Symmetrical structural pattern matching

[0001] Cross-reference is made to U.S. patent applications Ser. No. 09/______, entitled Bi-Valuation Of Programming Statements (Attorney Docket No. D/A0858Q), which is assigned to the same assignee as the present invention and incorporated herein by reference.

[0002] 1. Field of the Invention

[0003] The invention generally relates to pattern matching techniques in programming languages and in particular to a method and system for determining whether a computer-storable expression matches a filter.

[0004] 2. Description of the Related Art

[0005] Presently, several techniques have been developed for string pattern matching. The most fundamental underlying theory, i.e. that of regular languages and recognition automatons, has been widely applied in areas ranging from text processing to DNA sequence searches. Expressions describing string patterns are today still loosely integrated with general purpose scripting languages such as Python or Perl, because of their convenience for solving recurrent string related problems such as pattern extraction or replacement.

[0006] With the increasing importance of structured information processing, and especially of complex structural transformations involved in data and document interchanges, the growing trend is to propose specialized query or transformation languages that strongly integrate pattern matching facilities. As long as complex structures are considered, e.g. trees or even graphs, language theorists and designers are again looking for expressive, powerful, simple, clear and precise formalisms in order to capture the most fundamental matching operations. However, the richness of complex data structures induces richer transformation requirements, and general solutions for specifying general pattern matching at the right abstraction level have not yet been proposed. Such general pattern matching technique would however be required for application to a broad class of programming languages.

[0007] Given the problems of the existing technologies, it would therefore be advantageous to provide a method and system that enable the specification of matching operations on various and arbitrary complex data structures while keeping the amount of basic building blocks reasonably small.

[0008] It would further be advantageous to provide a pattern matching technique that is based on explicit and restricted assumptions, so that it can be embedded into existing or future, specialized or general purpose programming languages.

[0009] Further it would be advantageous to provide a pattern matching method and system having underlying syntax and semantics defined on an unambiguous basis and within an extensible and flexible theoretic framework.

[0010] Moreover, it would be advantageous to provide an improved pattern matching technique of great simplicity and high expressive power.

[0011] The present invention has been made in consideration of the above situation and provides a method, and article of manufacture therefor, of operating a computer system for determining whether a computer-storable expression matches a filter. A first code structure that represents the expression is evaluated for determining a value of the expression. A second code structure that represents the filter is analyzed for determining the characteristics of the filter. The first code structure is constructed from a plurality of first programming language code structure elements and the second code structure is constructed from a plurality of second programming language code structure elements, where each second structure elements corresponds to one of the first structure elements. The value of the expression is then filtered according to the determined filter characteristics.

[0012] The present invention further provides a computer system capable of determining whether a computer-storable expression matches a filter. The computer system comprises processing means for evaluating a first code structure that represents the expression for determining a value of the expression, analyzing a second code structure that represents the filter for determining the characteristics of the filter, and filtering the value according to the filter characteristics. The first code structure is constructed from a plurality of first programming language code structure elements and the second code structure is constructed from a plurality of second programming language code structure elements. Each second structure element corresponds to one of the first structure elements.

[0013] The accompanying drawings are incorporated into and form a part of the specification to illustrate several embodiments of the present invention. These drawings, together with the description, serve to explain the principles of the invention. The drawings are only for the purpose of illustrating alternative examples of how the invention can be made and used and are not to be construed as limiting the invention to only the illustrated and described embodiments. Further features and advantages will become apparent from the following and more particular description on the various embodiments of the invention as illustrated in the accompanying drawings, wherein:

[0014] FIG. 1 illustrates a computer system according to the invention;

[0015] FIG. 2 is a general flow chart illustrating the pattern-matching technique of the invention;

[0016] FIG. 3 is a more particular flow chart illustrating the pattern matching technique of the invention;

[0017] FIG. 4 is a flowchart illustrating the evaluation statement process of the invention; and

[0018] FIG. 5 illustrates an example of the hierarchy of types in a programming language based on the techniques of the invention.

[0019] Referring now to the drawings and particularly to FIG. 1, which illustrates a computer system of the present invention, processing means 100 is provided that is connected to an input section 110 and an output section 120. The processing means 100 can be of any type and has access to a program code storage 130 that stores the programming language statements and expressions that are operated on by the invention. The system further comprises in the example shown in FIG. 1, a statement evaluation section 140 and a pattern matching section 150 including instructions that allow the processing means 100 to operate as discussed in more detail below.

[0020] General Syntax and Operational Semantics

[0021] Before going into the details of the invention, an example of a general abstract syntax for terms of the underlying language is provided, where the notation of e encompasses imperative and declarative statements. Operational semantics is described by using the SOS style described in G. D. Plotkin's article A structural approach to operational semantics, Technical Report DAIMI-FN-19, Computer Sciences Dept., rhus university, Denmark, 1981, considering small step transition in order to cover an eventual extension to concurrent languages with interleaving semantics.

[0022] The first set of definitions is now provided that describe basic constructs that can be considered as universal.

	e :: n s true false	constants: numerics, strings, booleans
	none unit	distinguished values
	e e	basic operations ( , , *, /)
	if e1 then e2 else e3	boolean choice
	var x e.e	variable declaration
	x	variables
	e :: true false	boolean literals
	e e e e	equality and difference comparisons

[0023] The construction for the introduction of local variables

[0024] var xe in e

[0025] is often noted

[0026] let xe in e

[0027] in functional language communities.

[0028] More sophisticated computation structures, e.g. sequencing, loops and assignment, with widely understood semantics are described through the following grammar extension:

	e :: e;e	sequence
	x : e	simple assignment to a variable
	*(e)	closure computation

[0029] A transition system describes an interpreter of the language through a transition relation which defines the computation of any expression e into a new expression e, and is noted ee. Several computation steps such as eee can be abbreviated by using the notation ee. By definition, terminating computations will reduce e to a normal value noted v which cannot be reduced anymore; this particular computation step is noted eo v, and a derivation chain that reduces to a normal value is noted eo v. The relation o is formally described through: $\frac{e->e^{}{e}^{}{e}^{}}{e->e^{}}\mathrm{or}\frac{e->e^{}\left(e^{}->e^{}\right)}{e->e^{}}\left[\mathrm{norm}\right].$

[0030] The formal semantics of basic boolean, string and arithmetic operations (noted above) will not be described here, being considered as widely understood.

[0031] The computing environment, as well as the use and notation of variables and references will now be described in more detail. The computing environment, noted S, is a mapping from names (of variables) into reduced values: $S=\left\{x_i^v\right\}.$

[0032] with x_i^v being a denotation for the mapping of a unique label x_i into a unique reduced value v_i. Reference handling requires also another execution structure, a heap $H=\left\{r_i^v\right\}$

[0033] using the same notation provided that r denotes references. The introduction of recursive procedures or functions with non-lazy evaluation would simply require a stack of mappings in order to handle local environments. The entire environment H, S is sometimes abbreviated by .

[0034] In another notation, the computing environment is noted as the mapping from variable names into reduced values: $=\left\{x_0^v,,x_k^v\right\}$

[0035] Further, ,x^v is an abbreviation of H,Sx^v, or x^v, respectively, and similarly , r^v is an abbreviation of Hr^v, S. The transition relation becomes now ee in the general case, although invariant execution structures can be skipped for clarity when required. This relation means expression e computed in environment H, S becomes e in environment H, S, understood that H and S can be equal to or different from H and S. The following equations describe the semantics of variable access var and declaration d1,d2:

[0036] ti ,x^vx,x^vv var $\begin{array}{c}\frac{e_1->{}^{}{e}_1^{}}{\mathrm{var}x=e_1{e}_2->{}^{}\mathrm{var}x=e_1^{}{e}_2}\left[\mathrm{d1}\right]\\ \frac{e_1->v}{\mathrm{var}x=e_1{e}_2->,x^v{e}_2}\left[\mathrm{d2}\right]\end{array}$

[0037] Reference creation and dereferencing involve intermediate distinguished values r (references) on which no other operations are defined, as opposed to pointers and pointer arithmetics. $\begin{array}{cc}\frac{e->{}^{}{e}^{}}{@e->{}^{}@e^{}}\left[\mathrm{ref1}\right]& \frac{e->{}^{}{e}^{}}{!e->{}^{}!e^{}}\left[\mathrm{dref1}\right]\end{array}$ v,r^vr ref2 , r^vr, r^vv dref2

[0038] In ref2, it is to be mentioned that a new unique reference r is created in the Heap and that this reference is returned as result of the computation of v.

