# Patent

## Title
Input partitioning and minimization for automaton implementations of capturing group regular expressions

## Abstract

A method for submatch extraction may include receiving an input string, receiving a regular expression, and converting the regular expression with capturing groups into a plurality of finite automata to extract submatches. The method further includes using a first automaton to determine whether the input string is in a language described by the regular expression, and to process the input string, and using states of the first automaton in a second automaton to extract the submatches. In addition, input partitioning and automaton minimization techniques may be employed to reduce the storage area consumed by the plurality of finite automata.

## Background
RELATED FIELD
This disclosure generally relates to the field of content search systems, and specifically relates to regular expression search operations.
BACKGROUND OF RELATED ART
Regular expressions provide a concise and formal way of describing a set of strings over an alphabet. Regular expression search and/or match operations are employed in various applications including, for example, intrusion detection systems (IDS), anti-virus products, policy-based routing functions, internet and text search operations, document comparisons, and so on. A regular expression can simply be a word, a phrase or a string of characters. For example, a regular expression including the string “gauss” would match data containing gauss, gaussian, degauss, etc. More complex regular expressions include metacharacters that provide certain rules for performing the match. Some common metacharacters are the wildcard “.”, the alternation symbol “|’, and the character class symbol “[ ].” Regular expressions can also include quantifiers such as “*” to match 0 or more times, “+” to match 1 or more times, “?” to match 0 or 1 times, {n} to match exactly n times, {n,} to match at least n times, and {n,m} to match at least n times but no more than m times. For example, the regular expression “a.{2}b” will match any input string that includes the character “a” followed exactly 2 instances of any character followed by the character “b” including, for example, the input strings “abbb,” adgb,” “a7yb,” “aaab,” and so on.
Given a regular expression and a string, the regular expression matches the string if the string belongs to the set described by the regular expression. Regular expression matching may be used, for example, by command shells, programming languages, text editors, and search engines to search for text within a document. Known techniques for regular expression matching can have long worst-case matching times. In addition, regular expression matching technique typically consume a significant amount of memory space to store the states and transitions associated with one or more automaton representative of the regular expression.



BRIEF DESCRIPTION OF DRAWINGS
Features of the present disclosure are illustrated by way of example and not limited in the following figure(s), in which like numerals indicate like elements, in which:
FIG. 1 illustrates an architecture of a submatch extraction system, according to an example of the present disclosure;
FIG. 2 illustrates an architecture of a pattern creation module of the submatch extraction system, according to an example of the present disclosure;
FIG. 3 illustrates rules for the construction of an automaton M1, according to an example of the present disclosure;
FIGS. 4A and 4B respectively illustrate a deterministic finite automaton (DFA) M1 for ((a)*|b)(ab|b), and a nondeterministic finite automaton (NFA) M2 for the same regular expression, according to an example of the present disclosure;
FIG. 5 illustrates an automaton M3 for the regular expression ((a)*|b)(ab|b), where v={16}, w={13, 14}, x={8}, y={5}, and z={5,12}, according to an example of the present disclosure;
FIG. 6 illustrates relationships among q, r, P, p, and τ in the construction of the automaton M4, according to an example of the present disclosure;
FIGS. 7A and 7B illustrate the automaton M4 for the regular expression ((a)*|b)(ab|b), showing inputs from Q3 and outputs in T*, respectively, and 7C illustrates a simplified diagram of an input string, automata M3 and M4, and boundaries, according to an example of the present disclosure;
FIG. 8 illustrates a method for submatch extraction, according to an example of the present disclosure;
FIG. 9 illustrates a further detailed method for submatch extraction, according to an example of the present disclosure;
FIG. 10 illustrates an architecture of a pattern creation module of the submatch extraction system, according to another example of the present disclosure;
FIG. 11 depicts a pseudo code for implementing Hoperoft's minimization technique;
FIG. 12 depicts a pseudo code for creating initial partitions for application of a DFA minimization technique to the automata M3, according to an example of the present disclosure;
FIG. 13 depicts a pseudo code for creating initial partitions for application of a DFA minimization technique to the automata M4, according to an example of the present disclosure;
FIG. 14 shows an illustrative flow chart depicting an exemplary operation, according to an example of the present disclosure; and
FIG. 15 illustrates a computer system, according to an example of the present disclosure.



