Type-3 grammars generate the regular languages. Such a grammar restricts its rules to a single nonterminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed by a single nonterminal (right regular). Alternatively, the right-hand side of the grammar can consist of a single terminal, possibly preceded by a single nonterminal (left regular). These generate the same languages.
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A context-free grammar for the language consisting of all strings over {a,b} containing an unequal number of a's and b's: : : : : Here, the nonterminal T can generate all strings with more a's than b's, the nonterminal U generates all strings with more b's than a's and the nonterminal V generates all strings with an equal number of a's and b's. Omitting the third alternative in the rules for T and U doesn't restrict the grammar's language.
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Note that this algorithm is highly sensitive to the nonterminal ordering; optimizations often focus on choosing this ordering well.
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In computer science, terminal and nonterminal symbols are the lexical elements used in specifying the production rules constituting a formal grammar. Terminal symbols are the elementary symbols of the language defined by a formal grammar. Nonterminal symbols (or syntactic variables) are replaced by groups of terminal symbols according to the production rules. The terminals and nonterminals of a particular grammar are two disjoint sets.
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What follows is an implementation of a recursive descent parser for the above language in C. The parser reads in source code, and exits with an error message if the code fails to parse, exiting silently if the code parses correctly. Notice how closely the predictive parser below mirrors the grammar above. There is a procedure for each nonterminal in the grammar. Parsing descends in a top-down manner, until the final nonterminal has been processed.
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A grammar is left-recursive if and only if there exists a nonterminal symbol A that can derive to a sentential form with itself as the leftmost symbol. Notes on Formal Language Theory and Parsing, James Power, Department of Computer Science National University of Ireland, Maynooth Maynooth, Co. Kildare, Ireland.JPR02 Symbolically, : A \Rightarrow^+ A\alpha, where \Rightarrow^+ indicates the operation of making one or more substitutions, and \alpha is any sequence of terminal and nonterminal symbols.
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Every context-free grammar can be transformed into an equivalent nondeterministic pushdown automaton. The derivation process of the grammar is simulated in a leftmost way. Where the grammar rewrites a nonterminal, the PDA takes the topmost nonterminal from its stack and replaces it by the right-hand part of a grammatical rule (expand). Where the grammar generates a terminal symbol, the PDA reads a symbol from input when it is the topmost symbol on the stack (match).
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To eliminate each rule :A → X1 ... a ... Xn with a terminal symbol a being not the only symbol on the right-hand side, introduce, for every such terminal, a new nonterminal symbol Na, and a new rule :Na → a. Change every rule :A → X1 ... a ... Xn to :A → X1 ... Na ... Xn. If several terminal symbols occur on the right-hand side, simultaneously replace each of them by its associated nonterminal symbol. This does not change the grammar's produced language.
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"I think you could learn all you need to know in a month," he said. "Orson Welles said four hours. But he was being outrageous".Author of 'Terminal Man' Building Nonterminal Career: CRICHTON GELMIS, JOSEPH.
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A further extension of conjunctive grammars known as Boolean grammars additionally allows explicit negation. The rules of a conjunctive grammar are of the form :A \to \alpha_1 \And \ldots \And \alpha_m where A is a nonterminal and \alpha_1, ..., \alpha_m are strings formed of symbols in \Sigma and V (finite sets of terminal and nonterminal symbols respectively). Informally, such a rule asserts that every string w over \Sigma that satisfies each of the syntactical conditions represented by \alpha_1, ..., \alpha_m therefore satisfies the condition defined by A.
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Informally, Kayne's theory states that if a nonterminal category A asymmetrically c-commands another nonterminal category B, all the terminal nodes dominated by A must precede all of the terminal nodes dominated by B (this statement is commonly referred to as the "Linear Correspondence Axiom" or LCA). Moreover, this principle must suffice to establish a complete and consistent ordering of all terminal nodes -- if it cannot consistently order all of the terminal nodes in a tree, the tree is illicit. Consider the following tree: center (S and S' may either be simplex structures like BP, or complex structures with specifiers and complements like CP.) In this tree, the set of pairs of nonterminal categories such that the first member of the pair asymmetrically c-commands the second member is as follows: {, , }. This gives rise to the total ordering: .