[0039] In the testing operation, environment modifications are allowed during evaluation of the boolean part. This enables the use of a matching operation as a condition for the test. $\frac{e_1->{}^{}{e}_1^{}}{\mathrm{if}{e}_1\mathrm{then}{e}_2\mathrm{else}{e}_3->{}^{}\mathrm{if}{e}_1^{}\mathrm{then}{e}_2\mathrm{else}{e}_3}\left[\mathrm{if}\right]$

[0040] if true then e₂ else e₃e₂ if1 if false then e₂ else e₃e₃ if2

[0041] The semantics of the basic numerical operations , /, *, is well known in the art. The operator is polymorphic, i.e. it applies in various semantics to numerals, strings, sequences, concatenations, multisets, disjunctive unions, i.e. m₁m₂m₂m₁, and dictionaries (non-commutative, right priority).

[0042] The pattern matching technique, which will now be described in more detail, evaluates a first code structure that represents an expression, analyzes a second code structure that represents the filter to be matched by the expression, and filters the value determined by evaluating the first code structure according to filter characteristics determined by the second code structure. The first code structure, hereafter denoted data structure and the second code structure, hereafter denoted pattern structure, are constructed symmetrically so that matching operations can be specified on various and arbitrary complex data structures, such as strings, sequences, sets, dictionaries and records, but also on trees, DAGs (Directed Acyclic Graphics) and general graphs. This will be shown in more detail hereafter.

[0043] In the following, it is assumed that it is possible to declare local variables which will potentially be assigned to parts of or to the entire filtered structures. Expressions, such as e, e_i can be literal constants, variables or basic operations, such as ee, e*e. References are considered because they allow the sharing of substructures and the modeling of DAGs and graphs.

[0044] Pattern matching operations make it possible to check a given expression e for whether the expression contains a structural pattern, and to extract a part of its content, in a single operation. A pattern matching operation is made up of three parts: The first part is an expression e against which the filter is to be checked. This part is called the subject. The second and third parts are the pattern-matching operator and the filter f. The subject can be a variable, a constant, or a more complex expression that will be evaluated prior to matching.

[0045] Thus, a pattern matching operation looks like

[0046] ef.

[0047] These operations return a boolean value. If the operation succeeds, meaning that the expression matches the filter and, optionally, that variables have been assigned to a part of the subject's content, then it returns true. If it fails, meaning that the expression did not match the filter structure or that it did not contain a given value, then false is returned.

[0048] By convention, normal values are noted v or n, s if they are numeric or strings, respectively. As shown above, a derivation chain that reduces to a normal value is noted eo v. The general semantics of the matching operation ef requires in the following order: the evaluation (step 300 in FIG. 3) of the left operand e (normalized to v), the evaluation (step 310) of the right operand, i.e. the filter f, and the application (step 320) of the filter to the normalized value, which returns either true or false. It will be appreciated that the term value does not necessarily relate to a numeric value as the expression may be a non-numeric expression. Moreover, the environment might be modified by the matching operation, whatever result is obtained. Normalized filters are noted as bold letters. The formal semantics of matching operation is twofold. The first stage computes a reduced form first for the subject (match1 iterated) and then for the filter (f-match1 iterated): $\frac{e->{}^{}{e}^{}}{ef->{}^{}{e}^{}f}\left[\mathrm{match1}\right]\frac{f->{}^{}{f}^{}}{vf->{}^{}v{f}^{}}\left[f\text{-}\mathrm{match1}\right]$

[0049] The last stage, i.e. the filter application itself, depends on the filter and data structure considered and will be defined case by case in the following.

[0050] Examples of structure constructors are now presented in order to illustrate the symmetry between them and the corresponding filter constructors described below. The structure constructors and filter constructors are indicator elements indicating the respective data type:

e1, e2, . . .	Multiset: a collection type in which
	elements are not ordered and which can
	contain multiple instances of the same
	value
e1, e2, . . .	Sequence: an ordered collection of
	elements of a common type
<e1, e2, . . . >	Tuple: an ordered collection of elements
	which has a fixed size. Elements need
	not be of a common type
<name1 e1, name2 e2, . . .	Record: an unordered collection of
	named elements
key1 e1, key2 e2, . . .	Dictionary: an unordered collection of
	elements that are each accessed by a
	key. Each key must be unique
(e)	References: a reference to a value (itself
	computed from an expression e) stored in
	memory
	De-referencing operator, used to access
	a referenced value

[0051] Further, there is provided a operator that can take different semantics depending on the data structure, e.g. arithmetic addition, string concatenation, set union, etc.

[0052] Filters are expressed using a small number of operators and can describe many different structures. Besides this, filters closely resemble the structures to be matched, making them rather easy to specify. Filters can match sequences, multisets, tuples, records, dictionaries, strings and any combination of these structures. All filters are built using three basic filter operators, in addition to filter constructors which use the same notation as their data constructor counterparts. The three basic filter operators are the test operator, the existence operator and the assignment operator.

[0053] The test operator, noted % in the present example, is used in conjunction with a constant, variable or complex expression, which will be evaluated prior to matching. It tests the occurrence of the value given by its operand at some point in the structure. For instance,

[0054] e %circus

[0055] tests whether e is a string equal to circus or not.

[0056] e %(22), %3

[0057] checks that e is a sequence having one item of value 4 and one item of value 3 in that order.

[0058] The existence operator, noted , is defined as match any element. For instance,

[0059] e <,>

[0060] matches any tuple that has exactly two elements, like

[0061] <circus, 47>.

[0062] The assignment operator, noted x, where x is a variable name, is used to extract a part of the subject and assign this part to x. For instance,

[0063] e <%circus,,y>

[0064] will succeed if e is a tuple containing three elements. The first element has to be the string circus, the second one can be anything, and the value of the third element is assigned to variable y. The same pattern matching operation on a sequence would be very similar:

[0065] e %circus,,y

[0066] changing only the constructor from tuple to sequence.

[0067] In addition to these basic filter operators, there is provided a concatenation operator, noted , that combines compatible elements. It can be seen as the dual operator of the concatenation operator (noted , see above) of data structures. For instance, the string circus can be thought of as the concatenation of three substrings

[0068] circus.

[0069] In a symmetrical way, the filter appears as

[0070] e % ci %r %cus.

[0071] For finding strings containing character r, the filter

[0072] %r

[0073] can be used, which means zero or more elements, plus the letter r, and a further zero or more element. This operator can be used with any structure, making it possible to take other elements into account. For instance, e x matches only sequences consisting of exactly one element, and assigns this element to x. To express the fact that a sequence e must contain at least one element, without any ordering, combined with the operator can be used:

[0074] e x .

[0075] For illustrating the advantages of the present structural pattern matching scheme, more complex examples are now given for strings, tuples, sequences, records, multisets and dictionaries.

[0076] e %r x

[0077] will match any string containing r and assign the substring beginning after the first occurrence of r to variable x; if e-circus, then x will be assigned cus; if ered, then x will be assigned ed.

[0078] e <f1x>

[0079] will match any sequence containing one record as the first element. This record must contain a field f1; the value of this field will be assigned to variable x.

[0080] <f110>,<f2s>,<f13> <f1x>y

[0081] will succeed; x will be assigned 10 and y will be assigned <f2s>, <f13>.

[0082] e <, x>

[0083] will match any multiset containing at least one sequence whose elements are tuples made up of two elements and will assign the second element of the last tuple of a sequence in this multiset to variable x. It will select the last tuple of a sequence because the pattern structure includes <, x> instead of <, x> which would have selected the first tuple.

[0084] ab, cd, ef %ef,%ab,%cd

[0085] will match since the order is irrelevant in sets.

[0086] Pattern matching can further be used to retrieve values from a dictionary. Suppose a dictionary which maps some strings to some sequences,

[0087] dict k1

[0088] will succeed if dict contains a key k1, and

[0089] dict k2x

[0090] will succeed if dict contains key k2 and will assign the corresponding sequence to variable x.

[0091] Since pattern matching operations return boolean values, they can be composed using logical connectors. It is then possible to express filters which succeed only if the subject does not match the filter, or complex filters. There are three composition operators, not, or and and.

[0092] not will succeed if the subject does not match the filter. For instance,

[0093] e not ( %r )

[0094] will succeed only if string e does not contain the character r.

[0095] or takes two filter arguments, which are matched from left to right. For instance

[0096] e %c or %r

[0097] will succeed only if string e begins with the character c or if it contains the character r.

[0098] and works in a similar way as or. It succeeds if the expression matches both filters. For instance,

[0099] e <,> and x

[0100] allows checking that e is a two-element tuple and assigns e to x.