DETAILED DESCRIPTION
The present disclosure is described below in the context of automata for performing regular expression matching operations and regular expression submatch extraction operations for simplicity only. It is to be understood that the present disclosure is equally applicable to automata used for other purposes. In the following description, numerous specific details are set forth such as examples of specific components, circuits, and processes to provide a thorough understanding of the present disclosure. The term “coupled” as used herein means connected directly to or connected through one or more intervening components or circuits. Also, in the following description and for purposes of explanation, specific nomenclature is set forth to provide a thorough understanding of the present embodiments. However, it will be apparent to one skilled in the art that these specific details may not be required to practice the present embodiments. In other instances, well-known circuits and devices are shown in block diagram form to avoid obscuring the present disclosure. Any of the signals provided over various buses described herein may be time-multiplexed with other signals and provided over one or more common buses. Additionally, the interconnection between circuit elements or software blocks may be shown as buses or as single signal lines. Each of the buses may alternatively be a single signal line, and each of the single signal lines may alternatively be buses, and a single line or bus might represent any one or more of a myriad of physical or logical mechanisms for communication between components. The present embodiments are not to be construed as limited to specific examples described herein but rather to include within their scopes all embodiments defined by the appended claims.
Throughout the present disclosure, the terms “a” and “an” are intended to denote at least one of a particular element. As used herein, the term “includes” means includes but not limited to, the term “including” means including but not limited to, and the term “based on” means based at least in part on. Further, the terms “accepting states” and “final states” refer to states of an automaton that indicate a match with an input string, and therefore may be interchangeable herein unless otherwise indicated. Similarly, the terms “non-accepting states” and “non-final states” refer to states of an automaton that are intermediate states and that do not result in a match or output value of the automaton, and therefore may be interchangeable herein unless otherwise indicated.
1. Overview
Regular expressions are a formal way to describe a set of strings over an alphabet. Regular expression matching is the process of determining whether a given string (for example, a string of text in a document) matches a given regular expression, that is, whether it is in the set of strings that the regular expression describes. Given a string that matches a regular expression, submatch extraction is a process of extracting substrings corresponding to specified subexpressions known as capturing groups. This feature provides for regular expressions to be used as parsers, where the submatches correspond to parsed substrings of interest. For example, the regular expression (.*)=(.*) may be used to parse key-value pairs, where the parentheses are used to indicate the capturing groups.
A submatch extraction system and a method for extracting submatches from a string that matches a regular expression are described herein. The system and method provide reduced submatch extraction times in the worst case. The submatch extraction system may include an input module to receive a regular expression. The regular expression may be used to create a pattern by a pattern creation module. In order to create the pattern, automata M1, M2, M3 and M4 may be respectively created by automata generation modules. An automaton is defined as an abstract machine that can be in one of a finite number of states and includes rules for traversing the states. The automata may be stored in the system as machine readable instructions. A comparison module may receive input strings, and match the input strings to the regular expression. If an input string does not match the regular expression, submatches are not extracted. However, if an input string matches the regular expression, the pattern created may be used to extract submatches by an output module. In this manner, the regular expression may be compiled by the automata generation modules, and then matched to many different input strings to extract submatches.
In an example, the submatch extraction system may generally include a memory storing a module comprising machine readable instructions to receive an input string, receive a regular expression, and convert the regular expression into a plurality of finite automata to extract submatches. The extracting may include using a first automaton to determine whether the input string is in a language described by the regular expression, and to process the input string, and using states of the first automaton as input to a second automaton to extract the submatches. The system may include a processor to implement the module.
In an example, the method for submatch extraction may include receiving an input string, receiving a regular expression, and converting the regular expression with capturing groups into a plurality of finite automata to extract submatches. The method may further include using a first automaton to determine whether the input string is in a language described by the regular expression, and to process the input string, and using states of the first automaton in a second automaton to extract the submatches.
A non-transitory computer readable medium may have stored thereon machine readable instructions for submatch extraction is also described. The machine readable instructions that when executed may cause a computer system to receive an input string, receive a regular expression, and convert the regular expression into a plurality of finite automata to extract submatches. The extracting includes using a first automaton to determine whether the input string is in a language described by the regular expression, and to process the input string, and using states of the first automaton in a second automaton to extract the submatches.
For regular expressions, the reluctant closure operator, denoted *?, is a variant of the standard greedy closure operator for regular expressions, denoted *, with different submatching behavior. Where other rules do not apply, shorter submatches to a subexpression E*?take priority over longer ones, whereas for E* the reverse is true.
For example, consider matching the string a=b=c first against the regular expression (.*?)=(.*), where the meta-character ‘.’ matches any character in the alphabet, and then against the regular expression (.*)=(.*?). In the first case, the two capturing groups (which are delimited by parentheses) should match a and b=c, respectively, while in the second case the respective submatches should be a=b and c. If the two capturing groups are both greedy or both reluctant, then matching behavior could in principle be governed by a standard related to precedence, although no suitable standard exists.
With regard to parsing, parsing using regular expressions may be used as a building block for security applications such as security information and event management (SIEM) systems. SIEM systems perform real-time analysis of event logs and security alerts generated by hardware and software systems in an enterprise network. Since each source generates its logs in a different format, a SIEM system may use submatch extraction to parse common fields, such as, for example, device name and attack source from the logs. In such a setting, a relatively small number of regular expressions, which are known in advance, may be matched to a large number of input strings in real time. In this regard, the submatch extraction system and method provide for efficient submatch extraction when matching a string to a regular expression that may contain reluctant operators, where the expression may be compiled in advance into a form that will speed up matching and submatching.
For the example of a submatch extraction system whose construction is described in detail herein, the syntax of regular expressions with capturing groups and reluctant closure on an fixed finite alphabet Σ, for example the standard ASCII set of characters, is:

E::=ε∪α∪EE∪E|E∪E*∪E*?∪[E]∪[E]  Equation (1)

For Equation (1), a stands for an element of Σ, and ε is the empty string. The square brackets [, ] are used to group terms in a regular expression that are not capturing groups. The parentheses (, ) are reserved for marking capturing groups. Grouping terms is not explicitly needed when the order of operations is clear. Specifically, (,) and [,] have a higher priority than * and *?, which have a higher priority than concatenation, which has a higher priority than /. The submatch extraction system may use this syntax. Other examples of the submatch extraction system may perform submatch extraction for regular expressions written in a syntax that uses different notation to denote one or more of the operators introduced in Equation (1); or that does not include either or both of the operators * or *? in Equation (1); or that includes additional operators, such as, for example, special character codes, character classes, boundary matchers, quotation, etc.

Indices may be used to distinguish the capturing groups within a regular expression. Given a regular expression E containing c capturing groups marked by parentheses, indices 1, 2, . . . c may be assigned to each capturing group in the order of their left parentheses as E is read from left to right. The notation idx(E) may be used to refer to the resulting indexed regular expression. For example, if E=((a)*/b)(ab/b) then idx(E)=((a)*2/b)1(ab/b)3.
If X, Y are sets of strings, XY is used to denote {xy: xεX, yεy}, and X/Y to denote X∪Y. If β is a string and B a set of symbols, β/B denotes the string in B* obtained by deleting from β all elements that are not in B. A set of symbols T={St, Et: 1≦t≦c} are introduced and may be referred to as tags. The tags may be used to encode the start and end of capturing groups. The language L(F) for an indexed regular expression F=idx(E), where E is a regular expression written in the syntax given by Equation (1), is a subset of (Σ∪T)*, defined by L(ε)={ε}, L(a)={a}, L(F1F2)=L(F1)L(F2), L(F1/F2)=L(F1)∪L(F2), L(F*)=L(F*?)=L(F)*, L([F])=L(F), and L((F)t)={StαEt: αεL(F)}, where ( )t denotes a capturing group with index t. There are standard ways to generalize this definition to other commonly-used regular expression operators, so that it can be applied to cases where the regular expression E is written in a commonly-used regular expression syntax different from the syntax given in Equation (1).
A valid assignment of submatches for regular expression E with capturing groups indexed by {1, 2, . . . c} and input string α is a map sub: {1, 2, . . . c}→Σ*∪{NULL} such that there exists βεL(E) satisfying the following three conditions:

    
    
        (i) β/Σ=α;
        (ii) if St occurs in β then sub(t)=βt/Σ, where βt is the substring of β between the last occurrence of St and the last occurrence of Et; and
        (iii) if St does not occur in β then sub(t)=NULL.
    
