
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                 end for 
               
            
           
           
               
               
            
               
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                 end for 
               
               
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                 create all feasible combinations of error-extended ranges from 
               
               
                   
                 error-extended endpoint values 
               
               
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                 for all combinations of error-extended ranges do 
               
               
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                 propagate the error-extended range through b to result in a set of 
               
               
                   
                 error-extended output ranges 
               
               
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                 end for 
               
               
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                 for all endpoints of the output signal ranges without error do 
               
            
           
           
               
               
            
               
                 12: 
                 determine the min and max values among all of the associated 
               
               
                   
                 error-extended endpoint values 
               
               
                 13: 
                 compute the error at the endpoint by subtracting these min and 
               
               
                   
                 max values 
               
            
           
           
               
               
            
               
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                 end for 
               
               
                   
               
            
           
         
       
     
     In other examples, a range of signal value error can be calculated at intermediate values other than or in addition to the maximum and minimum values for the range of the output signal by following method  300 . For example, a range of signal value error can be computed at zero or at a midpoint value in the range of the output signal. In an example, these signal value errors can be captured by introducing intervals in the range and using the interval endpoints to define their respective set of ranges for signal value error. 
     Method  300  can be performed for all functional blocks in a model in order to propagate range and associated signal value error through a model. For example, method  300  can be used for multiple or all blocks in a model for a system under test. Using method  300  range values and error values associated with the range values can be propagated through a plurality of functional blocks corresponding to the model of a system under test by calculating in topological order output signal ranges and minimum output signal error ranges and a maximum output signal error ranges for the plurality of functional blocks receive. This process can be used to perform block-centric analysis on the error extended input signal ranges of one or more functional blocks in the model to determine if error extended anomalies are possible. In an example, if an error extended anomaly is determined for one or more functional blocks an error (e.g., a warning) can be output to alert a user that the model may include an error extended anomaly. 
       FIG. 5  illustrates an example of a model  400  for a system under test in which signal value error is propagated through the model  400 . Model  400  includes a first input signal  402  and a second input signal  404  with ranges of [−2000.0 1000.0] and [100.0 499.0], respectively. For simplicity, model  400  uses floating-point types of all ranges (T=Float32), and does not show type information. The signal value error associated with the minimum and maximum signal values are given from the below the ranges. For example, the signal value error associated with the range of input signal  402  is [−0.2, 0.2] at the min value of −2000 and [−0.1, 0.1] at the max value of 1000. So output signal values can be expected ranging anywhere from −2000.2 to 1000.1. Similarly, the signal value error associated with the range of input signal  404  is [−0.3, 0.2] at the min value of 100 and [−0.4, 0.1] at the max value of 499. 
     Computing the signal value error at the output of functional block  406 , which implements a sum function, involves first computing the range of the output signal without signal value error. This is [1, 400]. Next the set of error extended input signal values are computed. These are [99.7, 498.6], [99.7, 499.1], [100.2, 498.6], [100.2, 499.1] from input signal  404  and [−99.0099], [−98.9901] for the constant  408 . All permutations of these two sets of error extended values are propagated through the functional block  406 . Hence, the set of ranges for the error extended output signal is: RMFsum([99.7, 498.6], [−99.0099])=[0.6901, 399.5901], RMFsum([99.7, 499.1], [−99.0099])=[0.6901, 400.0901], RMFsum([100.2, 498.6], [−99.0099])=[1.1901, 399.5901], RMFsum([100.2, 499.1], [−99.0099])=[1.1901, 400.0901], RMFsum([99.7, 498.6], [−98.9901])=[0.7099, 399.6099], RMFsum([99.7, 499.1], [−98.9901])=[0.7099, 400.1099], RMFsum([100.2, 498.6], [−98.9901])=[1.2099, 399.6099], RMFsum([100.2, 499.1], [−98.9901])=[1.2099, 400.1099], where RMFsum is the range mapping function of functional block  406 . In this case RMFsum can specify that the inputs&#39; minimum values can be added together to get the output minimum value and the inputs&#39; maximum values can be added together to get the output maximum value. 