[0101] There are also provided Kleene operators * and that work with the known semantics, zero or more elements and at least one element, respectively. For instance,

[0102] e %abb (%c)* %dee

[0103] will match any of the following strings: abbdee, abbcdee, and abbcccdee. The same filter using the second operator,

[0104] e %abb (%c) %dee

[0105] will match the same set of strings except abbdee. The Kleene operators can be used with sequences, sets and dictionaries.

[0106] As shown above, the described pattern matching technique enables the symmetrical constructions of both pattern matching structures and data structures, so that they reach the same arbitrary level of structural complexity. Further, the matching operation is invoked explicitly, through a dedicated operator. The general form of the structure to be matched is tested as well as the contained (sub-)structures and the assignment of part(s) of the matching structure to variable(s) of the execution environment.

[0107] The technique relates to a number of fields including those of language construction, control abstractions and transformations models, rewriting systems theory, term rewriting systems, transformation languages or systems for compilation and language processing, structured document transformation, tree pattern matching, explicit or automatic document transformation systems, and so on.

[0108] Further, the technique may serve for building transformation models which are less abstract and more general, than rewrite systems, which perform implicit pattern matching and apply built-in strategies for rule application. The technique is on the other hand abstract enough to simplify and extend general purpose programming languages. Thus, the technique may play an important role in the design of new transformation techniques or the extension of existing ones.

[0109] Four basic topics are addressed in a single approach different to and independent from existing ones that are currently and conceptually less efficient: the definition of a data model, the definition of matching operations, the definition of a transformation model, and the integration of the transformation model in a programming language.

[0110] The definition of a data model is to give a precise form to the notion of data structures, which need to be rich and clear for the programmer but at the same time kept as simple as possible. The computational properties of each kind of structure should be stressed to reach completeness at minimal cost. For instance, sequences are ordered structures that are useful for handling stacks, fifo's, or to memorize intermediate nodes when walking through a tree, whereas multisets are useful for managing bags of unorganized data.

[0111] The technique described above proposes a universal data model based on a synthesis of the most commonly used data types. These are chosen in order to avoid redundancy and to enable the construction of heterogeneous structures of arbitrary complexity while being kept simple and explicit. However, applying the technique does not require using exactly the proposed type set, but only applying the symmetrical construction principle, both to data structures and to pattern structures.

[0112] The definition of matching operations is clearly related to the previous point. This covers three fundamental issues: identifying and testing the form of the structure, (step 200 in FIG. 2); check the value of the structure (or a part of it) (step 210); and extracting a part of the information stored in the structure (step 220).

[0113] The technique allows for building pattern matching structures in such a way that the required form is made explicit, thus clear to the programmer who specifies it. Parts of the subject structure which are not of interest are abstracted through a basic existence filter noted . Values inside the structure can be checked as equal to arbitrary complex values, thanks to a basic testing filter noted %, e being a potentially complex expression. Finally, information located in the subject structure can be extracted and copied to any variable that is available in the execution environment, by using a basic assignment filter, noted x, where x is a variable name.

[0114] The definition of a transformation model starts from the finding that matching operations are just intermediate steps of realistic structure transformation processes and should therefore be put in a more general context: How are transformation steps described, chained and ordered How are structures scanned and what is the result of the overall transformation How is the contextual information extracted that is used during matching operations: is it simply transferred into the output structure(s) and/or does it drive the transformation itself

[0115] The explicit invocation of the matching operation, according to the present technique, allows one to see it as a boolean evaluation. The transformation model can therefore take an useful form, depending on the requirements. The transformations can cascade matching operations by using rules like e₁ f₁e₂ f₂ . . . , or composed sequentially or through complex networks of if-then-else statements, possibly comprising computation on the extracted context. Output structures, the result of the transformation, are constructed in a coherent way by using the available data models and contextual information, possibly transformed by other computational means.

[0116] Depending on the definition of the transformation model and the expected computational power of the transformation, one can consider either to extend the pattern matching with language constructs to specify internal computation, as a complement to matching operations, or to integrate the transformation model itself into a more general programming language, as an extension which increases expressiveness. Both options are made possible by this technique, thanks to the explicit matching operator and the few general, and modular, assumptions regarding the underlying language.

[0117] Considering the above observations, it will be appreciated that the described pattern matching technique is in particular suited for supporting general typing mechanisms. Type checking improves the global reliability of programming languages, and provides efficient means for runtime optimization.

[0118] Pattern Matching Syntax and Operational Semantics

[0119] As noted above, the general abstract syntax uses e for expressions and f for pattern matching filters. Matching operations are invoked through the operator e f where the left expression e is the subject and the right operant f is the filter or pattern.

[0120] There are provided four basic operators plus combinators (or filter connectors):

	f ::	existence test
	x	existence test and assignment to variable x
	%e	equality check
	f	ref. filter
	f :: f f	filter composition
	f* f*	kleene-like operators
	f and f f or f not f	boolean (ordered) connectives

[0121] The first two filters and x are under normal form. The computation of the equality check and of reference filters requires the computation of their embedded expression: $\frac{e->e^{}}{\%e->\%e^{}}\left[f\text{-}\mathrm{anti}\right]\frac{f->f^{}}{@f->@f^{}}\left[f\text{-}@\right]$

[0122] By using a meta symbol , and, or, we define a family of equations valid for all binary combinators: $\frac{f_1->f_1^{}}{f_1{f}_2->f_1^{}{f}_2}\left[f\text{-}\mathrm{left}\right]\frac{f_2->f_2^{}}{f_1{f}_2->f_1{f}_2^{}}\left[f\text{-}\mathrm{right}\right]$

[0123] The reduction of the kleene-like operator f* preserves the structure of the filter (f*) whereas the computation of f is a pure rewriting (f): $\frac{f->f^{}}{f^{*}->f^{{}^{*}}}\left[f^{*}\right]$ ff f* f

[0124] In the semantics of filter application (matching), v and w denote reduced terms:

[0125] ti vtrue m-free2 , x^vwx, x^w true m-free1 $\frac{,r^{v}vf->{}^{},r^v\mathrm{true}}{,r^{v}r@f->{}^{},r^v\mathrm{true}}\left[m\text{-}@\right]\frac{v_1=v_2->\mathrm{true}\text{/}\mathrm{false}}{v_1\%v_2->\mathrm{true}\text{/}\mathrm{false}}\left[m\text{-}\%\right]$

[0126] It is to be mentioned that m-% uses again the structural equality on structures. This last is defined for all considered structures as s₁s₂ iff s₁s₂ and s₁s₂. The inclusion itself is straight forward for strings, sequences and multisets; for dictionaries, it corresponds to the inclusion of key/value pairs.

[0127] The composition combinator applies only to strings, sequences, multisets or dictionaries. It can be abstracted over these various data structures, noted s, by using the operator which models concatenation for strings and sequences, disjoint union for multisets and non-commutative union for records and dictionaries. As keys must be unique in a dictionary, the key/value pairs of the right hand operand override the key/value pairs of the left hand operand, if required. This does not impact the semantics of the related matching operation. The following definition uses again the equality relation on structures: $\begin{array}{c}\begin{array}{c}\left[m\text{-}\mathrm{fadd1}\right]\\ \frac{s_1,s_2|s_1+s_2=s{s}_1{f}_1->\mathrm{true}{s}_2{f}_2->\mathrm{true}}{s{f}_1++f_2->\mathrm{true}}\end{array}\\ s_1,s_2\mathrm{such}\mathrm{that}{s}_1+s_2=s\{\begin{array}{c}\frac{s_1{f}_1->\mathrm{false}}{s{f}_1++f_2->\mathrm{false}}\left[m\text{-}\mathrm{fadd2}\right]\\ \frac{s_1{f}_1->\mathrm{true}{s}_2{f}_2->\mathrm{false}}{s{f}_1++f_2->\mathrm{false}}\left[m\text{-}\mathrm{fadd3}\right]\end{array}\end{array}$

[0128] Similarly, in the following definition which applies only to strings, sequences, sets and dictionaries, the term is an abstraction over empty string, empty sequence, empty set and empty dictionary: $\begin{array}{c}\left[m^{*}\right]\\ \mathrm{if}s\frac{s_1,s_2|s=s_1+s_2{s}_{1}f->\mathrm{true}{s}_2{f}^{*}->\mathrm{true}}{s{f}^{*}->\mathrm{true}}\end{array}$ f*true m*b

