BACKGROUND 
       [0001]    This disclosure relates generally to the field of verifying the equivalence of a regular expression to a transformation of the regular expression that takes advantage of a post-processor to optimize the storage efficiency, and more particularly to determining the equivalence of a finite state automaton representation of a regular expression to a transformed finite state automaton representation of the regular expression that is coupled with a pre-verified and pre-optimized post-processor. 
         [0002]    Packet content scanning is an essential part of network security and monitoring applications. Intrusion detection systems such as Snort (http://www.snort.org) rely heavily on regular expressions to express increasingly complex attack patterns. A typical way of matching regular expressions in a stream of input characters is by simulating the input on a Finite State Automaton (FSA), which may be a nondeterministic FSA (NFA) or a deterministic FSA (DFA), compiled from the regular expression. For example,  FIG. 1  shows an example of a DFA  100  that detects the regular expression “abc.*def*ghi” in an input data stream. The regular expression “abc.*def*ghi” is in perl compatible regular expression (PCRE) format. The DFA  100  is modeled as a directed graph. The DFA states are shown in circles, the state transitions are shown using directed edges, and the set of input characters resulting in the transitions (i.e., the transition rules) are given in the rectangular boxes. The initial state of the DFA is labeled as state  0 , with intermediate states numbered 1 to 8, leading up to a match of the regular expression at state number 9. The plurality of transition rules governs transitions between the states. Note that if the regular expression is non-anchored, additional transitions that point to state  0  and state  1  would be needed in  FIG. 1 . Similarly, if the regular expression is anchored, there has to be an explicit invalid state in the DFA and additional transitions pointing to the invalid state for state/input combinations without a valid next state. 
         [0003]    A FSA architecture may be programmed to recognize one or more regular expressions in an input data stream by loading a set of state transition rules into off-chip or on-chip memories. Performance of such architectures depends on the storage efficiency of the compiled set of state transition rules, as on-chip memory resources are usually limited, and off-chip memory accesses can be costly in terms of processing time. 
       BRIEF SUMMARY 
       [0004]    In one aspect, a method for determining correctness of a transformation between a first finite state automaton (FSA) and a second FSA, wherein the first FSA comprises a representation of a regular expression, and the second FSA comprises a transformation of the first FSA includes determining a third FSA, the third FSA comprising a cross product of the second FSA and a post-processor; determining whether the first FSA and the third FSA are equivalent; and in the event that the first FSA is determined not to be equivalent to the third FSA, determining that the transformation between the first FSA and the second FSA is not correct. 
         [0005]    In another aspect, a computer program product comprising a non-transitory computer readable storage medium containing computer code that, when executed by a computer, implements a method for determining correctness of a transformation between a first finite state automaton (FSA) and a second FSA, wherein the first FSA comprises a representation of a regular expression, and the second FSA comprises a transformation of the first FSA, wherein the method includes determining a third FSA, the third FSA comprising a cross product of the second FSA and a post-processor; determining whether the first FSA and the third FSA are equivalent; and in the event that the first FSA is determined not to be equivalent to the third FSA, determining that the transformation between the first FSA and the second FSA is not correct. 
         [0006]    In yet another aspect, A system for determining correctness of a transformation between a first finite state automaton (FSA) and a second FSA, wherein the first FSA comprises a representation of a regular expression, and the second FSA comprises a transformation of the first FSA, includes a computer configured to determine a third FSA, the third FSA comprising a cross product of the second FSA and a post-processor; determine whether the first FSA and the third FSA are equivalent; and in the event that the first FSA is determined not to be equivalent to the third FSA, determine that the transformation between the first FSA and the second FSA is not correct. 
         [0007]    Additional features are realized through the techniques of the present exemplary embodiment. Other embodiments are described in detail herein and are considered a part of what is claimed. For a better understanding of the features of the exemplary embodiment, refer to the description and to the drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0008]    Referring now to the drawings wherein like elements are numbered alike in the several FIGURES: 
           [0009]      FIG. 1  illustrates an example of a DFA. 
           [0010]      FIG. 2  illustrates an example of a FSA with a post-processor. 
           [0011]      FIG. 3  illustrates an example of a plurality of parallel FSAs with a post-processor. 
           [0012]      FIG. 4  illustrates an example of a DFA that is a transformation of the DFA of  FIG. 1 . 
           [0013]      FIG. 5  illustrates an example of an original DFA and a transformed DFA. 
           [0014]      FIG. 6  illustrates an embodiment of a method of determining equivalence of a FSA and a transformation of the FSA. 
           [0015]      FIG. 7  illustrates an embodiment of a method of optimizing a FSA through successive transformations. 
           [0016]      FIG. 8  illustrates an embodiment of a composite netlist built from a FSA and a transform of the FSA. 
           [0017]      FIG. 9  is a schematic block diagram illustrating an embodiment of a computer that may be used in conjunction with a method for determining equivalence of a FSA and a transformation of the FSA. 
       