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In computer science, a linear grammar is a context-free grammar that has at most one nonterminal in the right hand side of each of its productions. A linear language is a language generated by some linear grammar.
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Replace each rule :A → X1 X2 ... Xn with more than 2 nonterminals X1,...,Xn by rules :A → X1 A1, :A1 → X2 A2, :... , :An-2 → Xn-1 Xn, where Ai are new nonterminal symbols. Again, this does not change the grammar's produced language.
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Again from our example, hit is a child node of V. A nonterminal function is a function (node) which is either a root or a branch in that tree whereas a terminal function is a function (node) in a parse tree which is a leaf.
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The problem remains undecidable even if the language is produced by a "linear" context-free grammar (i.e., with at most one nonterminal in each rule's right-hand side, cf. Exercise 4.20, p. 105). nor whether it is an LL(k) language for a given k.
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Terminal and nonterminal symbols and production rules are defined in an object oriented flavor of the EBNF using operator overloading. The framework allows for the generation of an abstract syntax tree which can be traversed using the visitor pattern or evaluated using an interpreter.
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On the other hand, `ר` has two rules that can change it, thus it is nonterminal. A formal language defined or generated by a particular grammar is the set of strings that can be produced by the grammar and that consist only of terminal symbols.
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Every regular grammar is context-free, but not all context-free grammars are regular. The following context-free grammar, however, is also regular. : : : The terminals here are and , while the only nonterminal is . The language described is all nonempty strings of as and bs that end in a.
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By establishing a topological ordering on nonterminals, the above process can be extended to also eliminate indirect left recursion : :Inputs A grammar: a set of nonterminals A_1,\ldots,A_n and their productions :Output A modified grammar generating the same language but without left recursion :# For each nonterminal A_i: :## Repeat until an iteration leaves the grammar unchanged: :### For each rule A_i\rightarrow\alpha_i, \alpha_i being a sequence of terminals and nonterminals: :#### If \alpha_i begins with a nonterminal A_j and j: :##### Let \beta_i be \alpha_i without its leading A_j. :##### Remove the rule A_i\rightarrow\alpha_i. :##### For each rule A_j\rightarrow\alpha_j: :###### Add the rule A_i\rightarrow\alpha_j\beta_i. :## Remove direct left recursion for A_i as described above.
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Terminal affixes, when added to verb or noun themes, can complete words, while nonterminal affixes require additional affixation. The noun form rakhóhwalił, meaning 'he/she laughs at me', contains two inflectional affixes that modify the verb form rakhohw- shown above: -al is the nonterminal suffix that encodes a first person object, and -ił is the terminal suffix for a third person subject. Syntactic affixes, many of which are prefixes, also known as preverbs, are affixed to verb themes and often convey aspectual information. For example, in the phrase łekowa khúhnad, meaning 'finally it starts to get dark', the verb theme khuhn-, 'to get dark', is modified by two syntactic suffixes, łe- and kowa-.
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The algorithm in pseudocode is as follows: let the input be a string I consisting of n characters: a1 ... an. let the grammar contain r nonterminal symbols R1 ... Rr, with start symbol R1. let P[n,n,r] be an array of booleans. Initialize all elements of P to false.
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Proceedings of the Fourth European. IEEE, 2000. Some of the widely used formal metalanguages for computer languages are Backus–Naur form (BNF), extended Backus–Naur form (EBNF), Wirth syntax notation (WSN), and augmented Backus–Naur form (ABNF). These metalanguages have their own metasyntax each composed of terminal symbols, nonterminal symbols, and metasymbols.
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A unit rule is a rule of the form :A → B, where A, B are nonterminal symbols. To remove it, for each rule :B → X1 ... Xn, where X1 ... Xn is a string of nonterminals and terminals, add rule :A → X1 ... Xn unless this is a unit rule which has already been (or is being) removed.
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The Belgian parliament legalised euthanasia on 28 May 2002. A survey published in 2010 reported that those who died from euthanasia (compared with other deaths) were more often younger, male, cancer patients and more often died in their homes. In almost all cases, unbearable physical suffering were reported. Euthanasia for nonterminal patients was rare.
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Consider the expression `5^4^3^2`, in which `^` is taken to be a right-associative exponentiation operator. A parser reading the tokens from left to right would apply the associativity rule to a branch, because of the right-associativity of `^`, in the following way: # Term `5` is read. # Nonterminal `^` is read. Node: "`5^`".