[0129] In the following equation, the order of operands is important, and computing environments can be altered by unsuccessful operations. This enables more efficient implementations since no specific processing is required in order to preserve the original environment: $\begin{array}{c}\left[m\text{-}\mathrm{and1}\right]\\ \frac{v{f}_1->{}^{}\mathrm{true}{}^{}v{f}_2->{}^{}\mathrm{true}}{v{f}_1\mathrm{and}{f}_2->{}^{}\mathrm{true}}\end{array}$ $\begin{array}{c}\left[m\text{-}\mathrm{and2}\right]\\ \frac{v{f}_1->{}^{}\mathrm{false}}{v{f}_1\mathrm{and}{f}_2->{}^{}\mathrm{false}}\end{array}$ $\begin{array}{c}\left[m\text{-}\mathrm{and3}\right]\\ \frac{v{f}_1->{}^{}\mathrm{true}{}^{}v{f}_2->{}^{}\mathrm{false}}{v{f}_1\mathrm{and}{f}_2->{}^{}\mathrm{false}}\end{array}$

[0130] A similar behavior is defined for or filters: $\begin{array}{c}\left[m\text{-}\mathrm{or1}\right]\\ \frac{v{f}_1->{}^{}\mathrm{true}}{v{f}_1\mathrm{or}{f}_2->{}^{}\mathrm{true}}\end{array}$ $\begin{array}{c}\left[m\text{-}\mathrm{or2}\right]\\ \frac{v{f}_1->{}^{}\mathrm{false}{}^{}v{f}_2->{}^{}\mathrm{true}}{v{f}_1\mathrm{or}{f}_2->{}^{}\mathrm{true}}\end{array}$ $\begin{array}{c}\left[m\text{-}\mathrm{or3}\right]\\ \frac{v{f}_1->{}^{}\mathrm{false}{}^{}v{f}_2->{}^{}\mathrm{false}}{v{f}_1\mathrm{or}{f}_2->{}^{}\mathrm{false}}\end{array}$

[0131] Similarly, the not filter is: $\frac{v{f}_1->{}^{}\mathrm{true}\text{/}\mathrm{false}}{v\mathrm{not}{f}_1->{}^{}\mathrm{false}\text{/}\mathrm{true}}\left[m\text{-}\mathrm{not}\right]$

[0132] Of course, backtracking might be convenient in cases where the matching failed and still let the environment changed. Further, a solution by the mean of an explicit restoring operation v(f), may have semantics $\frac{f->f^{}}{V\left(f\right)->V\left(f^{}\right)}\left[f\text{-}V\right]$ $\frac{vf->{}^{}\mathrm{true}}{vV\left(f\right))->{}^{}\mathrm{true}}\left[m\text{-}\mathrm{Va}\right]$ $\frac{vf->{}^{}\mathrm{false}}{vV\left(f\right)->\mathrm{false}}\left[m\text{-}\mathrm{Vb}\right]$

[0133] The symmetrical construction of data and filter structures will now be described in more detail in the examples of tuples, sequences, multisets, dictionaries and records.

[0134] Tuples allow to handle heterogeneous structures of fixed length. The present description will present only constructors. Other common operations on tuple structures are: access to items through indexing notations such as ti, in order to fetch the i^th item, or assignment ti:e, in order to set the i^th item, provided that i corresponds to the actual size of the tuple. Tuples are particularly useful for computing Cartesian products such as done in relational algebra and related languages:

	e :: <e0, . . . , ek>	tuple construction (k 1 items)
	f :: <f0 , . . . , fk>	filter for tuples of k 1 elements

[0135] In the following part, {tilde over (e)} is noted as a shorthand for e₀, . . . , e_k. The computation of a tuple (or a filter tuple) is defined by the successive computation of all subterms, in the order of occurrence. A tuple or filter tuple is reduced when all subterms or sub-filters are reduced: $\frac{e_i->e_i^{}}{v_0,,e_i,,e_k->v_0,,e_i^{},,e_k}\left[\mathrm{tuple}\right]$ $\frac{f_i->f_i^{}}{f_0,,f_i,,f_k->f_0,,f_i^{},,f_k}\left[f\text{-}\mathrm{tuple}\right]$

[0136] Matching for tuples is performed by matching all subfilters to corresponding substructures. If the subject v is not a tuple, or if cardinalities are different, the operation returns false (m-tuple): $\frac{v_0{f}_0->{}^{\left(1\right)}\mathrm{true}{}^{\left(k-1\right)}{v}_k{f}_k->{}^{\left(k\right)}\mathrm{true}}{v_0,,v_k{f}_0,,f_k->{}^{\left(k\right)}\mathrm{true}}\left[m\text{-}\mathrm{tuple}\right]$

[0137] It is to be mentioned that potential modifications of execution context are propagated through the whole matching process.

[0138] Sequences, multisets and dictionaries are defined as shown below:

e :: e0, . . . , ek e0, . . . , ek	(ordered) sequences, multisets
:: e0 e0, . . . , ekek	dictionaries (unordered key/value
	pairs)
::	empty sequence, multiset, dictionary
f :: f0, . . . , fk f0, . . . , fk	sequence/multiset filters (k 0)
f0 f0, . . . , fk fk	dictionary filters (k 0)

[0139] $\begin{array}{cc}\mathrm{with}:\frac{e_i{e}_i^{}}{\left[v_0,,e_i,,e_k\right]\left[v_0,,e_i^{},,e_k\right]}& \left[\mathrm{seq}\right]\\ \frac{f_i{f}_i^{}}{\left[f_0,,f_i,,f_k\right]\left[f_0,,f_i^{},,f_k\right]}& \text{[f-seq]}\end{array}$

[0140] Equivalent equations for sets, called set, set2, f-set and f-set2, are strictly similar to the previous one: $\begin{array}{cc}\frac{e_i{e}_i^{}}{\left\{v_0,,e_i,,e_k\right\}\left\{v_0,,e_i^{},,e_k\right\}}& \left[\mathrm{set}\right]\\ \frac{f_i{f}_i^{}}{\left\{f_0,,f_i,,f_k\right\}\left\{f_0,,f_i^{},,f_k\right\}}& \text{[f-set]}\end{array}$

[0141] When matching sets S^0,k denotes the set v₀, . . . , v_k and SF^0,k the filter f₀, . . . , f_k. If i 0,k, the S^0,ki is equivalent to v₀, . . . , v_i1, v_i1^{, . . . v}_k, and similarly for SF^0,ki. Using this notation, the matching over sets is recursively defined as: $\begin{array}{cc}\frac{i,j\left[0,k\right]\mathrm{such}\mathrm{that}\{\begin{array}{cccc}& v_i\#f_j& & {}^{}\mathrm{true}\\ {}^{}& S^{\left[0,k\right]-\left\{i\right\}}\#{\mathrm{SF}}^{\left[0,k\right]-\left\{j\right\}}& & {}^{}\mathrm{true}\end{array}}{S^{\left[0,k\right]}\#{\mathrm{SF}}^{\left[0,k\right]}{}^{}\mathrm{true}}& \text{[m-set]}\end{array}$

[0142] This equation does not express any ordering in exploring the search space. It just requires that each value in S matches a corresponding filter in SF. Note that the computing environment can be modified throughout matching operations.

[0143] Dictionaries are computed by successive evaluation of key/value pairs, following the occurrence order: $\begin{array}{cc}\frac{e_i{e}_i^{}}{\left\{v_0=v_0^{},,e_i=e_i^{},,e_k=e_k^{}\right\}\left\{v_0=v_0^{},,e_i^{}=e_i^{},,e_k=e_k^{}\right\}}& \left[\mathrm{dic}\right]\\ \frac{e_i^{}{e}_i^{}}{\left\{v_0=v_0^{},,v_i=e_i^{},,e_k=e_k^{}\right\}\left\{v_0=v_0^{},,v_i=e_i^{},,e_k=e_k^{}\right\}}& \left[\mathrm{dic2}\right]\end{array}$

[0144] Computation of filter dictionaries is strictly similar, and therefore, corresponding equations f-dic, f-dic2 will not be shown here.

[0145] When matching dictionaries, D^0,k denotes the dictionary v₀V₀, . . . , v_kv_k and DF^0,k the filter f₀f₀, . . . , f_kf_k. Using this notation, the matching over dictionaries is recursively defined as: $\begin{array}{cc}\frac{i,j\left[0,k\right]\mathrm{such}\mathrm{that}\{\begin{array}{cccc}& v_i\#f_j& & {}^{}\mathrm{true}\\ {}^{}& v_i^{}\#f_j^{}& & {}^{}\mathrm{true}\\ {}^{}& D^{\left[0,k\right]-\left\{i\right\}}\#{\mathrm{DF}}^{\left[0,k\right]-\left\{j\right\}}& & {}^{}\mathrm{true}\end{array}}{D^{\left[0,k\right]}\#{\mathrm{DF}}^{\left[0,k\right]}{}^{}\mathrm{true}}& \text{[m-set]}\end{array}$

[0146] This equation does not express any ordering in exploring the search space. It just requires that each key/value pair matches a corresponding filter-key/filter-value pair. Note that the computing environment can be modified throughout matching operations, and that keys are matched prior to values.