    
    
     DETAILED DESCRIPTION 
       [0018]    Embodiments of systems and methods for determining equivalence of a FSA and a transformation of the FSA are provided, with exemplary embodiments being discussed below in detail. A transformation of a FSA, which may include a pre-verified and pre-optimized post-processor that contains local memory elements and a plurality of parallel FSA engines, can significantly reduce the size of the memory necessary to hold the transition rules of the transformed FSA, reducing time-consuming calls to off-chip memory. However, it is necessary to evaluate the output equivalence of a transformed FSA to its original FSA. Verifying that a transformation applied to a FSA preserves the meaning of the regular expression on which the original FSA is based is a challenging task, and automated methodologies for verifying the correctness of such transformations involving post-processor instructions are needed. 
         [0019]    A technique that is known to be effective in compressing the transition rule space of a FSA is the decomposition of a complex regular expression into a sequence of simpler regular expressions that communicate through bitwise instructions and counters in a post-processor that contains local storage. The resulting architecture is referred to as a finite state machine with datapath (FSMD).  FIG. 2  illustrates a FSMD  200  that combines a FSA  202  with a post-processor  203  that includes an instruction decoder  204 , registers  205 , combinational logic  206 , and registers  205 . An input data stream  201  is received by the FSA  202 , and a match is indicated at output  207 . 
         [0020]    To further optimize the memory usage in the presence of a complex regular expression, multiple parallel FSA engines may be used in conjunction with a common post-processor.  FIG. 3  shows an example of a FSMD  300  including a plurality of parallel FSA engines  302 A-L. The data stream at input  301  is input to each of the parallel FSA engines  302 A-L, and the post-processor  303  (which includes a set of internal components as shown with respect to post-processor  203  of  FIG. 2 ) indicates whether the regular expression is matched at output  304 . A FSMD may include any appropriate number of parallel FSAs. 
         [0021]      FIG. 4  shows an example of a DFA  400  expressed as a FSMD that is a transformation of the DFA  100  that was shown in  FIG. 1 . DFA  400  also detects the regular expression “abc.*def*ghi”. If the original regular expression is non-anchored, it may be split into independent subexpressions “abc”, “def”, and “ghi”, allowing the transformed DFA  400  to match each of the subexpressions independently. Starting at state  0  (zero), the leftmost column of states and transition rules detects the presence of “abc” in the input stream by proceeding through states  1  and  4  to state  7 . At state  7 , a register R 1  is set indicating that “abc” was matched. Then, proceeding through states  2  and  5  to state  8  detects a match of “def”. In state  8 , R 1  is tested, and if R 1  is set, then R 2  is set indicating the presence of “abc.*def”. Lastly, proceeding through states  3  and  6  to state  9  detects a match of “ghi”, and, in state  9 , R 2  is tested. If R 2  is set, then a match of the whole regular expression “abc.*def*ghi” is indicated. The transformed DFA  400  may be implemented in three parallel DFAs with a post-processor. R 1  and R 2  may be 1-bit registers located in the post-processor. The transformed DFA of  FIG. 4  includes a significantly smaller number of state transitions as compared to the initial DFA  100  of  FIG. 1  while implementing the same functionality, reducing the amount of memory needed to store the DFA  400 . 
         [0022]    However, a transformation of a DFA that matches the regular expression “abc.*cde” may be problematic. A transformed DFA may match the subexpressions “abc” and “cde” independently. However, setting a register R whenever “abc” matches and then reporting a match whenever “cde” matches only if the register R is set does not preserve the meaning of the regular expression “abc.*cde”. Such an approach would result in a match for an input string “abcde”, which does not belong to the language defined by the regular expression “abc.*cde”. A DFA transformation that does match “abc.*cde” may be achieved by applying further transformations to the DFA that make use of more complex instructions. For a general case, with a reference DFA M i  and a transformed DFA M i+1 , it is necessary to determine whether M i  and M i+1  are output equivalent, i.e., for any possible sequence of inputs, the same sequence of outputs are produced by the two DFAs M i  and M i+1 . 
         [0023]    Referring to  FIG. 5 , an embodiment of a reference DFA M ( 502 ), and a transformed DFA M T  ( 504 ) with a post-processor M P  ( 505 ) are shown. In other embodiments, DFA M ( 502 ) and transformed DFA M T  ( 504 ) may comprise an NFA and a transformed NFA. M C  ( 506 ) represents the FSMD that combines the transformed DFA M T  ( 504 ) and the post-processor M P  ( 505 ) as shown in  FIG. 5 . M C  ( 506 ) is equal to the cross product of M T  ( 504 ) and M P  ( 505 ). The post-processor  505  includes an instruction decoder, registers, and combinational logic as are shown in  FIG. 2  with respect to post-processor  203 . The post-processor  505  may be predetermined, i.e., the transformation must take in to account the logic and memory capabilities of the post-processor  505 . It must be determined whether M ( 502 ) and M C  ( 506 ) are output equivalent, i.e., that for any possible sequence of inputs  501 , the same sequence of outputs are given at the outputs  502 A and  502 B. The input  501  may be an 8-bit character per clock cycle. If the reference DFA M ( 502 ) implements a single regular expression only, a 1-bit input  501  is sufficient; one or both of outputs  502 A-B are equal to ‘1’ if there is a regular expression match in the current clock cycle, and are equal to zero otherwise. The notation introduced in Tables 1-5 below may be generalized to multiple regular expressions by increasing the size of the output set, e.g., using one output-bit per regular expression. Similarly, the input-length may be increased to support DFAs that can process multiple characters per clock cycle. The reference DFA M ( 502 ) may be a Moore machine or a Mealy machine in various embodiments. Below in Table 1 is a definition of a Mealy machine for the reference DFA M=(S; I; O; s 0 ; δ; λ): 
         [0000]    
       