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No matter which symbols surround it, the single nonterminal on the left hand side can always be replaced by the right hand side. This is what distinguishes it from a context-sensitive grammar. A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements.
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Instrumental affixes convey that action is performed using a device of some kind. Instrumental and benefactive affixes directly encode for the subject of the verb and thus do not appear with inflectional affixes for subject. Therefore, the most inflectional affixes a verb can possibly take is three. Inflectional affixes can be either terminal or nonterminal in nature.
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An item with a dot before a nonterminal, such as E → E + B, indicates that the parser expects to parse the nonterminal B next. To ensure the item set contains all possible rules the parser may be in the midst of parsing, it must include all items describing how B itself will be parsed. This means that if there are rules such as B → 1 and B → 0 then the item set must also include the items B → 1 and B → 0\. In general this can be formulated as follows: : If there is an item of the form A → v Bw in an item set and in the grammar there is a rule of the form B → w' then the item B → w' should also be in the item set.
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Context-free grammars are those grammars in which the left-hand side of each production rule consists of only a single nonterminal symbol. This restriction is non-trivial; not all languages can be generated by context-free grammars. Those that can are called context-free languages. These are exactly the languages that can be recognized by a non-deterministic push down automaton.
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In formal language theory, a context-free grammar is in Greibach normal form (GNF) if the right-hand sides of all production rules start with a terminal symbol, optionally followed by some variables. A non-strict form allows one exception to this format restriction for allowing the empty word (epsilon, ε) to be a member of the described language. The normal form was established by Sheila Greibach and it bears her name. More precisely, a context-free grammar is in Greibach normal form, if all production rules are of the form: :A \to a A_1 A_2 \cdots A_n or :S \to \varepsilon where A is a nonterminal symbol, a is a terminal symbol, A_1 A_2 \ldots A_n is a (possibly empty) sequence of nonterminal symbols not including the start symbol, S is the start symbol, and ε is the empty word.
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The canonical example of a context-free grammar is parenthesis matching, which is representative of the general case. There are two terminal symbols "(" and ")" and one nonterminal symbol S. The production rules are :, :, : The first rule allows the S symbol to multiply; the second rule allows the S symbol to become enclosed by matching parentheses; and the third rule terminates the recursion., Exercise 4.1b, p. 103.
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Conjunctive grammars are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with a conjunction operation. Besides explicit conjunction, conjunctive grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction can be used, in particular, to specify intersection of languages.
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A terminal symbol, such as a word or a token, is a stand-alone structure in a language being defined. A nonterminal symbol represents a syntactic category, which defines one or more valid phrasal or sentence structure consisted of an n-element subset. Metasymbols provide syntactic information for denotational purposes in a given metasyntax. Terminals, nonterminals, and metasymbols do not apply across all metalanguages.
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Type-1 grammars generate context-sensitive languages. These grammars have rules of the form \alpha A\beta \rightarrow \alpha\gamma\beta with A a nonterminal and \alpha, \beta and \gamma strings of terminals and/or nonterminals. The strings \alpha and \beta may be empty, but \gamma must be nonempty. The rule S \rightarrow \epsilon is allowed if S does not appear on the right side of any rule.
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Both examples above can be solved by letting the parser use the follow set (see LL parser) of a nonterminal A to decide if it is going to use one of As rules for a reduction; it will only use the rule A → w for a reduction if the next symbol on the input stream is in the follow set of A. This solution results in so-called Simple LR parsers.
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Straight-line grammar (with start symbol ß) for the second sentence of the United States Declaration of Independence. Each blue character denotes a nonterminal symbol; they were obtained from a gzip-compression of the sentence. Grammar-based codes or Grammar-based compression are compression algorithms based on the idea of constructing a context-free grammar (CFG) for the string to be compressed. Examples include universal lossless data compression algorithms.
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In computer programming, the interpreter pattern is a design pattern that specifies how to evaluate sentences in a language. The basic idea is to have a class for each symbol (terminal or nonterminal) in a specialized computer language. The syntax tree of a sentence in the language is an instance of the composite pattern and is used to evaluate (interpret) the sentence for a client. See also Composite pattern.