     Finally, the smallest and largest low and high error extended values are subtracted from the output range that was computed without error, as such [0.6901, 1.2099]−[1]=[−0.3099, 0.2099] and [399.5901, 400.1099]−[400]=[−0.4099, 0.1099] to result in an output signal with range [1, 400], low value error of [−0.3099, 0.2099], and high value error of [−0.4099, 0.1099]. 
       FIG. 5  shows the values without error as well as the corresponding low and high signal value errors for each signal as it is propagated through the model  400 . Note the large error that is the result of performing the computation of functional block  406  and feeding the results directly into the denominator of the functional block  410  which implements a divide function. The resulting high end error is [−898.4, 347.1]. This shows how particular multi-block structures can amplify error unexpectedly. The signal value error for functional block  412 , implementing a divide function, is mitigated somewhat due to the functional block  414  implementing a range limiter function. 
       FIG. 6  illustrates an example of a model  500  for a system under test having both continuous and discrete signals in which signal value error is propagated through the model  500 . In this model, the output of the functional block  502  implementing a divide function is provided directly to the functional block  504  implementing a floor function. The output of functional block  502  is also indirectly provided to the functional block  506  also implementing a floor function, via the functional block  508  implementing a sum function that adds a small offset value. The function of a floor block is to convert a real variable into a whole number by computing the greatest integer less than the input value. 
     By following the method  300 , the input signal range to functional block  504  can be computed as [8.0, 10.0] with range for the low signal value error of [−0.0008, 0.0008] and range for the high signal value error of [−0.001, 0.001]. 
     Following the method  300 , the range of the output signal without error is determined for functional block  504 . This range includes the whole numbers 8, 9, and 10. No values between the whole numbers are included. Next the set of error extended input signal values are computed. These are [7.9992], [8.0008], [9.999], [10.001]. All these error-extended values are propagated through the functional block  504 . Hence, RMFfloor([7.9992])=[7], RMFfloor([8.0008])=[8], RMFfloor([9.999])=[9], RMFfloor([10.001])=[10]. 
     Finally, the smallest and largest low and high error-propagated values are subtracted from the output range that was computed without error, as such [7]−[8]=[1]; [8]−[8]=[0]; [9]−[10]=[1]; [10]−[10]=[0]; to result in an output signal with range [8, 10], range for the low signal value error of [1, 0], and range for the high signal value error of [1, 0]. Note that the signal value error can be computed for the value nine of this range if the signal value error associated with this value is computed (e.g., by interpolation) at the inputs of model  500  and propagated through the model  500 . 
     As can be seen, the output of functional block  506  has an associated signal value error of [0, 0]. The reason is that since the small offset added to the output of the functional block  502  is greater than the signal value error at that point, it is not possible for the minimum signal value error to pull the signal value down to result in the next lower integer value. In the past the offset value has needed to be estimated. Using method  300 , the offset can be determined by computing the propagated error at the signal just prior to the functional block  508 . Then the value of the constant3 block should be set slightly greater than the high end max error value to eliminate all downstream error. 
     Propagation of signal value error through a model can be performed topologically from input signals to the model, following the direction of the connections between functional blocks. After the signal value error associated with the input ports of a functional block have been determined, the signal value error associated with the output ports can be computed. By following method  300 , it may be difficult for the signal value error to be computed in the case of feedback loops on all input ports for one or more functional blocks. This is because the signal value error at the input ports may be directly or indirectly dependent upon the signal value error associated with the output ports of the same or downstream functional blocks. Nested feedback structures are also possible. 
     One embodiment that addresses the issue of feedback signals in models is to develop a library of common feedback patterns that are comprised of structures of blocks and wires. For each pattern, a range mapping function can be written. After this library is created, it can be applied on any model as follows: (i) detect instances of any of the feedback patterns, (ii) to replace the detected structures with a single functional block in which the feedback is internal to the block. (iii) This new functional block can be handled the same way as other functional blocks. 
       FIGS. 7 and 8  illustrate this method.  FIG. 7  illustrates model  600  with feedback signals  602  and  603 . In this model  600 , it may not be possible to compute the signal value error on all of the inputs of the functional block  604 , implementing a switch function, using only a topological algorithm. This is because one of the input signals for functional block  604  depends (indirectly through the functional block  606  (implementing a sum function) and the functional blocks  608  (implementing a delay function)) on its output signal  603 . Similarly, it is not possible to compute the signal value error on all of the inputs of the functional block  610  that implements a logical OR function due to feedback signal  602 . In one embodiment, a pattern recognizer can be used to detect the existence of the particular structure of a plurality of functional blocks (e.g., functional blocks  604 ,  606 ,  608 ,  610 ,  612 ) and replace the pattern with a single functional block. 