         
               
             
           
               
                 TABLE 1 
               
               
                   
               
             
             
               
                 S is a finite non-empty set of states 
               
               
                 I ∈ {0, 1} N  is the set of inputs (N is the length of the input in bits) 
               
               
                 0 ∈ {0, 1} K  is the set of outputs (K is the length of the output in bits) 
               
               
                 s 0  ∈ S is the initial state 
               
               
                 δ: S X I → S is the transition function 
               
               
                 λ: S X I → 0 is the output function 
               
               
                   
               
             
          
         
       
     
         [0024]    The transformed DFA M T  ( 504 ) can be a Moore machine of a Mealy machine in various embodiments. Below in Table 2 is a definition of a Mealy machine for the transformed DFA M T =(S T ; I; P; s T   0 ; δ T ; λ T ): 
         [0000]    
       
         
               
             
           
               
                 TABLE 3 
               
               
                   
               
             
             
               
                 R is a finite non-empty set of states 
               
               
                 P is a finite non-empty set of instructions supported by the post-processor 
               
               
                 (505) 
               
               
                 O ∈ {0, 1} K  is the set of outputs (K is the length of the output in bits) 
               
               
                 r 0  ∈ R is the initial state of R 
               
               
                 δ R : R X P → R is the transition function 
               
               
                 λ R : R X P → O is the output function 
               
               
                   
               
             
          
         
       
     
         [0025]    The post-processor ( 505 ) can also be modeled as a Moore machine or a Mealy machine. Below in Table 3 is a definition of a Mealy machine for the post-processor M P =(R; P; O; r 0 ; δ R ; λ R ): 
         [0000]    
       
         
               
             
           
               
                 TABLE 2 
               
               
                   
               
             
             
               
                 S T  is a finite non-empty set of states 
               
               
                 I ∈ {0, 1} N  is the set of inputs (N is the length of the input in bits) 
               