[0147] Records are unordered and heterogeneous collection of values which can be statically designed by a name called member. Standard operations on such data structures s are member access, e.g. a:s.m, or member assignment, e.g. s.m:10. Record provide useful mnemonics in order to select a part of complex structures and are well known for their expressive richness in data modeling:

e :: <m0 e0, . . . , mk ek>	records (unordered member-name/
	value pairs) (k 0)
f :: <m0 f0, . . . , mk fk>	record filters (k )

[0148] The reduction of data and filters follows the order of occurrence: $\begin{array}{cc}\frac{e_i{e}_i^{}}{m_0=v_0,,m_i=e_i,,m_k=e_k{m}_0=v_0,,m_i=e_i^{},,m_k=e_k}& \text{[e-rec]}\end{array}$

[0149] Filter records are similarly defined: $\begin{array}{cc}\frac{f_i{f}_i^{}}{m_0=f_0,,m_i=f_i,,m_k=f_k{m}_0=f_0,,m_i=f_i^{},,m_k=f_k}& \text{[f-record]}\end{array}$

[0150] Matching is not order sensitive, i.e. the ordering of filters can differ from the ordering of the subject, but filters are applied through their definition order. Member names must be exactly identical: $\begin{array}{cc}\mathrm{distinct}{i}_0,,i_k\left[0k\right]\mathrm{such}\mathrm{that}\frac{\{\begin{array}{c}{m}_0^{}=m_{\mathrm{i0}}\mathrm{and}{v}_{\mathrm{i0}}\#f_0{}^{\left(1\right)}\mathrm{true}\\ \\ m_k^{}=m_{\mathrm{ik}}\mathrm{and}{}^{\left(k-1\right)}{v}_{\mathrm{ik}}\#f_k{}^{\left(k\right)}\mathrm{true}\end{array}}{m_0=v_0,,m_k=v_k\#m_0^{}=f_0,,m_k^{}=f_k{}^{\left(k\right)}\mathrm{true}}& \text{[m-record]}\end{array}$

[0151] The matching fails if one of the sub-matching fails: $\begin{array}{cc}\mathrm{distinct}{i}_0,,i_j\left[0k\right],jk\mathrm{such}\mathrm{that}\frac{\{\begin{array}{c}{m}_0^{}=m_{i_0}\mathrm{and}{v}_{i_0}\#f_0{}^{\left(1\right)}\mathrm{true}\\ \\ m_j^{}=m_{\mathrm{ij}}\mathrm{and}{}^{\left(j-1\right)}{v}_{\mathrm{ij}}\#f_j{}^{\left(j\right)}\mathrm{false}\end{array}}{m_0=v_0,,m_k=v_k\#m_0^{}=f_0,,m_k^{}=f_k{}^{\left(j\right)}\mathrm{false}}& \text{[m-record2]}\end{array}$

[0152] Recursive Filters

[0153] Recursive filters and the do operator will now be described in more detail.

[0154] Recursive filters enable the filtering of trees. For instance,

[0155] e rec F <%plus,F,F> or <%minus,F,F> or <>

[0156] succeeds if e is a tree with nodes that are labeled by strings plus and minus and with leaves that are tuples containing a unique element (of any structure).

[0157] The do operator takes a filter as the left argument and an instruction (or sequence of instructions) as the right argument. Instructions are executed only if the filter is successful.

[0158] For instance,

[0159] e <%12,> do i:i1

[0160] increments i only if e is a tuple made of two elements, the first one being 12. This operator can be particularly useful in recursive filters.

[0161] Using do it is possible, for instance, to extract all the numbers from a tree based on the previous example and append them to a list n. The extend filter A is

[0162] rec F <%plus,F,F> or <%minus,F,F>or (<x> do n:nx)

[0163] In a basic example such as

[0164] <plus,<minus,<1>,<2>>,<4>> A

[0165] where the tree encodes (12)4, the pattern matching operation succeeds and n is equal to n1,2,4 after application.

[0166] The syntax and operational semantics of recursive filters and the do operator is described hereafter:

	f :: rec F f F	recursive filter and recursion variable
	:: f do e	do operator (e is executed if f matches)

[0167] As described above, these filters are reduced by preserving their structures. Reduction of the f do e filter differs from reduction of the %e filter because the embedded expression e is not computed. It will be evaluated in the context of the matching operation, and only if required: $\begin{array}{cc}\frac{f{f}^{}}{\mathrm{rec}F=f\mathrm{rec}F=f^{}}& \text{[e-rec]}\\ \frac{f{f}^{}}{f\mathrm{do}e{f}^{}\mathrm{do}e}& \text{[e-do]}\end{array}$

[0168] Recursive matching is defined by using a substitution operation, as e.g. in the standard -reduction of the lambda calculus: $\begin{array}{cc}\frac{\#f\left[\mathrm{rec}F=f/F\right]->{}^{}\mathrm{true}/\mathrm{false}}{\#\mathrm{rec}F=f->{}^{}\mathrm{true}/\mathrm{false}}& \text{[m-rec]}\end{array}$

[0169] In the following m-do, the evaluation of e is done only if the filtering operation is successful. In that case, the new context is used, thus allowing side-effects such as the memorization of useful intermediate information: $\begin{array}{cc}\frac{\#f->{}^{}\mathrm{true}{}^{}e{}^{}w}{\#f\mathrm{do}e->{}^{}\mathrm{true}}& \text{[m-do]}\\ \frac{\#f->{}^{}\mathrm{false}}{\#f\mathrm{do}e->{}^{}\mathrm{false}}& \text{[m-do2]}\end{array}$

[0170] For illustration purposes, the technique described above is now applied to an example of managing a database of book references. The data is stored in a Set structure (unordered collection of records) describe hereafter.

	const DBase
	< year 1994 ,
	title TCP/IP illustrated ,
	author < last Stevens , first W. > ,
	publisher Addison-Wesley ,
	price 65.95
	>,
	< title Advanced Programming in the Unix environment ,
	year 1992 ,
	author < last Stevens , first W. > ,
	publisher Addison-Wesley ,
	price 65.95
	>,
	< year 2000 ,
	title Data on the Web ,
	author	< last Abiteboul , first Serge >,
		< last Buneman , first Peter >,
		< last Suciu , first Dan >
	,
	publisher Morgan Kaufmann Publishers ,
	price 39.95
	>,
	< year 1999 ,
	title The Economics of Technology and Content for Digital
	TV ,
	editor < last Gerbarg , first Darcy , affiliation
	CITI >,
	publisher Kluwer Academic Publishers ,
	price 129.95
	>

[0171] Now, examples are given to extract, filter and recombine book information by using the filtering primitives and structured pattern constructors.

[0172] For finding all books from Addison-Wesley that are published between 1990 and 1999, and for storing the titles, a recursive function F1 is defined that also returns the result in a Set:

	function F1 (x) is
	if
	x < titlet,publisher%Addison-
	Wesley,year%199>
	y
	then
	return F1 (y) < title t >
	else
	return

[0173] If F1 is called with the book database as parameter (written F1(DBase)) the result is

	< title Advanced Programming in the Unix environment
	>,
	< title TCP/IP Illustrated >

[0174] The rest of the information is not return in the result, e.g. the authors. The following variation, which is also perhaps more legible, does this job:

	function F1 (x) is
	if
	(xbook y) and
	(book< titlet,publisher%Addison-
	Wesley,year%199 >
	)
	then
	return F1 (y) book
	else
	return

[0175] This last example shows the interest of having explicit filtering operations: the first application is done on the parameter x and the second on book. The equivalent solution below shows the interest of the and filter; the previous and is just a standard boolean connective.

	function F1 (x) is
	if
	x
	(< titlet,publisher%Addison-Wesley,year%199
> )
	and book
	y
	then
	return F1 (y) book
	else
	return

[0176] For finding all publishers known in the database, the filters are used in another construction: the for loop, has for instance the following syntax

[0177] for f in e1 do e2

[0178] where f is a filter, e₁ a computable expression that evaluates to a Set or a Sequence, and e₂ any programming statement. This loop applies the filter to all elements of the Set/Sequence, and if successful, executes e₂. The solution is then:

[0179] for <publisherp> in DBase do (R : Rp)

[0180] This result of this program is found in R, with R being supposed to be empty before the execution: $\begin{array}{c}R={\{}^{}\mathrm{Addison}\text{-}{\mathrm{Wesly}}^{},\\ {}^{}\mathrm{Addison}\text{-}{\mathrm{Wesley}}^{},\\ {}^{}\mathrm{Morgan}\mathrm{Kaufmann}{\mathrm{Publishers}}^{},\\ {}^{}\mathrm{Kluwer}\mathrm{Academic}{\mathrm{Publishers}}^{}\}\end{array}$

[0181] As can be seen, the Addison-Wesley publisher is repeated twice. One might want to have a real projection of the information, and for that, can use an additional filter:

	for <publisherp> in DBase do (
	if not (R%p) then R:Rp
	)

[0182] The result of this program is as expected: $\begin{array}{c}R={\{}^{}\mathrm{Addison}\text{-}{\mathrm{Wesly}}^{},\\ {}^{}\mathrm{Morgan}\mathrm{Kaufmann}{\mathrm{Publishers}}^{},\\ {}^{}\mathrm{Kluwer}\mathrm{Academic}{\mathrm{Publishers}}^{}\}\end{array}$

[0183] For finding all authors having last names that begin with A, the following construction can be used:

	for <authors> in DBase do (
	for <last(%A and auth)> in s do (
	if not (R%auth) then R : Rauth
	)
	)

[0184] The result will be Abiteboul.