    


If αεΣ*, α matches E if and only if α=β/Σ for some βεL(E). For a regular expression without capturing groups, this coincides with the standard definition of the set of strings matching the expression. By definition, if there is a valid assignment of submatches for E and α, then α matches E. It may be proved by structural induction on E that the converse is also true, that is, whenever E matches α, there is at least one valid assignment of submatches for E and α. The submatch extraction system may take as input a regular expression and an input string, and output a valid assignment of submatches to the capturing groups of the regular expression if there is a valid assignment, or report that the string does not match if there is no valid assignment.
The operators (, ) and [, ] have the same effect as [, ] on the set of strings that match a regular expression. The difference is that (, ) marks a subexpression whose submatch is to be reported. Similarly, the difference between the operators * and *? is not apparent in the set of valid assignments of submatches, but is apparent in which of these valid assignments is reported.
2. System
FIG. 1 illustrates an architecture of a submatch extraction system 100, according to an example. Referring to FIG. 1, the system 100 may include an input module 101 to receive a regular expression. The regular expression may be used to create a pattern by a pattern creation module 102. The pattern creation module 102 is described in further detail below with reference to FIGS. 2 and 10. A comparison module 103 may receive input strings, and match the input strings to the regular expression. If an input string does not match the regular expression, submatches are not extracted. However, if an input string matches the regular expression, the pattern created by the pattern creation module 102 may be used to extract submatches by an extraction module 104. Referring to FIG. 2, in order to create the pattern by the pattern creation module 102, automata M1, M2, M3 and M4 may be respectively created by automata generation modules 105, 106, 107 and 108. Thus, the regular expression may be compiled by the modules 105-108 to create the pattern by the pattern creation module 102. In this manner, the regular expression may be compiled by the modules 105-108, and then matched to many different input strings to extract submatches.
The modules 101-108, and other components of the system 100 may comprise machine readable instructions stored on a computer readable medium. In addition, or alternatively, the modules 101-108, and other components of the system 100 may comprise hardware or a combination of machine readable instructions and hardware.
The components of the system 100 are described in further detail with reference to FIGS. 1-7C.
Referring to FIG. 1, for a regular expression E received by the input module 101, the regular expression E may be fixed and indices may be assigned to each capturing group to form idx(E). In order to create the pattern, the pattern creation module 102 may convert the regular expression E into two deterministic finite automata, denoted M3 and M4. The finite automata M3 and M4 may be used to match a string. For the real-time operation, the input string may be reversed and consumed by the first automaton M3, and the states visited as this happens may be journaled. Once all the symbols in the string have been processed, the journaled states may be used in reverse order as input to the second automaton M4, which is used to determine the start and end locations of each capturing group. These start and end locations may be used by the extraction module 104 to obtain the submatches.
In order to create the pattern, the M1 and M2 automata generation modules 105, 106 of the pattern creation module 102 may be used to construct two finite automata, M1 and M2. The automaton M1 is described by the tuple (Q1, Σ1, Δ1, s1, f1), where Q1 is a set of states identified by the integers in the set {1, 2 . . . f}, Σ1 is the alphabet Σ∪{+, −}∪T, where + and − are two special alphabet characters whose use is described below, Δ1 is a transition function, s1=1 is the start state and f1=f is the unique final state. Δ1 is built using structural induction on the indexed regular expression, idx(E), following the rules illustrated by the diagrams of FIG. 3. For this example it is assumed that the syntax of the regular expression is that given in Equation (1). In FIG. 3, the initial state of the automaton is marked with > and the final state with a double circle. A dashed arrow with label F or G is used as shorthand for the diagram corresponding to the indexed expression F or G. For example, the automaton M1 for ((a)*/b)(ab/b) is shown in FIG. 4A. The automata of FIGS. 4A, 4B, 5 and 7A-7C are illustrated as a graphical representation of state machines by way of example and not limitation.
The automaton M1 uses separate transitions with labels St and Et to indicate the start and end of a capturing group with index t, in addition to transitions labeled with + and − to indicate submatching priorities.
The automaton M1 may be considered as a directed graph. If x is any directed path in M1, ls(x) denotes its label sequence. Let π: Q1×Q1→T* be a mapping from a pair of states to a sequence of tags, to be used below in the construction of M4, defined as follows. For any two states q, pεQ1, consider a depth-first search of the graph of M1, beginning at q and searching for p, using only transitions with labels from T∪{+, −}, and such that at any state with outgoing transitions labeled ‘+’ and ‘−’, the search explores all states reachable via the transition labeled ‘+’ before following the transition labeled ‘−’. If this search succeeds in finding successful search path λ(q, p), then π(q, p)=ls(λ(q, p))/T is the sequence of tags along this path. If the search fails, then π(q, p) is undefined. π(p, p) is defined to be the empty string. It can be shown that this description of the search uniquely specifies λ(q, p), if it exists.
The automaton M1 may be converted into the nondeterministic finite automaton (NFA) M2, described by the tuple (Q2, Σ, Δ2, S2, F2), by the M2 automaton generation module 106. The set Q2 includes the final state of M1 together with any state in M1 that has an outgoing transition labeled with a symbol in Σ, i.e.:

Q2={f}∪{q: ∃aεΣ,pεQ1,(q,a,p)εΔ1}  Equation (2)

If p, qεQ2 and aεΣ, there is a transition (p, a, q)εΔ2 if and only if there exists a state rεQ1 such that (p, a, r)εΔ1 and π(r, q) is defined. S2 is a set of initial states, corresponding to those states pεQ2 for which π(1, p) is defined. For example, the automaton M2 for ((a)*/b)(ab/b) is shown in FIG. 4B.