               
                 P is a finite non-empty set of instructions supported by the post-processor 
               
               
                 (505) if a post processor is used, otherwise P is equal to the output 
               
               
                 set 0 ∈ {0, 1} K   
               
               
                 s T   0  ∈ S T  is the initial state of S T   
               
               
                 δ T : S T  X I → S T  is the transition function 
               
               
                 λ T : S T  X I → P is the output function 
               
               
                   
               
             
          
         
       
     
         [0026]    The combined architecture M C  ( 506 ) containing the transformed DFA M T  ( 504 ) and the post-processor M P  ( 505 ) is a cross product of M T  and M P . Below in Table 4 is a definition of a Mealy machine for the combined architecture M C =((S T , R); I; O; (s T   0 , r 0 ); (δ T , δ R ); λ R ): 
         [0000]    
       
         
               
             
           
               
                 TABLE 4 
               
               
                   
               
             
             
               
                 S T  is a finite non-empty set of states 
               
               
                 R is a finite non-empty set of states 
               
               
                 I ∈ {0, 1} N  is the set of inputs (N is the length of the input in bits) 
               
               
                 O ∈ {0, 1} K  is the set of outputs (K is the length of the output in bits) 
               
               
                 s T   0  ∈ S T  is the initial state of S T   
               
               
                 r 0  ∈ R is the initial state of R 
               
               
                 δ T : S T  X I → S T  is the transition function of S T   
               
               
                 δ R : R X S T  X I → R is the transition function of R 
               
               
                 λ R : R X S T  X I → O is the output function 
               
               
                   
               
             
          
         
       
     
         [0027]    The architecture shown in  FIG. 5  models R using the physical registers located in the post-processor  505 . The transition function δ R  is implemented using post-processor instructions P that are input to the post-processor from the DFA engine to modify the contents of the post-processor registers. The combinational logic located in the post-processor acts as the state transition logic inside the post-processor and is activated by the received instructions P. The instructions in P may be modeled as a function of S T  and I. Similarly, the output O of M C  can be formulated as a function of R and P. 
         [0028]    An additional way of optimizing the state space in the presence of multiple regular expressions is the use of multiple parallel FSA engines for M T  in conjunction with the post-processor function M P  (such as is shown in  FIG. 3 ). To take advantage of the optimizations related to the use of parallel FSA engines and the optimizations related to the use of a post-processor at the same time, an architecture that supports parallel FSA engines interfaced to a common post-processor is needed. Assuming that the architecture supports up to L parallel DFAs sharing a post-processor, the automaton that combines the parallel DFAs with the post-processor can be formulated as a Mealy machine M C =((S 1 , S 2  . . . S L , R); I; O; (s 1   0 , s 2   0  . . . s L   0 , r 0 ); (δ 1 , δ 2 , . . . δ L , δ R ); λ T ) where S i  represents states of parallel DFA engine i, s i   0  represents the initial state of parallel DFA engine i, and δ i  represents the transition function of the parallel DFA engine i, shown below in Table 5: 
         [0000]    
       
         
               
             
           
               
                 TABLE 5 
               
               
                   
               
             
             
               
                 S i  is a finite non-empty set of states 
               
               
                 R is a finite non-empty set of states 
               
               
                 I ∈ {0, 1} N  is the set of inputs (N is the length of the input in bits) 
               
               
                 O ∈ {0, 1} K  is the set of outputs (K is the length of the output in bits) 
               
               
                 s i   o  ∈ S i  is the initial state of S i , i ∈ {1, 2, . . . , L} 
               
               
                 r o  ∈ R is the initial state of R 
               
               
                 δ i : S i  X I → S i  is the transition function of S i , i ∈ {1, 2, . . . , L} 
               
               
                 δ R : R X S 1  X . . . X S L  X I → R is the transition function of R 
               
               
                 λ T : R X S 1  X . . . X S L  X I → O is the output function 
               
               
                   
               
             
          
         
       
     