[0185] It will be appreciated that the presented pattern matching technique is suitable for either designing programming languages specialized in data structure transformation, or for facilitating extensions to existing languages in order to handle such transformations. The construction of data structures and pattern matching structures, i.e. filters, of arbitrary complexity uses symmetrical constructors, thereby allowing for great simplicity. The definition of basic filters, structured filters, recursive filters and logical combinators provides any programmer with means that have a high expressive power and a good level of simplicity. A boolean operator is provided that allows to apply explicitly the pattern matching structure to the subject data structure. Further, a formal operational semantics is provided that defines precisely the nature of filters and of the matching operation as well the relationship with any associated computational model. Moreover, this formalization offers a flexible theoretical framework that can help further integration.

[0186] Bi-Valuation of Programming Statements

[0187] By means of the statement evaluation section 140, the computer system can perform the process which is depicted in FIG. 4. A programming language statement may include a first and a second sub-statement. In step 400, the first sub-statement is evaluated, and depending on the decision in step 410, the second sub-statement may likewise be evaluated. As will be shown in more detail below, statements exist that require the evaluation of the second sub-statement whereas on the other hand, there are also statements where evaluation of the second sub-statement is not necessary in each case.

[0188] When the first and/or the second sub-statement is evaluated in steps 400, 420, an evaluation success result is determined if evaluation succeeds or a distinguished value is returned if evaluation fails. In the following discussion, the distinguished value is noted none.

[0189] The provision of a distinguished value in addition to the set of possible evaluation success results is called bi-valuation.

[0190] The bi-valuation technique makes it possible to combine imperative and declarative statements in one programming language. Basically, both imperative and declarative statements are evaluated when executed and either return unit, a value, or none. The value unit is returned for imperative statements since imperative statements always succeed. A value is returned for declarative statements which succeed, and for declarative statements which fail return none.

[0191] It is therefore possible to combine both kinds of statements using so-called imperative connectors, like Then or Else having semantics that are based on the evaluation of the statements. Conditional, imperative, declarative statements and imperative connectors are described below in more detail. Further, it is demonstrated how imperative connectors are used for mixing statements.

[0192] In the following, s and s_i denote one or more imperative and/or declarative statements combined together by imperative connectors. The term to evaluate s means to execute the statements in s and return the evaluation of the last statement in s. Expressions like e, e_i can be literal constants, variables or basic operations like ee, e*e.

[0193] Conditional statements, or if-then-else statements, are noted

[0194] if (e) then (s₁) else (s₂)

[0195] where e is a boolean expression. Depending on whether e evaluates to true or false, s₁ or s₂ is evaluated, respectively.

[0196] As opposed to declarative statements, imperative statements always succeed, and evaluate to unit. One example is the assignment, noted x:e, where x is a variable identifier and e is an expression. It assigns the result of the evaluation of e to x. Another example is the closure loop, noted *(s). The loop ends when s evaluates to none. Its semantics is as follows:

[0197] if (s none) then *(s) else unit

[0198] Declarative statements are based on conditions. The evaluation of a declarative statement returns a value if the condition(s) is verified, none otherwise. Examples are rules, ordered action systems and unordered action systems.

[0199] The Rule is noted es where condition e can be a boolean or a pattern-matching expression. If the condition is verified, meaning that the pattern has been matched or that the boolean expression has evaluated to true, then the rule fires, triggering the evaluation of the right hand side s. Rules are commonly used in rewriting systems by combining them in constructs similar to action systems.

[0200] The (ordered) action system is noted s₁, s₂, . . . , s_n and can be compared with a powerful Switch construct. It is used to combine rules. The action system evaluates them one by one until it finds one that does not evaluate to none, i.e. a rule which fires and that has right hand side statements that do not evaluate to none. The action system itself returns the result of the evaluation of these statements. Action systems can also contain an imperative statement in the last position which will act as a default case since imperative statements always return unit.

[0201] The unordered action system is noted s₁. s₂, . . . , s_n and does not guarantee the order of evaluation of rules, i.e. the order of declaration has not semantic meaning.

[0202] There are three imperative connectors which make it possible to mix imperative and declarative statements. An example set of imperative connectors is given below:

[0203] The sequencing operation ; separates instructions that are to be executed in sequence.

[0204] s₁; s₂; . . . ; s_n

[0205] will execute every statement and return the evaluation of s_n.

[0206] The concurrent operators {circumflex over ()} and ^v separate instructions that are to be executed concurrently. They have respectively logical-And-like and logical-Or-like semantics. Basically, s₁{circumflex over ()}s₂ concurrently evaluates s₁ and s₂ returning none when at least one of the s_i evaluates to none; unit is returned when both operands evaluate to unit. s₁^vs₂ behaves in the following way: it concurrently evaluates s₁ and s₂ returning unit when at least one of the s_i evaluates to unit, and returning none when both operands evaluate to none. A more formal definition of the semantics is provided below.

[0207] The Else operator s₁ Else s₂ evaluates s₁. If s₁ evaluates to none, then s₂ is evaluated. The semantics is as follows:

[0208] var v_fs₁. if (v_f none) then s₂ else v_f.

[0209] v_f is a fresh variable, i.e. doesn't occur in the current context nor in s₁ or s₂.

[0210] The Then operator s₁ Then s₂ evaluates s₁. If s₁ does not evaluate to none, then s₂ is evaluated. The semantics is as follows:

[0211] if (s₁ none) then (s₂) else (none)

[0212] The Or operator s₁ Or s₂ evaluates s₁. If s₁ evaluates to none, the evaluation of s₂ is returned. If s₁ evaluates to unit, s₂ is still evaluated but unit is returned, no matter what s₂ evaluates to. The semantics is as follows:

[0213] if (s₁ none) then (s₂) else(s₂ ; unit).

[0214] The And operator s₁ And s₂ evaluates s₁. If s₁ evaluates to unit, the evaluation of s₂ is returned. If s₁ evaluates to none, s₂ is still evaluated but none is returned. The semantics is as follows:

[0215] if (s₁ none) then (s₂) else (s₂ ; none)

[0216] For operators And, Or, {circumflex over ()} and ^v, s₁ and s₂ have to evaluate to unit or none, meaning that they have to be imperative expressions.

[0217] Some introductory examples are now provided to get familiar with the notation, and then the expressiveness brought by the technique is demonstrated with more complex examples. Variables i and str1 are supposed to be respectively of type Integer and String.

[0218] A simple sequence is made of an assignment statement followed by a while loop containing a rule: this loop increments i until it reaches 10.

[0219] i :0;

[0220] *((i<10)i:i1)

[0221] In an action system containing a rule, a nested action system is made of rules and an expression:

	(i<5) (str1 : cir ; 50 3),
	(i 100) abcd,
	(i 200) efgh
	,
	3.1415

[0222] If i is less than 5, then str1 is assigned the string cir and the action system returns 47. Otherwise, if i is equal to 100 or 200, the action system respectively returns strings abcd or efgh. Otherwise, it returns 3.1415. This last expression acts as a default case. If it were not present and if i was not previously assigned a value less than 5 or equal to 100 or 200, the action system would have evaluated to none.

[0223] It is to be noted that action system components do not necessarily contain a simple expression like 50-3 or abcd as the last statement. They can also contain imperative expressions; in that case, the action system returns unit.