The automaton M2 may be converted into the deterministic finite automaton (DFA) M3 by the M3 automaton generation module 107, and is specified by the tuple (Q3, Σ, Δ3, s3, F3). The construction of M3 from M2 is a standard powerset construction of a DFA from a reversed NFA. Specifically, each state in Q3 corresponds to a subset of states in the powerset of Q2. The initial state s3 is {f}. Q3 is initialized to {{f}}, and states r in Q3 are explored by constructing for each aεΣ the following set:

P(r,a)={pεQ2: (p,a,q)εΔ2 for some qεr},  Equation (3)

Equation (3) represents the set of states from which there is a transition labeled a to an element of r. If this set is not empty, it is added to Q3 and the transition (r, a, P(r, a)) is added to Δ3. Each state in Q3 previously not explored is explored until there are no states in Q3 left to explore. The set of final states in M3, F3, is the set of all states q in Q3 such that q∩S2 is not empty. As discussed above, FIG. 4B illustrates the automaton M2 for the regular expression ((a)*|b)(ab|b). Further, FIG. 5 illustrates the automaton M3 for the same regular expression, where v={16}, w={13, 14}, x={8}, y={5}, and z={5,12}.

M2 and M3 may be used to construct another automaton, M4 by the M4 automaton generation module 108. The automaton M4 is a DFA except that the transition function is a four-tuple, i.e. Δ4⊂Q2×Q3×Q2×T*. The DFA is similar to M2 with one extra state, where the input alphabet is Q3 instead of Σ, and some edges are deleted. The description of automaton M4 that follows will use some notation concerning Σ1 and M1. Let  be the lexicographic partial ordering on the set of strings in Σ1* generated by the relation {(−,+)}∪{(a,a): aεΣ1} on Σ1.
Next a new state labeled ‘0’ is introduced, which is the start state of M4. To define Δ4, let (q, P, p, τ) be in Δ4 if and only if there exist q, rεQ2, PεQ3, pεP, aεΣ, such that (q, a, r)εΔ1, π(r, p) is defined, and




  
    
      
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It can be proved that these maximal elements exist, and are unique.

FIG. 6 illustrates the relationships among q, r, P, p, and τ in the construction of the transition function for M4. Referring to FIG. 6, for the construction of M4, q, r, piεQ1, P={p1, . . . , pn}εQ3, each path λ(r, pi) has label sequence τi and τ=π(r, p)=(max{τ1, . . . , τn})/T. FIGS. 7A and 7B illustrate the automaton M4 for the regular expression ((a)*/b)/(ab/b), showing the inputs from Q3 and the outputs in T*, respectively, and FIG. 7C illustrates a simplified diagram of an input string, automata M3 and M4, and boundaries.
As discussed above, if an input string matches the regular expression, the pattern created by the pattern creation module 102 may be used to extract submatches by the extraction module 104 as follows.
Extracting the submatches for a string a1 . . . alεΣ* may occur as follows. The extraction process is subdivided in steps 1-3 for facilitating the description of the submatch extraction system 100. First, for step 1, the string alal−1 . . . a1 is processed using M3. As it is processed, the states ql, ql−1, . . . visited during the processing are journaled, where ql is {f}, the initial state of M3. If the processing terminates before the whole input string has been processed, or terminates with q0∉F3, it is reported that the string does not match, and the submatch extraction terminates. It can be proved that if M3 has been constructed in the way described in the example, this will happen if and only if the string does not match.
Next, for step 2, if the submatch extraction did not terminate in the previous step, the system 100 may run M4 on input q0, q1, . . . ql, using an additional data structure along the way in order to discover the submatch values for each capturing group. The data structure may include an array of length 2c, indexed by elements of T, all initialized to NULL. While processing the ith transition, namely (qi, P, qi+1, τ)εΔ4, for each tag in τεT*, the system 100 may write i in the array entry corresponding to the tag, overwriting the current entry. It can be proved that if M4 is constructed in the way described in this example, this process will not terminate before all of g0, q1, . . . ql have been processed by M4.
Next, for step 3, the submatch extraction system 100 uses the resulting array to read off the submatches from the input string, as follows. If the array entries for the tags Sj and Ej are sj and ej, respectively, then the system reports that the submatch for capturing group j is asj+1 . . . aej. If the array entries for Sj and Ej are NULL, then the system reports that there is no submatch for the jth capturing group. It can be proved that if M3 and M4 are the automata described in the example, then in the case that the array entries for Sj and Ej are NULL, there is indeed no submatch. It can also be proved that if M3 and M4 are the automata described in this example, the assignment of submatches that is reported by the system is valid.
Referring to FIGS. 5 and 7A-7C, an example of processing an input string aaab for the regular expression ((a)*/b)/(ab/b) is described. As discussed above, FIG. 5 illustrates the automaton M3 for the regular expression ((a)*|b)(ab|b), where the states in FIG. 5 correspond to sets of states in FIG. 4B as follows: v={16}, w={13, 14}, x={8}, y={5}, and z={5,12}. Referring to FIG. 5, in step 1, the reversed input string is processed as baaa using M3. As shown in FIG. 5, the processing begins at v, then proceeds from v to w (processing the symbol b), then proceeds from w to z (processing a), then proceeds from z to y (processing another a), then remains at y (processing another a). The states journaled during this processing are v, w, z, y and y, i.e. states {16}, {13,14}, {5,12}, {5}, {5} respectively. These states are then input in reverse order into the automaton M4. Referring to FIG. 7A, the states visited are {16}, {13, 14}, {5, 12}, {5}, {5}, and respectively correspond to v, w, z, y and y. For the processing of these states in this example, referring to FIGS. 7A and 7B the state of M4 is initially 0, then transitions to 5 as a result of receiving the input {5}, giving output S1S2, then remains at 5 after receiving input {5}, giving output E2S2, then remains at 5 after receiving input {5,12}, giving output E2S2, then transitions to 14 as a result of receiving input {13,14}, giving output E2E1S3, and transitions to 16 as a result of receiving input {16}, giving output E3. Thus in step 2, the submatch extraction system 100 runs automaton M4 with input ({5}, {5}, {5, 12}, {13, 14}, {16}), writing entries in the array with each transition. The resulting array reads as follows:

[S1,E1,S2,E2,S3,E3]=[0,3,2,3,3,4]  Equation (7)

In step 3, the extraction module 104 reads from the array that the three capturing groups have respective submatches aaa, a, and b. For example, referring to FIG. 7C, it can be seen that the last instance of each S and each E is kept. The submatches to the capturing groups are read off from the input string aaab using this array, resulting in submatch aaa to capturing group 1, a to capturing group 2 and b to capturing group 3.

3. Method

FIGS. 8 and 9 illustrate flowcharts of methods 200 and 300 for submatch extraction, corresponding to the example of a submatch extraction system whose construction is described in detail above. The methods 200 and 300 may be implemented on the submatch extraction system with reference to FIGS. 1-7C by way of example and not limitation. The methods 200 and 300 may be practiced in other systems.
Referring to FIG. 8, at block 201, the example method includes receiving a regular expression.
At block 202, the example method includes converting the regular expression with capturing groups into a plurality of finite automata to extract submatches. In this example method, these are the automata M1, M2, M3 and M4 whose construction is described above.
At block 203, the example method includes receiving an input string.
At block 204, the example method includes using a first automaton (i.e., M3) to determine whether the input string is in a language described by the regular expression, and to process the input string.
At block 205, the example method includes using states of the first automaton (i.e., M3) in a second automaton (i.e., M4) to extract the submatches. This includes reversing the input string and processing the reversed input string by the automaton M3, and using the states visited during the processing of the reversed input string in reverse order as input to the automaton M4 to extract the submatches. Indices may be assigned to the capturing groups for the regular expression, the start and end locations of each capturing group may be determined, and the start and end locations may be used to extract the submatches.
Referring to FIG. 9, the further detailed method 300 for submatch extraction is described. At block 301, the example method includes receiving a regular expression and an input string.
At block 302, the example method includes generating the finite automaton M1, whose construction is described above.
At block 303, the example method includes converting the automaton M1 into another automaton, the NFA M2, whose construction is described above.
At block 304, the example method includes converting the automaton M2 into the DFA M3, whose construction is described above.
At block 305, the example method includes using M1, M2 and M3 to construct another automaton, M4, whose construction is described above. Thus, the example method includes converting the regular expression with capturing groups into the automata M3 and M4.
At block 306, the example method includes reversing the input string and processing the reversed input string by the automaton M3, and using the states visited during the processing of the reversed input string in reverse order as input to the automaton M4 to extract the submatches. The method also includes assigning indices to the capturing groups for the regular expression, determining start and end locations of each capturing group, and using the start and end locations to extract the submatches.
As described above, using a plurality of automata to implement regular expression matching operations allows for fast submatch extraction, for example, compared with backtracking algorithms. Unfortunately, automaton-based submatch extraction techniques typically consume a significant amount of storage space, and thus implementation for some applications may be constrained by memory space limitations. In particular, the transition function is typically represented by a two dimensional array, which may be denoted “transitionArray,” whose size is the product of the number of states and the number of inputs. In addition, the automaton-based submatch extraction techniques described in this disclosure utilize a second array, which may be denoted “tagArray,” to indicate where submatches occur. The size of tagArray is also the product of the number of states and the number of inputs. Thus, in accordance with the present disclosure, the storage space needed to implement the automata described above may be reduced using input partitioning techniques and/or minimization techniques. As described in more detail below, input partitioning techniques described herein may be used to replace a plurality of state transitions having identical behavior with a single state transition (e.g., thereby reducing storage area by eliminating duplicative state transition information), and minimization techniques described herein may be applied to an automaton to find an equivalent automaton that has a fewer number of states (e.g., thereby reducing storage area and increasing processing speeds).
Referring to FIG. 10, input partitioning and minimization operations may be performed on automata M3 and M4 by input partitioning modules 1001-1002 and by minimization modules 1003-1004, respectively. Modules 1001-1004 may be formed as part of pattern creation module 102 (as depicted in FIG. 10), or by another suitable module within or otherwise associated with system 100 of FIG. 1.
The modules 1001-1004, and other components of the system 100 may comprise machine readable instructions stored on a computer readable medium. In addition, or alternatively, the modules 1001-1004, and other components of the system 100 may comprise hardware or a combination of machine readable instructions and hardware.
4. Input Partitioning
More specifically, for a given automaton, if a group of input values to a state in the automata result in a state transition to the same next state such that the state transition behavior is the same for all such input values, then the group of input values may be mapped to and replaced by a single “coded” value that forms an input partition for the automaton. For example, for the automaton M3, if providing input characters “a” and “b” as inputs to a current state P in M3 always results in a transition to the same next state P′ in M3, then the two previously separate state transition entries for input characters “a” and “b” at state P in M3 may be mapped to and replaced by a single state transition entry that embodies the input character group {a, b}. In this manner, the input characters “a” and “b” form an input partition for M3 that may be stored as a single state transition entry rather than as two separate state transition entries. Similarly, for the automaton M4, if providing journaled states S1 and S2 as inputs to a current state Q in M4 always results in a transition to the same next state Q′ in M4, then the two previously separate state transition entries for journaled states 51 and S2 at state Q in M4 may be mapped to and replaced by a single journaled state transition entry that embodies the journaled state group {S1, S2}. In this manner, the journaled states 51 and S2 form an input partition for M4 that may be stored as a single state transition entry rather than as two separate state transition entries. Accordingly, creating input partitions for automata M3 and M4 may advantageously reduce the number of state transition entries that represent the transition functions for automata M3 and M4, which in turn reduces the amount of storage area consumed by automata M3 and M4.
Accordingly, for the present disclosure, an input partition may be described as a map Π: A→B, where A is the input alphabet (e.g., Σ for M3 and Q3 for M4), and B is a set of equivalent character groups or input partitions. Here, the input alphabet may refer to the set of state transitions of the corresponding automaton. Then, the transitions Δ3 of M3 and the transitions Δ4 of M4 may be redefined in terms of the equivalent groups B instead of the input alphabet A, thereby mapping the input values to one or more input partitions. Thus, when traversing automata such as M3 and M4, rather than implementing the conventional pseudo code:

    
    
        State nextState=transitionFunction[currentState][input]
        Tags tags=tagFunction[currentState][input], where (currentState; input; nextState; tags)εΔ4,

a new pseudo code may be implemented as follows:

    
    


Partition b=map[input];
State nextState=transitionFunction[currentState][b];
Tags tags=tagFunction[currentState][b].
In this manner, the amount of storage area to represent the arrays “transitionFunction” and “tagFunction” may be significantly reduced.
Because some examples of the present disclosure utilize two inter-dependent deterministic automata M3 and M4 to implement regular expression match operations and submatch extraction operations, respectively (e.g., where M3 is a DFA that processes a reversed input string to find matches, and M4 is a DFA that processes journaled states of M3 to identify the start and end of the input string submatches corresponding to the capturing groups of the regular expression), input partitions are created for one of M3/M4 to be consistent with the other of M3/M4 so that the relationship between the inter-dependent automata M3 and M4 remains constant. This makes creating input partitions for M3 and M4 much more difficult that creating input partitions for only one automaton. In other words, because M3 and M4 process different types of input values in different directions (e.g., M3 processes reversed input strings and M4 processes journaled states of M3), only a selected subset of possible input partitions created for M3 may be suitable for M4 and, similarly, only a selected subset of possible input partitions for M4 may be suitable for M3. Thus, it is important to identify the selected subsets of input partitions that may be applied to both M3 and M4 without altering their inter-dependent relationship.
Input Partitioning for M3 
Input partitions may be created for the automaton M3 as follows. Two input character state transitions ai and aj may be grouped into the same input partition if for every state P in M3 there is another state P′ such that when either ai or aj is provided as an input value to state P, the automaton M3 transitions to the same next state P′. If true, then the separate state transition entries for ai and aj may be mapped to a single state transition entry associated with state P of M3. The input partition creation technique for M3 may be expressed as Statement 1:

    
    
        aiεΣ and ajεΣ are in the same input partition if for all PεQ3, there exists a P′εQ3 such that if (P, ai, P′)εΔ3 then (P, aj, P′)εΔ3.

Input Partitioning for M4 

    
    


Input partitions may be created for the automaton M4 as follows. Two journaled states Pi and Pj of M3 may be grouped into the same input partition for M4 if for every state q in M4 there is another state q′ such that when either Pi or Pj is provided as an input value to state q, the automaton M4 transitions to the same next state q′ and outputs the same tag value (t). If true, then the state transition entries for Pi and Pj may be mapped to a partitioned state transition entry associated with state q of M4. The input partition creation technique for M4 may be expressed as Statement 2:

    
    
        PiεQ3 and PjεQ3 are in the same input partition if for all qεQ4, there exists a q′εQ4 and tεT* such that if (q, Pi, q′, t)=Δ4 then (q, Pj, q′, t)=Δ4.

5. Minimization

    
    


As mentioned above, it is desirable to minimize the number of states in M3 and M4. This may be accomplished by transforming automata M3 and M4 into respective equivalent automata M3′ and M4′ that have a minimum number of states. Normal minimization algorithms work by creating an initial state partition (not to be confused with the input partitions described above), and then refining that initial state partition recursively, for example, using Hoperoft's algorithm. Normally, that initial state partition corresponds to splitting the states into two sets: those states that are final states and those that are non-final states. In this disclosure, minimization may be performed by applying a standard minimization algorithm to a modified initial state partition created in accordance with the present disclosure. Then, the redefined initial partitions for M3 and M4 may be processed by a suitable DFA minimization technique to derive equivalent automata M3′ and M4′. For example, some applications of the present disclosure may use Hoperoft's algorithm to process the redefined initial partitions of M3 and M4 to derive equivalent DFAs having fewer numbers of states, thereby reducing the memory area for storing the two arrays “transitionFunction” and “tagFunction” associated with the pseudo code described above with respect to creating input partitions. Of course, for other applications, other suitable DFA minimization techniques may be used to derive equivalent automata M3′ and M4′.
Minimizing M3 
For the automaton M3, two states in M3 may be grouped together in the same redefined initial partition if (1) both states are either non-final states or final states and (2) both states generate the same tag outputs when provided as journaled state entries to a given state q in M4. This may be expressed as Statement 3:

    
    
        Let states PiεQ3 and PjεQ3 be in the same initial state partition if (1) Pi and Pj are both in F or Pi and Pj are both not in F and (2) Pi and Pj are in the same input partition of M4.
    
    


FIG. 12 depicts a pseudo code 1200 that may be employed to create the redefined initial partitions for M3. Thus, to create the redefined initial partitions for M3, the states in M3 are grouped according to whether they are final (e.g., accepting) states or non-final (e.g., non-accepting) states and also according to whether the states are in the same journaled state groups in M4. In other words, each redefined initial partition of M3 includes only states that (1) are either all final states or all non-final states and (2) belong to the same journaled state partition of M4.
For example, if M3 has five states S1-S5 for which states S1-S3 are grouped into a first input partition Z1 and states S4-S5 are grouped into a second input partition Z2 using Statement 1, and if states S1-S2 are both non-final states and S3 is a final state, then the first input partition Z1 is divided into two separate initial partitions Z1a containing states S1-S2 and Z1b containing state S3, for example, as may be implemented by the pseudo code 1200 of FIG. 12.
Thereafter, the redefined initial partitions created using the pseudo code 1200 of FIG. 12 may be processed by Hoperoft's algorithm (or any other suitable DFA minimization algorithm) to find an equivalent DFA M3′ having a fewer number of states than original DFA M3.
Minimizing M4 
For the automaton M4, which does not have final and non-final states but rather has states that either output tag values or don't output tag values, two states may not be grouped in the same input partition if there is at least one input value that results in M4 outputting different tag values. In other words, if the two states produce different output tags in response to the same journaled state input value, then the two states cannot be in the same input partition. Otherwise, grouping these two states together in the same partition may undesirably result in M4 becoming a non-deterministic automaton. This may be expressed as Statement 4:

    
    
        Let states qiεQ4 and qjεQ4 be in in different partitions if for all PεQ3, there exist distinct tiεT* and tjεT* such that there exists at least two relations (qi, P, q′, τi)εΔ4 and (qj, P, q′, τj) for some qi′εQ3 and qj′εQ3.
    
    


Thus, to define the initial partitions of M4, the technique begins with the input partition being the entire set of states Q4. Then for each input, iεQ3, a partition of the states is derived based on their tag strings. This is possible because there is only one relation (q, i, q′, τ) for a given state q and input i. Using Statement 4, the input partition may be iteratively refined to derive the initial partitions for M4, for example, using the pseudo code 1300 shown in FIG. 13. Thereafter, the initial partitions created according to the pseudo code 1300 of FIG. 13 may be processed by Hoperoft's algorithm (or any other suitable DFA minimization algorithm) to find an equivalent DFA M4′ having a fewer number of states than M4.
Note that the initial partitions may be created by comparing every pair of states and then sequentially assigning them to the next available partition if they are in conflict. However, such an approach would require a processing time of O(n2) for n states, and would thus undermine the efficiencies of Hoperoft's algorithm over other quadratic time minimization algorithms. Thus, to make the minimization technique asymptotically efficient, the following observation may be made. Suppose there are two partitions π1 and π2 of the set of states Q4. Let φi(q) be the index of the set in πi that contains state q. Then, the intersection of π1 and π2 is a new partition π12 such that any two states q1 and q2 are in the same set of π12 if and only if they have the same values for φ1 and φ2.
For other embodiments, a one-pass automaton may be used to process an input string to find matches with certain regular expressions. The one-pass automaton may be defined by the tuple (Q, Σ, Δ, q0, F), where Q is some set of states, Σ is an alphabet of input symbols, Δ is a set of relations in Q×Σ×Δ×T called the transition function, q0εQ3 is an initial state, and F⊂Q is a set of accepting states. For such automata, the pseudo code 1300 of FIG. 13 may be applied to create the input partition, except that instead of starting with the entire set of states as the initial partition (e.g., P:={Q}), we start with states partitioned into accepting states and non-accepting states, where P:={{F}, {Q−F}.
FIG. 14 is a illustrative flow chart depicting an exemplary operation for creating input partitions for M3 and M4 and for minimization the number of states in M3 and M4. As mentioned above, the automaton M3 determines whether the input string matches the regular expression by processing characters of the input string in a reversed order, and the automaton M4 determines boundaries of the submatches by processing states transitioned by the first automaton in a reversed order. To create input partitions for and to minimize M3 and M4, a plurality of first state transition entries of the M3 are grouped into an M3 input partition if all of the first state transition entries cause M3 to transition from a first current state to the same first next state (1401). Then, a plurality of selected states of M3 are grouped into an M4 input partition if (1) all of the selected states cause M4 to transition from a second current state to the same second next state and (2) all of the selected states cause M4 to output the same tag value, wherein the tag value indicates a start and an end of a corresponding extracted submatch (1402).
Next, to minimize M3, the selected states of M3 are grouped into a first initial partition if (1) the selected states are either all final states or all non-final states and (2) all the selected states cause M4 to generate the same tag value when provided as inputs to the second current state of M4 (1403). Then, in response to the first initial partition, a first equivalent automaton M3′ is derived for M3 such that M3′ has a minimum number of states (1404).
Next, to minimize M4, certain states are removed from the M4 input partition to define an M4 initial partition if the certain states cause M4 to generate different tag values when provided as inputs to the second current state of M4 (1405). Then, in response to the second initial partition, a second equivalent automaton M4′ is derived for M4 such that M4′ has a minimum number of states (1406).
6. Computer Readable Medium
FIG. 15 shows a computer system 400 that may be used with the examples described herein. The computer system 400 represents a generic platform that includes components that may be in a server or another computer system. The computer system 400 may be used as a platform for the system 100. The computer system 400 may execute, by a processor or other hardware processing circuit, the methods, functions and other processes described herein. These methods, functions and other processes may be embodied as machine readable instructions stored on computer readable medium, which may be non-transitory, such as hardware storage devices (e.g., RAM (random access memory), ROM (read only memory), EPROM (erasable, programmable ROM), EEPROM (electrically erasable, programmable ROM), hard drives, and flash memory).
The computer system 400 includes a processor 402 that may implement or execute machine readable instructions performing some or all of the methods, functions and other processes described herein. Commands and data from the processor 402 are communicated over a communication bus 404. The computer system 400 also includes a main memory 406, such as a random access memory (RAM), where the machine readable instructions and data for the processor 402 may reside during runtime, and a secondary data storage 408, which may be non-volatile and stores machine readable instructions and data. The memory and data storage are examples of computer readable mediums. The memory 406 may include modules 420 including machine readable instructions residing in the memory 406 during runtime and executed by the processor 402. The modules 420 may include the modules 101-108 of the system 100 shown in FIG. 1 and/or modules 1001-1004 of FIG. 10.
The computer system 400 may include an I/O device 410, such as a keyboard, a mouse, a display, etc. The computer system 400 may include a network interface 412 for connecting to a network. Other known electronic components may be added or substituted in the computer system 400.
In the foregoing specification, the present disclosure has been described with reference to specific examples. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the disclosure as set forth in the appended claims. The specification and drawings are, accordingly, to be regarded in an illustrative sense rather than a restrictive sense. For example, method steps depicted in the flow charts of FIGS. 8-9 and 14 may be performed in other suitable orders and/or one or more methods steps may be omitted.