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Nonterminal symbols are those symbols which can be replaced. They may also be called simply syntactic variables. A formal grammar includes a start symbol, a designated member of the set of nonterminals from which all the strings in the language may be derived by successive applications of the production rules. In fact, the language defined by a grammar is precisely the set of terminal strings that can be so derived.
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A reserved word is one that "looks like" a normal word, but is not allowed to be used as a normal word. Formally this means that it satisfies the usual lexical syntax (syntax of words) of identifiers – for example, being a sequence of letters – but cannot be used where identifiers are used. For example, the word `if` is commonly a reserved word, while `x` generally is not, so `x = 1` is a valid assignment, but `if = 1` is not. Keywords have varied uses, but primarily fall into a few classes: part of the phrase grammar (specifically a production rule with nonterminal symbols), with various meanings, often being used for control flow, such as the word `if` in most procedural languages, which indicates a conditional and takes clauses (the nonterminal symbols); names of primitive types in a language that support a type system, such as `int`; primitive literal values such as `true` for Boolean true; or sometimes special commands like `exit`.
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Boolean grammars, introduced by , are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context- free grammars, with conjunction and negation operations. Besides these explicit operations, Boolean grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction and negation can be used, in particular, to specify intersection and complement of languages.
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LALR generators calculate lookahead sets by a more precise method based on exploring the graph of parser states and their transitions. This method considers the particular context of the current parser state. It customizes the handling of each grammar occurrence of some nonterminal S. See article LALR parser for further details of this calculation. The lookahead sets calculated by LALR generators are a subset of (and hence better than) the approximate sets calculated by SLR generators.
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The '$' sign is used to denote 'end of input' is expected, as is the case for the starting rule. This is not the complete item set 0, though. Each item set must be 'closed', which means all production rules for each nonterminal following a '•' have to be recursively included into the item set until all of those nonterminals are dealt with. The resulting item set is called the closure of the item set we began with.
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Another wayHopcroft et al. (2006) to define the Chomsky normal form is: A formal grammar is in Chomsky reduced form if all of its production rules are of the form: : A \rightarrow\, BC or : A \rightarrow\, a, where A, B and C are nonterminal symbols, and a is a terminal symbol. When using this definition, B or C may be the start symbol. Only those context-free grammars which do not generate the empty string can be transformed into Chomsky reduced form.
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Side effects of laudanum are generally the same as with morphine, and include euphoria, dysphoria, pruritus, sedation, constipation, reduced tidal volume, respiratory depression, as well as psychological dependence, physical dependence, miosis, and xerostomia. Overdose can result in severe respiratory depression or collapse and death. The ethanol component can also induce adverse effects at higher doses; the side effects are the same as with alcohol. Long-term use of laudanum in nonterminal diseases is discouraged due to the possibility of drug tolerance and addiction.
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An ABNF specification is a set of derivation rules, written as rule = definition ; comment CR LF where rule is a case-insensitive nonterminal, the definition consists of sequences of symbols that define the rule, a comment for documentation, and ending with a carriage return and line feed. Rule names are case-insensitive: ``, ``, ``, and `` all refer to the same rule. Rule names consist of a letter followed by letters, numbers, and hyphens. Angle brackets (`<`, `>`) are not required around rule names (as they are in BNF).
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On May 1, 1985, electrical engineer Larry McAfee became completely paralyzed and ventilator dependent following a motorcycle crash. After he quickly exhausted his $1 million insurance deductible, he was shunted into a series of nursing homes for Medicare and Medicaid recipients unaccustomed to working with young, nonterminal patients. He devised a switch which would allow him to turn off his own ventilator, but found the process too painful to pursue unaided. Seeing no end to this existence, he petitioned the state for his right to die.
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Type-2 grammars generate the context-free languages. These are defined by rules of the form A \rightarrow \alpha with A being a nonterminal and \alpha being a string of terminals and/or nonterminals. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton. Context-free languages—or rather its subset of deterministic context-free language—are the theoretical basis for the phrase structure of most programming languages, though their syntax also includes context-sensitive name resolution due to declarations and scope.
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A context-sensitive grammar (CSG) is a formal grammar in which the left-hand sides and right-hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols. Context-sensitive grammars are more general than context-free grammars, in the sense that there are languages that can be described by CSG but not by context-free grammars. Context- sensitive grammars are less general (in the same sense) than unrestricted grammars. Thus, CSG are positioned between context-free and unrestricted grammars in the Chomsky hierarchy.