       FIG. 8  illustrates a model  700  in which the pattern of the feedback loop from  FIG. 7  has been replaced with the functional block  702  implementing a feedback counter function. Note in  FIG. 8 , the internals of the functional block  702  are shown, but in other examples the internals are not shown. The model  700  of  FIG. 8  no longer has a feedback loop. Hence the method  300  can be successfully applied to the model  700   
     In one example, the range mapping function for the functional block  702  can be dependent upon the number of iterations through the loop. The range mapping function can statically compute the bound of the number of iterations through the loop in order to compute a tight range and error bound for the model  700 . 
     For nested feedback structures, start with the leaf-level feedback structures and perform pattern identification and replacement. Then move up to the next higher level, and so on. 
     Another embodiment that addresses the issue of feedback signals in models is to automatically break the feedback loop by determining the feed back signals and setting a default (and conservative) error value to those signals. This error value may be dependent upon a bound of the number of iterations through the feedback structure. 
     In an example, certain classes of functional blocks perform computations that take continuous input signals and produce output signals that vary in discrete increments. For example, rounding operators such floor, round, ceiling, and fix convert a continuous signal into a discrete signal that varies in integer increments. Likewise comparison operators, such as greaterEq1  510  and greaterEq2  512  in  FIG. 6 , may compare two continuous signals and produce a Boolean output signal. For this class of blocks, the impact of error may be much greater than the mere propagation of the error to the output results; the error can cause non-deterministic and/or anomalous behavior. 
     For these classes of blocks, analysis of the model can be implemented for several classes of blocks to report any potential for non-determinism. The analysis can be performed by a number of methods, in particular (i) extended-type checking and (ii) evaluating block-specific predicates over the set of intervals and errors. 
     For example, the error margins at the decisions points that impact the block behavior can be analyzed to determine if they can potentially affect the result. In the example of  FIG. 6 , the software would check at both greater-than blocks  510  and  512  to determine if the error of either or both of the input signal ranges can individually or in conjunction cause an otherwise true result to evaluate to false or an otherwise false result to evaluate to true. This analysis can determine that for the block greaterEq1  510 , the error in the signal from floor  1  block  504  can cause an otherwise true result to evaluate to false (i.e., when v=[10] and e=[1]). 
     Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement, which is calculated to achieve the same purpose, may be substituted for the specific embodiments shown. Therefore, it is manifestly intended that this invention be limited only by the claims and the equivalents thereof. 
     Example Embodiments 
     Example 1 includes a method comprising: providing an input signal range corresponding to a range of expected values for an input signal to a functional block; providing a minimum value error range corresponding to a range of error for a minimum value endpoint of the input signal range and a maximum value error range corresponding to a range of error for a maximum value endpoint of the input signal range; mapping the input signal range to one or more output signal ranges as a function of a range mapping function corresponding to the functional block; calculating a set of error extended input signal ranges by: adding a min endpoint of the minimum value error range to the minimum value of the input signal range; adding a max endpoint of the minimum value error range to the minimum value of the input signal range; adding the min endpoint of the maximum value error range to the maximum value of the input signal; and adding the max endpoint of the maximum value error range to the maximum value of the input signal range; mapping the set of error extended input signal ranges to a set of error extended output signal ranges as a function of the range mapping function; and calculating a minimum output error range and a maximum output error range as a function of a difference between the set of error extended output signal ranges and the output signal ranges. 
     Example 2 includes a system comprising: at least one processing device; at least one memory device coupled to the at least one processing device, the at least one memory device having instructions thereon for execution by the at least one processing device, wherein the instructions, when executed by the at least one processing device, cause the at least one processing device to: receive an input signal range corresponding to a range of expected values for an input signal to a functional block; receive a minimum value error range corresponding to a range of error for a minimum value endpoint of the input signal range and a maximum value error range corresponding to a range of error for a maximum value endpoint of the input signal range; map the input signal range to one or more output signal ranges as a function of a range mapping function corresponding to the functional block; calculate a set of error extended input signal ranges by: adding a min endpoint of the minimum value error range to the minimum value of the input signal range; adding a max endpoint of the minimum value error range to the minimum value of the input signal range; adding the min endpoint of the maximum value error range to the maximum value of the input signal; and adding the max endpoint of the maximum value error range to the maximum value of the input signal range; map the set of error extended input signal ranges to a set of error extended output signal ranges as a function of the range mapping function; and calculate a minimum output error range and a maximum output error range as a function of a difference between the set of error extended output signal ranges and the output signal ranges. 