         [0029]      FIG. 6  illustrates a flowchart of an embodiment of a method  600  of determining equivalence of a FSA and a transformation of the FSA. Method  600  may be used to determine whether a FSA M i  and an associated transformed FSA M i+1  are equivalent as part of a sequence of FSA transformations as described above with respect to method  700  of  FIG. 7 , i.e., in block  703 , which is discussed below. Alternately, method  600  may be executed in a stand-alone equivalence verifying module for a FSA transformation. The transformed FSA M i+1  may include one or more parallel FSA engines and a post-processor, such as was shown above with respect to  FIG. 3 . Referring to  FIG. 5 , M i  may be equivalent to M, and M i+1  may be equivalent to M C , which is the cross product of M T  and M P . In block  601  of  FIG. 6 , two FSAs, M i  and M i+1 , are received. The two FSAs M i  and M i+1  are each converted into hardware description language (HDL) code. The hardware description language code of M i  is referred to as H i , and the hardware description language code of M i+1  is referred to as H i+1 . The hardware description language may be VHSIC HDL (VHDL) in some embodiments. A partial example of computer code for conversion of the DFA  400  of  FIG. 4  into VHDL is shown below with respect to the code segment in Table 6. In block  602 , H i  and H i+1  are synthesized into gate-level netlists N i  and N i+1 . A netlist is a list of logic gates and interconnections that make up a circuit, and is generated from the HDL code. In block  603 , a composite netlist N′ that includes N i  and N i+1  is built. The composite netlist includes a common input to both N i  and N i+1 , and an exclusive- or logic connected to the outputs of both N i  and N i+1 . An embodiment of a composite netlist N′ is shown in  FIG. 7 , and discussed in further detail below. In block  604 , a test data stream is supplied at the input to the composite netlist N′, and it is determined if any output of the composite netlist N′ can be asserted (i.e., is a logical ‘1’). If any output of N′ can be asserted, then N i  and N i+1  have differing outputs for the same test input, indicating that M i  and M i+1  are not output equivalent. In block  605 , if M i  and M i+1  are determined not to be equivalent, counterexample traces may be generated from the portion of the test data stream that generated the asserted output of the composite netlist N′. These counterexample traces may be used to correct M i+1  so that the corrected version of M i+1  is equivalent to M i . 
         [0030]    For a FSA based on a complex regular expression, multiple transformations may be applied to the FSA in order to optimize the compiled transition rule space of the FSA.  FIG. 7  illustrates a flowchart of an embodiment of a method  700  for optimizing a FSA. In block  701 , a FSA M i  is identified, and i is initialized to 0 (zero). In block  702 , M i  is transformed to produce M i+1 . This may be performed according to any appropriate FSA transformation method. M i+1  may include one or more parallel FSA engines and a predetermined post-processor, such as was shown in  FIG. 5 . The transformation may take into account the predetermined post-processor architecture. In block  703 , it is determined whether M i  and M i+1  are equivalent by performing the method  600  illustrated in  FIG. 6 , which was discussed above. In block  704 , if M i  and M i+1  are not determined to be equivalent in block  703  then M i+1  is not safe, and M i+1  reverts to M i . Otherwise, i is incremented by 1. In block  705 , it is determined if M i  is adequately optimized. This determination may be based on, for example, whether the sets of transition rules that make up M i  fit in the on-chip memories available for parallel FSA engines in the FSMD architecture. If M i  is not adequately optimized, flow returns to block  702 , and blocks  702 - 705  are repeated until the FSA is adequately optimized. 
         [0031]      FIG. 8  shows an embodiment of a composite netlist N′  800  that is generated in block  603  of  FIG. 6 . A test data stream is received at input  801 , and is processed by both netlist N i  ( 802 ) and netlist N i+1  ( 803 ). The outputs of both N i  ( 802 ) and N i+1  ( 803 ) are received by exclusive OR (XOR) logic  804 . The output  805  of N′ will only asserted to a logical ‘1’ if the corresponding outputs of N i  ( 802 ) and N i+1  ( 803 ) are differing values. If the output  805  is asserted, N i  ( 802 ) and N i+1  ( 803 ), and therefore their underlying FSAs M i  and M i+1 , are not equivalent. If every output  805  of N′ ( 800 ) is proven as un-assertable, N i  ( 802 ) and N i+1  ( 803 ), and hence M i  and M i+1 , are determined to be equivalent. A sequential equivalence checking tool may be used to generate the test data stream received at input  801 . For example, the Sixth Sense Software by International Business Machines (IBM) may be used to generate test data stream  801 . Synthesizing the two FSAs M i  and M i+1  into a form that equivalence checking software such as Sixth Sense can handle, i.e., the composite netlist N′  800  as shown in  FIG. 7 , greatly simplifies FSA equivalence checking The initial state of composite netlist N′  800  is the cross-product of the initial states of N i  ( 802 ) and N i+1  ( 803 ). Counterexample traces, such as may be generated in block  605  of  FIG. 6 , may be obtained by determining a portion of the test data stream  801  that generates an asserted output  805  of N′ ( 800 ). 
         [0032]    The code segment below in Table 6 illustrates an example of a method of generating partial VHDL code of the DFA  400  of  FIG. 4 . The state transitions may be implemented using an encapsulating case statement and additional if statements nested inside the case statement. For brevity, only the VHDL code for states  4 ,  5 , and  6  of DFA  400  of  FIG. 4  are shown in the code segment below. Again for brevity, an explicit post-processor module is not instantiated in the code, and the post-processor instructions are implemented directly inside the “if” statements. Note that the overall state of the DFA  400  is defined by the contents of the registers CS, C_REG, and C_MATCH. 
         [0000]    
       