[0224] The next more complex example makes use of the pattern matching features described above. Basically, pattern matching expressions can be used in place of boolean expressions in the left hand side of rules; they enable the recognition of patterns and extraction of parts of a data structure. A pattern matching expression evaluates to true if the subject matches and then the rule fires. Otherwise it evaluates to false and the rule returns none.

[0225] In the present example, some structural data about an XML file stored in string variable str1 are extracted. To this end, several numeric variables are created which will count:

	nb_open_tags	number of openings tags
	nb_close_tags	number of closing tags
	nb_emp-tags	number of empty tags

[0226] The first three tests identify the kind of the encountered tag, i.e. opening, closing, empty, whereas the three others examine the content of that tag. Since the last three tests are independent from the previous ones and relevant only when a new tag is encountered, the tests are split in two action systems combined with a Then connector. The variable tags is a string in which each tag name is appended. When all tags have been tested, the results are displayed.

*(
	str1 % < str2 % > strl
	nb_open_tags : nb_open_tags1,
	str1 % </ str2 % > strl1
	nb_close_tags : nb_close_tags1,
	str1 % < str2 %/> strl1
	nb_emp_tags : nb_emp_tags1
	Then
	str2 % :
	nb_prefixed_tags : nb_prefixed_tags 1,
	str2 % xrce nb_xrce_tags : nb_xrce_tags 1,
	nb_other_tags : nb_other_tags 1
	Then
	(tags : tags str2)
);
print(number of opening tags : str(nb_open_tags) ) ;
print(number of closing tags : str(nb_close_tags) ) ;
print(number of empty tags : str(nb_emp_tags) ) ;
print(number of prefixed tags : str(nb_prefixed_tags) ) ;
print(number of xrce tags : str(nb_xrce_tags) ) ;
print(number of other tags : str(nb_other_tags) ) ;
print (tags)

[0227] In this example, an action system, i.e. a declarative statement, is composed with another action system and with an assignment instruction, i.e. an imperative statement, and the three are nested in a closure loop, i.e. an imperative statement.

[0228] To illustrate the compactness and readability provided by the proposed technique, the same program coded with only the closure loop and if-then-else statements is presented in the following. For simplicity purposes, it is assumed that boolean expressions in conditions can be replaced by pattern-matching expressions.

*(
	if (length(str1) > 0) then (
	if (str1 % < str2 % > str1) then (
	if (str2 % : ) then (
	nb_open_tags : nb_open_tags 1;
	nb_prefixed_tags : nb_prefixed_tags 1
	)
	else (
	if (str2 % xrce ) then (
	nb_open_tags : nb_open_tags 1;
	nb_xrce_tags : nb_xrce_tags 1
	)
	else (
	nb_open_tags : nb_open_tags 1;
	nb_other_tags : nb_other_tags 1
	)
	);
	tags : tags str2
	)
	else (
	if (str1 % </ str2 % > str1) then
(
	if (str2 % : ) then (
	nb_close_tags : nb_close_tags 1;
	nb_prefixed_tags : nb_prefixed_tags 1
	)
	else (
	if (str2 % xrce ) then (
	nb_close_tags : nb_close tags 1;
	nb_xrce_tags : nb_xrce_tags 1
	)
	else (
	nb_close_tags : nb_close tags 1;
	nb_other_tags : nb_other tags 1
	)
	);
	tags : tags str2
	)
	else (
	if ( % < str2 % /> str1) then (
	if (str2 % : ) then (
	nb_emp_tags : nb_emp_tags 1;
	nb_prefixed_tags : nb_prefixed_tags 1
	)
	else (
	if (str2 % xrce ) then (
	nb_emp_tags : nb_emp_tags 1;
	nb_xrce_tags : nb_xrce_tags 1
	)
	else (
	nb_emp_tags : nb_emp_tags 1;
	nb_other_tags : nb_other_tags 1
	)
	);
	tags : tags str2
	)
	)
	)
	)
	else (none)
);
print(number of opening_tags : str(nb_open_tags) ) ;
print(number of closing_tags : str(nb_close tags) ) ;
print(number of empty_tags : str(nb_emp_tags) ) ;
print(number of prefixed_tags : str(nb_prefixed_tags) ) ;
print(number of xrce_tags : str(nb_xrce_tags) ) ;
print(number of other_tags : str(nb_other_tags) ) ;
print (tags)

[0229] Using if-then-else statements, 339 cases have to be dealt with separately. In languages like C or Java, a Switch construct would have produced a more readable solution, but unfortunately, Switch can only be used with primary integer and char/byte types, and test values have to be statically defined.

[0230] It will be appreciated that the above described technique enables the definition of basic imperative and declarative statements, the combination of both kinds of operations in meaningful and arbitrarily complex control structures, and the typing of such constructions.

[0231] Main applications are language construction, refining control abstractions and programmation models toward more computational expressiveness, rewriting systems theory, classical conditional term rewriting systems, explicit strategies, transformation languages and systems for compilation and language processing, structured document transformation, tree pattern matching, event management, explicit or automatic document transformation systems, and others.

[0232] The technique provides material for building languages which are less abstract and more general than rewriting systems but abstract enough to simplify and extend general purpose programming languages. The technique could hence play an important role in the design of new transformation techniques or the extension of existing ones. Further, the technique may solve data and document interchange problems which will lead to low costs and high quality solutions.

[0233] Moreover, the technique provides a programming language framework through the definition of generic syntactic constructs, their operational semantics and the associated formal type system. The important difficulties which underlie the ambitious goal of bringing together declarative and imperative programming styles are solved all together, in a conceptually new and generic approach.

[0234] It will be appreciated that the bi-valuation technique enables and complements the symmetrical structural pattern matching technique that has been described above. The technique is further particular suited for supporting general typing mechanisms. Type checking improves the global reliability of programming languages, and provides efficient means for optimization, thus, increasing the runtime performance. This will be described in more detail below.

[0235] By means of bi-valuation of programming statements, and by using imperative connectors, the design of new programming languages or the extension of existing programming languages are rendered possible with constructions that provide innovative expressiveness. Resulting languages are located at an intermediate level of abstraction, between declarative, functional and imperative languages. This kind of abstraction could play an important role in general structure transformation technologies. When building modern programming languages with adequate control abstraction level, the major difficulty is to find the best trade-off between simplicity and power, conciseness and expressiveness. It will therefore be appreciated that the present technique provides programming constructs in such a way that the constructs offer compositional capabilities. Further, connectors are provided that allow the composition of such constructs with relevant semantics. Further, a sound type system is provided which enables to check out composition errors.

[0236] Statement Evaluation Syntax and Operational Semantics

[0237] In the following, the syntax and semantics of the basic conditional and imperative constructs are described in more detail.

[0238] In the testing operation, environment modifications are allowed during evaluation of the boolean part. This enables the use of complex operations with effects on context as a condition for the test: $\begin{array}{cc}\frac{e_1->{}^{}{e}_1^{}}{\mathrm{if}{e}_1\mathrm{then}{e}_2\mathrm{else}{e}_3->{}^{}\mathrm{if}{e}_1^{}\mathrm{then}{e}_2\mathrm{else}{e}_3}& \left[\mathrm{if}\right]\end{array}$

[0239] if true then e₂ else e₃e₂ if1

[0240] if false then e₂ else e₃ e₃ if2

[0241] The sequencing operator is defined as shown hereafter: $\begin{array}{cc}\frac{e_1->{}^{}{e}_1^{}}{e_1;e_2->{}^{}{e}_1^{};e_2}& \left[\mathrm{seq1}\right]\\ \frac{e_1->}{e_1;e_2->e_2}& \left[\mathrm{seq2}\right]\end{array}$

[0242] Further, an assignment operator is provided that finally reduces to a distinguished value unit. In the prior art, similar mechanisms for typing imperative statements exist for integrating imperative extensions in the functional framework. However, the result of such an assignment operation x :e is just the result of the expression e itself: $\begin{array}{cc}\frac{e->{}^{}{e}^{}}{x:=e->{}^{}x:=e^{}}& \left[\mathrm{assg1}\right]\\ \frac{e->}{,x^{w}x:=e->,x^{}\mathrm{unit}}& \left[\mathrm{assg2}\right]\end{array}$

[0243] Moreover, an iterative construct is provided in the form of a closure operator, noted *(e), which allows to iterate an action e until it evaluates to a distinguished value none. Thus, this action must be a declarative action, since applying the closure to a pure imperative action like an assignment would produce endless computation. This property is checked by the typing rules described below. The semantics of the * operator is simply defined by

[0244] *(e)if (enone) then *(e) else unit *

[0245] A closure operation, returning always the distinguished value unit, is considered as a pure imperative (mono-valuated) statement.