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The states and transitions give all the needed information for the parse table's shift actions and goto actions. The generator also needs to calculate the expected lookahead sets for each reduce action. In SLR parsers, these lookahead sets are determined directly from the grammar, without considering the individual states and transitions. For each nonterminal S, the SLR generator works out Follows(S), the set of all the terminal symbols which can immediately follow some occurrence of S. In the parse table, each reduction to S uses Follow(S) as its LR(1) lookahead set.
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In formal language theory, a grammar is in Kuroda normal form if all production rules are of the form: :AB -> CD or :A -> BC or :A -> B or :A -> a where A, B, C and D are nonterminal symbols and a is a terminal symbol. Some sources omit the A -> B pattern. It is named after Sige-Yuki Kuroda, who originally called it a linear bounded grammar—a terminology that was also used by a few other authors thereafter. Every grammar in Kuroda normal form is noncontracting, and therefore, generates a context-sensitive language.
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The rest of the item sets can be created by the following algorithm : 1. For each terminal and nonterminal symbol A appearing after a '•' in each already existing item set k, create a new item set m by adding to m all the rules of k where '•' is followed by A, but only if m will not be the same as an already existing item set after step 3. : 2. shift all the '•'s for each rule in the new item set one symbol to the right : 3.
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Fodor and Lyn Frazier proposed a new two-stage model of parsing human sentences and the syntactic analysis of these sentences. The first step of this new model is to “assign lexical and phrasal nodes to groups of words within the lexical string that is received”. The second step is to add higher nonterminal nodes and combines these newly created phrases into a sentence. Fodor and Frazier suggest this new method because it can transcend the complexities of language by parsing only a few words at a time.
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The transition table of this automaton is defined by the shift actions in the action table and the goto actions in the goto table. The next terminal is now '1' and this means that the parser performs a shift and go to state 2: : [0 E 3 '+' 6 '1' 2] Just as the previous '1' this one is reduced to B giving the following stack: : [0 E 3 '+' 6 B 8] The stack corresponds with a list of states of a finite automaton that has read a nonterminal E, followed by a '+' and then a nonterminal B. In state 8 the parser always performs a reduce with rule 2. The top 3 states on the stack correspond with the 3 symbols in the right-hand side of rule 2. This time we pop 3 elements off of the stack (since the right-hand side of the rule has 3 symbols) and look up the goto state for E and 0, thus pushing state 3 back onto the stack : [0 E 3] Finally, the parser reads a '$' (end of input symbol) from the input stream, which means that according to the action table (the current state is 3) the parser accepts the input string.
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If some nonterminal symbol S is used in several places in the grammar, SLR treats those places in the same single way rather than handling them individually. The SLR generator works out `Follow(S)`, the set of all terminal symbols which can immediately follow some occurrence of S. In the parse table, each reduction to S uses Follow(S) as its LR(1) lookahead set. Such follow sets are also used by generators for LL top-down parsers. A grammar that has no shift/reduce or reduce/reduce conflicts when using follow sets is called an SLR grammar.
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John Corcoran considers this terminology unfortunate because it obscures the use of schemata and because such "variables" do not actually range over a domain.. The convention is that a metavariable is to be uniformly substituted with the same instance in all its appearances in a given schema. This is in contrast with nonterminal symbols in formal grammars where the nonterminals on the right of a production can be substituted by different instances.. Attempts to formalize the notion of metavariable result in some kind of type theory.Masahiko Sato, Takafumi Sakurai, Yukiyoshi Kameyama, and Atsushi Igarashi. "Calculi of Meta-variables" in Computer Science Logic.
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The active site of OmpT resembles that of other omptins, and is characterized by conserved residues at Asp84, Asp86, Asp206, and His208. The most common bond cleavage by OmpT is between two arginine residues because their positive charge can favorably interact with the negatively charged species at the active site during substrate binding. Because of the specificity of the active site, OmpT does not act on peptides with a negatively charged residue adjacent to the scissile bond. Also, OmpT is specifically identified an endopeptidase because it does not cleave peptides at the N- or C-terminus, but only between nonterminal amino acids.