         
               
             
           
               
                 TABLE 6 
               
               
                   
               
             
             
               
                 1: entity transformed is 
               
               
                 2: port (CLOCK, RESET : in STD_LOGIC; 
               
               
                 3:  C_IN: in STD_LOGIC_VECTOR(8 downto 1); 
               
               
                 4:  MATCH: out STD_LOGIC); 
               
               
                 5: end transformed; 
               
               
                 6: 
               
               
                 7: architecture BEHAVIORAL of transformed is 
               
               
                 8: type STATE_TYPE is (S0, S1, S2, S3, S4, S5, S6, S7, S8, S9); 
               
               
                 9: signal CS, NS: STATE_TYPE; 
               
               
                 10: signal C_REG, N_REG: STD_LOGIC_VECTOR(2 downto 1); 
               
               
                 11: signal C_MATCH, N_MATCH: STD_LOGIC; 
               
               
                 12: begin 
               
               
                 13:  SYNC_PROC: process (CLOCK, RESET) 
               
               
                 14:  begin 
               
               
                 15:    if (RESET=’1’) then 
               
               
                 16:      CS &lt;= S0; C REG &lt;= b”00”; C_MATCH &lt;= ’0’; 
               
               
                 17:    elsif (CLOCK’event and CLOCK = ’1’) then 
               
               
                 18:      CS &lt;= NS; C REG &lt;= N REG; C_MATCH &lt;= 
               
               
                         N_MATCH; 
               
               
                 19:    end if; 
               
               
                 20:  end process; -End SYNC_PROC 
               
               
                 21:  COMB PROC: process (CS, C IN) 
               
               
                 22:  begin 
               
               
                 23:    MATCH &lt;= C_MATCH; 
               
               
                 24:    case CS is 
               
               
                 25:    when S1 =&gt; ... 
               
               
                 26:    when S2 =&gt; ... 
               
               
                 27:    when S3 =&gt; ... 
               
               
                 28:    when S4 =&gt; 
               
               
                 29:      if (C IN = x”63”) then - Set first bit 
               
               
                 30:        N_REG &lt;= C_REG or b”01”; NS &lt;= S7; 
               
               
                           N_MATCH &lt;= ’0’; 
               
               
                 31:      else 
               
               
                 32:        N_REG &lt;= C_REG; NS &lt;= S0; N_MATCH &lt;= 
               
               
                           ’0’; 
               
               
                 33:      end if; 
               
               
                 34:    when S5 =&gt; 
               
               
                 35:      if (C_IN = x”66”) then 
               
               
                 36:        if (C_REG(1) = ’1’) then - Test first bit 
               
               
                 37:          N_REG &lt;= C_REG or b”10”; - Set second 
               
               
                             bit 
               
               
                 38:        else 
               
               
                 39:          N_REG &lt;= C_REG; 
               
               
                 40:        end if; 
               
               
                 41:        NS &lt;= S8; N_MATCH &lt;= ’0’; 
               
               
                 42:      else 
               
               
                 43:        N_REG &lt;= C_REG; NS &lt;= S0; N_MATCH 
               
               
                           &lt;= ’0’; 
               
               
                 44:      end if; 
               
               
                 45:    when S6 =&gt; 
               
               
                 46:      if (C_IN = x”69”) then 
               
               
                 47:        if (C_REG(2) = ’1’) then - Test second bit 
               
               
                 48:          N_MATCH = ’1’; - Match! 
               