## Claims

1. In a system including a first automaton for finding a match between an input string and a regular expression and including a second automaton for extracting submatches corresponding to capturing groups in the regular expression, a method for creating input partitions for the first and second automata, the method comprising:
grouping a plurality of first state transition entries of the first automaton into a first automaton input partition when all of the first state transition entries cause the first automaton to transition from a first current state to the same first next state; and
grouping a plurality of selected states of the first automaton into a second automaton input partition when (1) all of the selected states cause the second automaton to transition from a second current state to the same second next state and (2) all of the selected states cause the second automaton to output a same tag value, wherein the tag value indicates a start and an end of a corresponding extracted submatch.

2. The method of claim 1, wherein:
the first automaton determines whether the input string matches the regular expression by processing characters of the input string in a reversed order; and
the second automaton determines boundaries of the submatches by processing states transitioned by the first automaton in a reversed order.

3. The method of claim 1, wherein the first automaton transitions from the first current state to the first next state in response to a transition function defined by the first current state and the first automaton input partition.

4. The method of claim 1, wherein the second automaton transitions from the second current state to the second next state in response to a transition function defined by the second current state and the second automaton input partition.

5. The method of claim 1, further comprising:
grouping the plurality of selected states of the first automaton into a first initial partition when (1) the selected states are either all final states or all non-final states and (2) all the selected states cause the second automaton to generate the same tag value when provided as inputs to the second current state of the second automaton; and
deriving, for the first automaton, a first equivalent automaton having a minimum number of states in response to grouping the plurality of selected states of the first automation into the first initial partition.

6. The method of claim 1, further comprising:
removing certain states from the second automaton input partition to define a second initial partition when the certain states cause the second automaton to generate different tag values when provided as inputs to the second current state of the second automaton; and
deriving, for the second automaton, a second equivalent automaton having a minimum number of states in response to defining the second initial partition.

7. The method of claim 1, wherein the plurality of first state transition entries represent a set of input values included in the input string, and wherein the first automaton input partition is defined by a single coded value representing the set of input values of the first automaton.

8. A system for creating input partitions for first and second automata, wherein the first automaton is to find a match between an input string and a regular expression and the second automaton is to extract submatches corresponding to capturing groups in the regular expression, the system comprising:
means for grouping a plurality of first state transition entries of the first automaton into a first automaton input partition when all of the first state transition entries cause the first automaton to transition from a first current state to the same first next state; and
means for grouping a plurality of selected states of the first automaton into a second automaton input partition when (1) when all of the selected states cause the second automaton to transition from a second current state to the same second next state and (2) all of the selected states cause the second automaton to output a same tag value, wherein the tag value indicates a start and an end of a corresponding extracted submatch.

9. The system of claim 8, wherein:
the first automaton determines whether the input string matches the regular expression by processing characters of the input string in a reversed order; and
the second automaton determines boundaries of the submatches by processing states transitioned by the first automaton in a reversed order.

10. The system of claim 8, wherein the first automaton is to transition from the first current state to the first next state in response to a transition function defined by the first current state and the first automaton input partition.

11. The system of claim 8, wherein the second automaton is to transition from the second current state to the second next state in response to a transition function defined by the second current state and the second automaton input partition.

12. The system of claim 8, further comprising:
means for grouping the selected states of the first automaton into a first initial partition when (1) the selected states are either all final states or all non-final states and (2) all the selected states cause the second automaton to generate the same tag value when provided as inputs to the second current state of the second automaton; and
means for deriving, for the first automaton, a first equivalent automaton having a minimum number of states in response to the first initial partition.

13. The system of claim 8, further comprising:
means for removing certain states from the second automaton input partition to define a second initial partition when the certain states cause the second automaton to generate different tag values when provided as inputs to the second current state of the second automaton; and
means for deriving, for the second automaton, a second equivalent automaton having a minimum number of states in response to the second initial partition.

14. The system of claim 8, wherein the plurality of first state transition entries represent a set of input values included in the input string, and wherein the first automaton input partition is defined by a single coded value representing the set of input values of the first automaton.

15. A system for creating input partitions for first and second automata, wherein the first automaton is to find a match between an input string and a regular expression and the second automaton is to extract submatches corresponding to capturing groups in the regular expression, the system comprising:
a processor; and
a memory coupled to the processor and having stored therein computer-executable instructions that when executed by the processor cause the system to:
group a plurality of first state transition entries of the first automaton into a first automaton input partition if all of the first state transition entries cause the first automaton to transition from a first current state to the same first next state; and
group a plurality of selected states of the first automaton into a second automaton input partition if (1) all of the selected states cause the second automaton to transition from a second current state to the same second next state and (2) all of the selected states cause the second automaton to output a same tag value, wherein the tag value indicates a start and an end of a corresponding extracted submatch.

16. The system of claim 15, wherein:
the first automaton determines whether the input string matches the regular expression by processing characters of the input string in a reversed order; and
the second automaton determines boundaries of the submatches by processing states transitioned by the first automaton in a reversed order.

17. The system of claim 15, wherein the instructions, when executed by the processor, further cause the system to:
group the selected states of the first automaton into a first initial partition if (1) the selected states are either all final states or all non-final states and (2) all the selected states cause the second automaton to generate the same tag value when provided as inputs to the second current state of the second automaton; and
derive, for the first automaton, a first equivalent automaton having a minimum number of states in response to the first initial partition.

18. The system of claim 15, wherein the instructions, when executed by the processor, further cause the system to:
remove certain states from the second automaton input partition to define a second initial partition if the certain states cause the second automaton to generate different tag values when provided as inputs to the second current state of the second automaton; and
derive, for the second automaton, a second equivalent automaton having a minimum number of states in response to the second initial partition.

19. The system of claim 15, wherein the plurality of first state transition entries represent a set of input values included in the input string, and wherein the first automaton input partition is defined by a single coded value representing the set of input values of the first automaton.