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The easiest description of GIGs is by comparison to Indexed grammars. Whereas in indexed grammars, a stack of indices is associated with each nonterminal symbol, and can vary from one to another depending on the course of the derivation, in a GIG, there is a single global index stack that is manipulated in the course of the derivation (which is strictly leftmost for any rewrite operation that pushes a symbol to the stack). Because of the existence of a global stack, a GIG derivation is considered complete when there are no non-terminal symbols left to be rewritten, and the stack is empty.
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For LR(1) for each production rule an item has to be included for each possible lookahead terminal following the rule. For more complex languages this usually results in very large item sets, which is the reason for the large memory requirements of LR(1) parsers. In our example, the starting symbol requires the nonterminal 'E' which in turn requires 'T', thus all production rules will appear in item set 0. At first, we ignore the problem of finding the lookaheads and just look at the case of an LR(0), whose items do not contain lookahead terminals.
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For one such construction the size of the constructed grammar is O(4) in the general case and O(3) if no derivation of the original grammar consists of a single nonterminal symbol, where is the size of the original grammar. This conversion can be used to prove that every context-free language can be accepted by a real-time (non-deterministic) pushdown automaton, i.e., the automaton reads a letter from its input every step. Given a grammar in GNF and a derivable string in the grammar with length , any top- down parser will halt at depth .
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A grammar mainly consists of a set of rules for transforming strings. (If it only consisted of these rules, it would be a semi-Thue system.) To generate a string in the language, one begins with a string consisting of only a single start symbol. The production rules are then applied in any order, until a string that contains neither the start symbol nor designated nonterminal symbols is produced. A production rule is applied to a string by replacing one occurrence of the production rule's left-hand side in the string by that production rule's right-hand side (cf.
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In formal language theory, the terminal yield (or fringe) of a tree is the sequence of leaves encountered in an ordered walk of the tree. Parse trees and/or derivation trees are encountered in the study of phrase structure grammars such as context-free grammars or linear grammars. The leaves of a derivation tree for a formal grammar G are the terminal symbols of that grammar, and the internal nodes the nonterminal or variable symbols. One can read off the corresponding terminal string by performing an ordered tree traversal and recording the terminal symbols in the order they are encountered.
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Indirect left recursion occurs when the definition of left recursion is satisfied via several substitutions. It entails a set of rules following the pattern :A_0 \to \beta_0A_1\alpha_0 :A_1 \to \beta_1A_2\alpha_1 :\cdots :A_n \to \beta_nA_0\alpha_n where \beta_0, \beta_1, \ldots, \beta_n are sequences that can each yield the empty string, while \alpha_0, \alpha_1, \ldots, \alpha_n may be any sequences of terminal and nonterminal symbols at all. Note that these sequences may be empty. The derivation :A_0\Rightarrow\beta_0A_1\alpha_0\Rightarrow^+ A_1\alpha_0\Rightarrow\beta_1A_2\alpha_1\alpha_0\Rightarrow^+\cdots\Rightarrow^+ A_0\alpha_n\dots\alpha_1\alpha_0 then gives A_0 as leftmost in its final sentential form.
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In informal terms, this algorithm considers every possible substring of the input string and sets P[l,s,v] to be true if the substring of length l starting from s can be generated from nonterminal variable R_v. Once it has considered substrings of length 1, it goes on to substrings of length 2, and so on. For substrings of length 2 and greater, it considers every possible partition of the substring into two parts, and checks to see if there is some production P \to Q \; R such that Q matches the first part and R matches the second part. If so, it records P as matching the whole substring.
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The resulting stack is: : [0 B 4] However, in state 4, the action table says the parser should now reduce with rule 3. So it writes 3 to the output stream, pops one state from the stack, and finds the new state in the goto table for state 0 and E, which is state 3. The resulting stack: : [0 E 3] The next terminal that the parser sees is a '+' and according to the action table it should then go to state 6: : [0 E 3 '+' 6] The resulting stack can be interpreted as the history of a finite state automaton that has just read a nonterminal E followed by a terminal '+'.
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In the formal language theory of computer science, left recursion is a special case of recursion where a string is recognized as part of a language by the fact that it decomposes into a string from that same language (on the left) and a suffix (on the right). For instance, 1+2+3 can be recognized as a sum because it can be broken into 1+2, also a sum, and {}+3, a suitable suffix. In terms of context-free grammar, a nonterminal is left-recursive if the leftmost symbol in one of its productions is itself (in the case of direct left recursion) or can be made itself by some sequence of substitutions (in the case of indirect left recursion).