               
                 49:        else 
               
               
                 50:          N_MATCH &lt;= ’0’; 
               
               
                 51:        end if; 
               
               
                 52:        N_REG &lt;= C_REG; NS &lt;= S9; 
               
               
                 53:      else 
               
               
                 54:        N_REG &lt;= C_REG; NS &lt;= S0; N_MATCH &lt;= 
               
               
                           ’0’; 
               
               
                 55:      end if; 
               
               
                 56:    when S7 =&gt; ... 
               
               
                 57:    when S8 =&gt; ... 
               
               
                 58:    when S9 =&gt; ... 
               
               
                 59:    end case; 
               
               
                 60:  end process; -End_COMB PROC 
               
               
                 61: end architecture BEHAVIORAL 
               
               
                   
               
             
          
         
       
     
         [0033]      FIG. 9  illustrates an example of a computer  900  which may be utilized by exemplary embodiments of a method for determining equivalence of a FSA and a transformation of the FSA as embodied in software. Various operations discussed above may utilize the capabilities of the computer  900 . One or more of the capabilities of the computer  900  may be incorporated in any element, module, application, and/or component discussed herein. 
         [0034]    The computer  900  includes, but is not limited to, PCs, workstations, laptops, PDAs, palm devices, servers, storages, and the like. Generally, in terms of hardware architecture, the computer  900  may include one or more processors  910 , memory  920 , and one or more input and/or output (I/O) devices  970  that are communicatively coupled via a local interface (not shown). The local interface can be, for example but not limited to, one or more buses or other wired or wireless connections, as is known in the art. The local interface may have additional elements, such as controllers, buffers (caches), drivers, repeaters, and receivers, to enable communications. Further, the local interface may include address, control, and/or data connections to enable appropriate communications among the aforementioned components. 
         [0035]    The processor  910  is a hardware device for executing software that can be stored in the memory  920 . The processor  910  can be virtually any custom made or commercially available processor, a central processing unit (CPU), a digital signal processor (DSP), or an auxiliary processor among several processors associated with the computer  900 , and the processor  910  may be a semiconductor based microprocessor (in the form of a microchip) or a macroprocessor. 
         [0036]    The memory  920  can include any one or combination of volatile memory elements (e.g., random access memory (RAM), such as dynamic random access memory (DRAM), static random access memory (SRAM), etc.) and nonvolatile memory elements (e.g., ROM, erasable programmable read only memory (EPROM), electronically erasable programmable read only memory (EEPROM), programmable read only memory (PROM), tape, compact disc read only memory (CD-ROM), disk, diskette, cartridge, cassette or the like, etc.). Moreover, the memory  920  may incorporate electronic, magnetic, optical, and/or other types of storage media. Note that the memory  920  can have a distributed architecture, where various components are situated remote from one another, but can be accessed by the processor  910 . 
         [0037]    The software in the memory  920  may include one or more separate programs, each of which comprises an ordered listing of executable instructions for implementing logical functions. The software in the memory  920  includes a suitable operating system (O/S)  950 , compiler  940 , source code  930 , and one or more applications  960  in accordance with exemplary embodiments. As illustrated, the application  960  comprises numerous functional components for implementing the features and operations of the exemplary embodiments. The application  960  of the computer  900  may represent various applications, computational units, logic, functional units, processes, operations, virtual entities, and/or modules in accordance with exemplary embodiments, but the application  960  is not meant to be a limitation. 
         [0038]    The operating system  950  controls the execution of other computer programs, and provides scheduling, input-output control, file and data management, memory management, and communication control and related services. It is contemplated by the inventors that the application  960  for implementing exemplary embodiments may be applicable on all commercially available operating systems. 
         [0039]    Application  960  may be a source program, executable program (object code), script, or any other entity comprising a set of instructions to be performed. When a source program, then the program is usually translated via a compiler (such as the compiler  940 ), assembler, interpreter, or the like, which may or may not be included within the memory  920 , so as to operate properly in connection with the O/S  950 . Furthermore, the application  960  can be written as an object oriented programming language, which has classes of data and methods, or a procedure programming language, which has routines, subroutines, and/or functions, for example but not limited to, C, C++, C#, Pascal, BASIC, API calls, HTML, XHTML, XML, ASP scripts, FORTRAN, COBOL, Perl, Java, ADA, .NET, and the like. 