[0246] Declarative constructs are defined as shown hereafter:

	e :: e e	rule
	e1, . . . , en	action system (ordered)
	e1, . . . , en	action system (unordered)

[0247] As mentioned above, the first universal construction is the rule, constituted of a left-hand condition and a right-hand action executed only if the condition can be verified. The proposed syntax, and the following type system, allows the cascading of rules of the form e₁e₂e₃, understood as e₁(e₂e₃):

[0248] e₁e₂ 43 if (e₁true) then e₂ else none Rule

[0249] In declarative environments, such rules are usually assembled in larger systems. The rule choice and application strategy varies a lot depending on the various proposed models, but most common tactics are try rules in order or try and rule and iterate application as much as possible. These standard models, and more sophisticated tactics as well, are easily emulated by combining the action systems with the closure operator presented above.

[0250] The ordered action system consists of a succession of actions to be tried in the given order. In the following definition, v_f denotes a fresh variable, i.e with a unique name: $\begin{array}{cc}n2\left[\left|e_1,,e_n\right|\right]->[\begin{array}{c}\mathrm{var}{}_f=e_1.\\ \mathrm{if}\left({}_f\mathrm{none}\right)\\ \mathrm{then}\left[\left|e_2,....,e_n\right|\right]\\ \mathrm{else}{}_f\end{array}& \left[\mathrm{Asysl}\right]\\ \left[\left|e\right|\right]->e& \left[\mathrm{Asys2}\right]\end{array}$

[0251] It is to be noted that this definiton implies that all actions e₁, . . . , e_n1 must be bi-valuated, i.e possibly evaluated to none; the last one can be mono-valuated. In that case the whole system becomes itself mono-valuated, since a value different from none will be finally returned.

[0252] Unordered action systems as similarly defined, up to the (random) choice of action e_i among all others: $\begin{array}{cc}n2,i\left\{1,,n\right\}\left\{|e_1,,e_n|\right\}->[\begin{array}{c}\mathrm{var}{}_f=e_i,\\ \mathrm{if}\left({}_f\mathrm{none}\right)\\ \mathrm{then}\left\{|e_1,,e_{i-1},e_{i+1},,e_n|\right\}\\ \mathrm{else}{}_f\end{array}& \left[\mathrm{Asysb1}\right]\\ \left\{\left|e\right|\right\}->e& \left[\mathrm{Asysb2}\right]\end{array}$

[0253] One consequence of this definition is that here, all actions must be bi-valuated, in the sense given above. Such an action system specifies permutation equivalent rule collection, allowing great perspectives for run-time optimizations. Moreover, it allows the programmer to avoid producing over-specified code, making the code more understandable to external code reviewers or maintainers.

[0254] Turning now to the imperative connectors, the terminology imperative connector has been chosen by reference to the so-called Boolean connectors, in order to underline the similarity. In both cases, values are bi-valuated. But the present, more subtle approach to the valuation domains allows to handle several data sets.

[0255] The first useful connector is the well-known sequence ; already presented.

[0256] Another useful set of operators are concurrent execution (binary) operators, noted e₁ {circumflex over ()} e₂ and e₁ ^v e₂, and also e₁ * e₂ for both, equipped with the following interleaving semantics: $\begin{array}{cc}\frac{e_1->{}^{}{e}_1^{}}{e_1{{}^{*}{e}_2->{}^{}{e}_1^{}}^{*}{e}_2}& \left[\mathrm{ParL}\right]\\ \frac{e_2->{}^{}{e}_2^{}}{e_1{{}^{*}{e}_2->{}^{}{e}_1}^{*}{e}_2^{}}& \left[\mathrm{ParR}\right]\end{array}$

[0257] The first operator {circumflex over ()}, besides its concurrent semantics, behaves like a logical and, commonly noted {circumflex over ()}, with respect to the management of the bi-valuation:

[0258] unit {circumflex over ()}e₂e₂ Par1a none {circumflex over ()}e₂e₂; none Par1b

[0259] e₁ {circumflex over ()}nonee₁; none Par1c e₁ {circumflex over ()}unite₁ Par1d

[0260] With ^v, the equivalent or behavior, commonly noted v, is expressed:

[0261] unit ^ve₂e₂; unit Par2a none ^ve₂e₂ Par2b

[0262] e₁ ^vnonee₁ Par2c e₁ ^vunite₁; unit Par2d

[0263] This definition describes a simple concurrent semantics, that can be extended to a wider scope. The important point here is to define this concurrent composition as a bi-valuated statement consistent with the global framework.

[0264] Besides concurrent imperative connectors, there are also provided sequential imperative connectors, Then, Else, And and Or.

[0265] Then and Else are defined by:

[0266] e₁ Then e₂ if (e₁none) then e₂ else none Then

[0267] e₁ Else e₂ var v_fe₁. if (v_fnone) then e₂ else v_f Else

[0268] v_f is a fresh variable, i.e. doesn't occur in the current context nor in e₁ or e₂. The following And and Or operators are quite similar, except that they impose the evaluation of both operands whatever result is computed. Another difference is that values computed by both operands can only range over unit, none. In that sense, e₁ and e₂ are imperative expressions:

[0269] e₁ And e₂ if (e₁none) then e₂ else (e₂; none) And

[0270] e₁ Or e₂ if (e₁none) then e₂ else (e₂; unit) Or

[0271] Type System

[0272] A minimal, formal, type system that illustrates and enables the use of the constructs is described hereafter. This type system is based on a subtyping relation using inclusion polymorphism brought by a non-discriminating union type constructor. An example of a type hierarchy is shown in FIG. 5.

[0273] The type system introduces other relevant tools, when required, to known techniques for building type systems. It is described through typing judgments logically organized by means of typing equations.

[0274] The typing judgement e : t states that expression e has type t in the typing context . The notion of tt asserts that type t is a sub-type of type t and a more original judgement e :: t, says that the expression e has minimal type t. Formally, this last relation is characterized by the following property:

[0275] (Minimal type) e :: t ifft, e : ttt

[0276] This notion of minimal type is useful in order to avoid overgeneralized inferences due to the so called subsumption rule (see Sub), while preserving the convenience of subtyping mechanism. The notation , x : t . . . expresses that the fact that x is a variable of type t is registered in the typing context. More precisely, , x : t is a typing context such that x:t. The following rules define the reflexive and transitive subtyping relation , for all well-formed types t, t, t₁, t₂, t₃ and the typing context . $\begin{array}{c}tt\left[\mathrm{Reff}\right]\\ \frac{t_1{t}_2{t}_2{t}_3}{t_1{t}_3}\left[\mathrm{Trans}\right]\\ \frac{et}{et}\left[\mathrm{Min}\right]\\ \frac{ett{t}^{}}{e{t}^{}}\left[\mathrm{Sub}\right]\\ \frac{t_{1}t{t}_{2}t}{t_1{t}_{2}t}\left[\mathrm{U1}\right]\\ \frac{t{t}_1}{t{t}_1{t}_2}\left[\mathrm{U2}\right]\\ \frac{t{t}_2}{t{t}_1{t}_2}\left[\mathrm{U3}\right]\end{array}$

[0277] Rules for defining well-formedness conditions on types will not be shown, and types are supposed well formed in all following definitions. An important type constructor is the non-discriminating union which provides inclusion polymorphism.

[0278] Basic typing of literals and variables is defined by the following axioms:

[0279] n :: int Num s :: string Str ,x:tx :: t Var

[0280] true :: bool Bool1 false :: bool Bool2

[0281] none :: None None unit :: Unit Unit

[0282] Operators are typed by: $\begin{array}{c}*\left\{+,*,-,/\right\}\\ \frac{e_1\mathrm{int}{e}_2\mathrm{int}}{e_1*e_2\mathrm{int}}\left[\mathrm{Add}1\right]\\ \frac{e_1\mathrm{string}{e}_2\mathrm{string}}{e_1+e_2\mathrm{string}}\left[\mathrm{Add}2\right]\\ \frac{e_{1}t{e}_2{t}^{}}{e_1==e_2\mathrm{bool}}\left[\mathrm{Eq}\right]\\ \frac{e_{1}t{e}_2{t}^{}}{e_1!=e_2\mathrm{bool}}\left[\mathrm{Neq}\right]\\ \frac{e\mathrm{bool}{e}_1{t}_1{e}_2{t}_2}{\mathrm{if}e\mathrm{then}{e}_1\mathrm{else}{e}_2{t}_1{t}_2}\left[\mathrm{If}\right]\\ \frac{x\mathrm{dom}\left(\right)e_{1}t,e_1:t{e}_2{t}_2}{\mathrm{var}x=e_1{e}_2{t}_2}\left[\mathrm{Var}\right]\end{array}$