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A slight variation of TDPL, known as Generalized TDPL or GTDPL, greatly increases the apparent expressiveness of TDPL while retaining the same minimalist approach (though they are actually equivalent). In GTDPL, in place of TDPL's recursive rule form A ← BC/D, we instead use the alternate rule form A ← B[C,D], which is interpreted as follows. When nonterminal A is invoked on some input string, it first recursively invokes B. If B succeeds, then A subsequently invokes C on the remainder of the input left unconsumed by B, and returns the result of C to the original caller. If B fails, on the other hand, then A invokes D on the original input string, and passes the result back to the caller.
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At the time of its publication, Syntactic Structures presented the state of the art of Zellig Harris's formal model of language analysis which is called transformational generative grammar. It can also be said to present Chomsky's version or Chomsky's theory because there is some original input on a more technical level. The central concepts of the model however follow from Louis Hjelmslev's book Prolegomena to a Theory of Language which was published in 1943 in Danish and followed by an English translation by Francis J. Whitfield in 1953. The book sets up an algebraic tool for linguistic analysis which consists of terminals and inventories of all different types of linguistic units, similarly to terminal and nonterminal symbols in formal grammars.
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A number of similar techniques exist, generally prefixing or suffixing an identifier to indicate different treatment, but the semantics are varied. Strictly speaking, stropping consists of different representations of the same name (value) in different namespaces, and occurs at the tokenization stage. For example, in ALGOL 60 with matched apostrophe stropping, `'if'` is tokenized as (Keyword, if), while `if` is tokenized as (Identifier, if) – same value in different token classes. Using uppercase for keywords remains in use as a convention for writing grammars for lexing and parsing – tokenizing the reserved word `if` as the token class IF, and then representing an if-then- else clause by the phrase `IF Expression THEN Statement ELSE Statement` where uppercase terms are keywords and capitalized terms are nonterminal symbols in a production rule (terminal symbols are denoted by lowercase terms, such as `identifier` or `integer`, for an integer literal).
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In a letter where he proposed a term Backus–Naur form (BNF), Donald E. Knuth implied a BNF "syntax in which all definitions have such a form may be said to be in 'Floyd Normal Form'", : \langle A \rangle ::= \, \langle B \rangle \mid \langle C \rangle or : \langle A \rangle ::= \, \langle B \rangle \langle C \rangle or : \langle A \rangle ::=\, a, where \langle A \rangle, \langle B \rangle and \langle C \rangle are nonterminal symbols, and a is a terminal symbol, because Robert W. Floyd found any BNF syntax can be converted to the above one in 1961. Here: p.354 But he withdrew this term, "since doubtless many people have independently used this simple fact in their own work, and the point is only incidental to the main considerations of Floyd's note." While Floyd's note cites Chomsky's original 1959 article, Knuth's letter does not.
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In formal language theory, a context-free grammar, G, is said to be in Chomsky normal form (first described by Noam Chomsky) Here: Sect.6, p.152ff. if all of its production rules are of the form: : A → BC, or : A → a, or : S → ε, where A, B, and C are nonterminal symbols, the letter a is a terminal symbol (a symbol that represents a constant value), S is the start symbol, and ε denotes the empty string. Also, neither B nor C may be the start symbol, and the third production rule can only appear if ε is in L(G), the language produced by the context-free grammar G. Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent onethat is, one that produces the same language which is in Chomsky normal form and has a size no larger than the square of the original grammar's size.
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An unrestricted grammar is a formal grammar G = (N, \Sigma, P, S), where N is a finite set of nonterminal symbols, \Sigma is a finite set of terminal symbols, N and \Sigma are disjoint,Actually, this is not strictly necessary since unrestricted grammars make no real distinction between the two. The designation exists purely so that one knows when to stop generating sentential forms of the grammar; more precisely, the language L(G) recognized by G is restricted to strings of terminal symbols P is a finite set of production rules of the form \alpha \to \beta where \alpha and \beta are strings of symbols in N \cup \Sigma and \alpha is not the empty string, and S \in N is a specially designated start symbol. As the name implies, there are no real restrictions on the types of production rules that unrestricted grammars can have.While Hopcroft and Ullman (1979) do not mention the cardinalities of N, \Sigma, P explicitly, the proof of their Theorem 9.3 (construction of an equivalent Turing machine from a given unrestricted grammar, p.