         [0040]    The I/O devices  970  may include input devices such as, for example but not limited to, a mouse, keyboard, scanner, microphone, camera, etc. Furthermore, the I/O devices  970  may also include output devices, for example but not limited to a printer, display, etc. Finally, the I/O devices  970  may further include devices that communicate both inputs and outputs, for instance but not limited to, a NIC or modulator/demodulator (for accessing remote devices, other files, devices, systems, or a network), a radio frequency (RF) or other transceiver, a telephonic interface, a bridge, a router, etc. The I/O devices  970  also include components for communicating over various networks, such as the Internet or intranet. 
         [0041]    If the computer  900  is a PC, workstation, intelligent device or the like, the software in the memory  920  may further include a basic input output system (BIOS) (omitted for simplicity). The BIOS is a set of essential software routines that initialize and test hardware at startup, start the O/S  950 , and support the transfer of data among the hardware devices. The BIOS is stored in some type of read-only-memory, such as ROM, PROM, EPROM, EEPROM or the like, so that the BIOS can be executed when the computer  900  is activated. 
         [0042]    When the computer  900  is in operation, the processor  910  is configured to execute software stored within the memory  920 , to communicate data to and from the memory  920 , and to generally control operations of the computer  900  pursuant to the software. The application  960  and the O/S  950  are read, in whole or in part, by the processor  910 , perhaps buffered within the processor  910 , and then executed. 
         [0043]    When the application  960  is implemented in software it should be noted that the application  960  can be stored on virtually any computer readable medium for use by or in connection with any computer related system or method. In the context of this document, a computer readable medium may be an electronic, magnetic, optical, or other physical device or means that can contain or store a computer program for use by or in connection with a computer related system or method. 
         [0044]    The application  960  can be embodied in any computer-readable medium for use by or in connection with an instruction execution system, apparatus, or device, such as a computer-based system, processor-containing system, or other system that can fetch the instructions from the instruction execution system, apparatus, or device and execute the instructions. In the context of this document, a “computer-readable medium” can be any means that can store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. The computer readable medium can be, for example but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. 
         [0045]    More specific examples (a nonexhaustive list) of the computer-readable medium may include the following: an electrical connection (electronic) having one or more wires, a portable computer diskette (magnetic or optical), a random access memory (RAM) (electronic), a read-only memory (ROM) (electronic), an erasable programmable read-only memory (EPROM, EEPROM, or Flash memory) (electronic), an optical fiber (optical), and a portable compact disc memory (CDROM, CD R/W) (optical). Note that the computer-readable medium could even be paper or another suitable medium, upon which the program is printed or punched, as the program can be electronically captured, via for instance optical scanning of the paper or other medium, then compiled, interpreted or otherwise processed in a suitable manner if necessary, and then stored in a computer memory. 
         [0046]    In exemplary embodiments, where the application  960  is implemented in hardware, the application  960  can be implemented with any one or a combination of the following technologies, which are well known in the art: a discrete logic circuit(s) having logic gates for implementing logic functions upon data signals, an application specific integrated circuit (ASIC) having appropriate combinational logic gates, a programmable gate array(s) (PGA), a field programmable gate array (FPGA), etc. 
         [0047]    The technical effects and benefits of exemplary embodiments include simplification of equivalence checking of two FSAs. 
         [0048]    The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a”, “an”, and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
         [0049]    The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.
A method for determining correctness of a transformation between a first finite state automaton (FSA) and a second FSA, wherein the first FSA comprises a representation of a regular expression, and the second FSA comprises a transformation of the first FSA includes determining a third FSA, the third FSA comprising a cross product of the second FSA and a post-processor; determining whether the first FSA and the third FSA are equivalent; and in the event that the first FSA is determined not to be equivalent to the third FSA, determining that the transformation between the first FSA and the second FSA is not correct.