     Example 3 includes a computer readable medium including instructions which, when executed by at least one processor, cause the at least one processor to: receive an input signal range corresponding to a range of expected values for an input signal to a functional block; receive a minimum value error range corresponding to a range of error for a minimum value endpoint of the input signal range and a maximum value error range corresponding to a range of error for a maximum value endpoint of the input signal range; map the input signal range to one or more output signal ranges as a function of a range mapping function corresponding to the functional block; calculate a set of error extended input signal ranges by: adding a min endpoint of the minimum value error range to the minimum value of the input signal range; adding a max endpoint of the minimum value error range to the minimum value of the input signal range; adding the min endpoint of the maximum value error range to the maximum value of the input signal; and adding the max endpoint of the maximum value error range to the maximum value of the input signal range; map the set of error extended input signal ranges to a set of error extended output signal ranges as a function of the range mapping function; and calculate a minimum output error range and a maximum output error range as a function of a difference between the set of error extended output signal ranges and the output signal ranges. 
     In Example 4 the subject matter of any of Examples 1-3 can optionally include wherein calculating a set of error extended input signal ranges includes calculating permutations of the min and max endpoints of the minimum value error interval added to the minimum value of the input signal range and the maximum value error range added to the maximum value of the input signal range. 
     In Example 5, the subject matter of any of Examples 1-4 can optionally include wherein each error extended output signal range has a minimum value and a maximum value; wherein calculating a minimum output error range includes calculating a difference between a smallest of the minimum values of the set of error extended output signal ranges and a minimum value of the output signal range, and calculating a difference between a largest of the minimum values of the set of error extended output signal ranges and the minimum value of the output signal range; and wherein calculating a maximum output error range includes calculating a difference between a smallest of the maximum values of the set of error extended output signal ranges and a maximum value of the output signal range, and calculating a difference between a largest of the maximum values of the set of error extended output signal ranges and the maximum value of the output signal range. 
     In Example 6, the subject matter of any of Examples 1-5 can optionally include: providing an intermediate value error range corresponding to a range of error for an intermediate value of the input signal range; mapping the intermediate value to a corresponding one or more output intermediate value as a function of the range mapping function; calculating a set of error extended intermediate signal valued by: adding the min endpoint of the intermediate value error range to the intermediate value of the input signal range; adding the max endpoint of the intermediate value error range to the intermediate value of the input signal range; mapping the set of error extended intermediate values to a set of error extended output intermediate values as a function of the range mapping function; and calculating an intermediate value output error range as a function of a difference between the set of error extended output intermediate values and the error extended intermediate values. 
     In Example 7, the subject matter of any of Examples 1-6 can optionally include wherein the input signal range consists of a constant value. 
     In Example 8, the subject matter of any of Examples 1-7 can optionally include wherein the input signal range includes two or more intervals, and wherein each interval has a corresponding minimum value error range and a maximum value error range, and calculating output signal error ranges corresponding to the two or more intervals. 
     In Example 9, the subject matter of any of Examples 1-8 can optionally include: performing block-centric analysis on the error extended input signal ranges to determine if error extended anomalies are possible. 
     In Example 10, the subject matter of any of Examples 1-9 can optionally include: propagating range values and error values associated with the range values through a plurality of functional blocks corresponding to a model of a system under test by calculating in topological order output signal ranges and minimum output signal error ranges and a maximum output signal error ranges for the plurality of functional blocks receive. 
     In Example 11, the subject matter of any of Examples 1-10 can optionally include wherein the model includes one or more feedback signals; detecting a feedback pattern; and replacing the feedback pattern with a functional block prior to propagating range values and error values. 
     In Example 12, the subject matter of any of Examples 1-11 can optionally include wherein the functional block receives a plurality of input signals, and wherein the input signal range corresponds to at least one of the plurality of input signals.