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In 2020, Virginia's legislature considered, but did not pass, a bill backed by Northam that would have created two additional categories of inmates eligible for geriatric release: those older than fifty- five who have served at least fifteen years of their sentences and those older than fifty who have served at least twenty years of their sentences. Although the parole board of Virginia cannot use terminal illness as a reason to release inmates, it can advise the governor on offering Executive Medical Clemency to certain inmates with less than three months to live. This compassionate release policy for terminally ill inmates is the second most restrictive in the country, and Virginia is the only state that does not offer any form of early release to inmates with complex, nonterminal illnesses or permanent disabilities. A bill to establish parole for inmates who are terminally ill or permanently incapacitated was supported by Northam during the 2020 legislative session and passed in the state Senate but was not passed in the House of Delegates.
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An intermediate class of grammars known as conjunctive grammars allows conjunction and disjunction, but not negation. The rules of a Boolean grammar are of the form A \to \alpha_1 \And \ldots \And \alpha_m \And \lnot\beta_1 \And \ldots \And \lnot\beta_n where A is a nonterminal, m+n \ge 1 and \alpha_1, ..., \alpha_m, \beta_1, ..., \beta_n are strings formed of symbols in \Sigma and N. Informally, such a rule asserts that every string w over \Sigma that satisfies each of the syntactical conditions represented by \alpha_1, ..., \alpha_m and none of the syntactical conditions represented by \beta_1, ..., \beta_n therefore satisfies the condition defined by A. There exist several formal definitions of the language generated by a Boolean grammar. They have one thing in common: if the grammar is represented as a system of language equations with union, intersection, complementation and concatenation, the languages generated by the grammar must be the solution of this system. The semantics differ in details, some define the languages using language equations, some draw upon ideas from the field of logic programming.
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The presence of a dot, •, in a configuration represents the current lookahead position, with the lookahead symbol shown to the right of the dot (and which always corresponds to a terminal symbol), and the current stack state to the left of the dot (and which usually corresponds to a nonterminal symbol). For practical reasons, including higher performance, the tables are usually extended by a somewhat large, auxiliary array of two-bit symbols, obviously compressed into four two-bit symbols, a byte, for efficient accessing on byte-oriented machines, often encoded as: ::00b represents ERROR ::01b represents SHIFT ::10b represents REDUCE ::11b represents STOP (STOP being a special case of SHIFT). The entire array generally includes mostly ERROR configurations, a grammar-defined number of SHIFT and REDUCE configurations, and one STOP configuration. In programming systems which support the specification of values in quaternary numeral system (base 4, two bits per quaternary digit), such as XPL, these are coded as, for example: ::"(2)…0…" represents ERROR ::"(2)…1…" represents SHIFT ::"(2)…2…" represents REDUCE ::"(2)…3…" represents STOP The SHIFT and the REDUCE tables are implemented separately from the array.
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This presents an opportunity to invoke an error recovery procedure, perhaps, in its most simplistic form, to discard the lookahead terminal symbol and to read the next terminal symbol, but many other programmed actions are possible, including pruning the stack, or discarding the lookahead terminal symbol and pruning the stack (and in a pathological case, it is usually possible to obtain ::⊥ • ⊥ where consists only of a "null statement"). In most cases, the stack is purposely pre-loaded, that is, initialized, with ::⊥ • ⊥ whereby the initial ⊥ is assumed to have already been recognized. This, then, represents the beginning of the program, and, thereby, avoids having a separate START configuration, which is, conceptually ::• ⊥ ⊥ ⊥ is a special pseudo-terminal symbol mechanically added to the grammar, just as is a special pseudo- nonterminal symbol mechanically added to the grammar (if the programmer did not explicitly include in the grammar, then would be automatically added to the grammar on the programmer's behalf). Clearly, such a parser has precisely one (implicit) START configuration and one (explicit) STOP configuration, but it can, and usually does have hundreds of SHIFT and REDUCE configurations, and perhaps thousands of ERROR configurations.
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