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These meta-symbols help us describe the structure of the grammar itself."}]},{"id":"card-7","type":"fill_blank","order":7,"blanks":[{"id":"def","isMath":false,"correctAnswer":"is defined as"},{"id":"or","isMath":false,"correctAnswer":"or"}],"sentence":"In BNF, `::=` means {{blank:def}}, and `|` means {{blank:or}}."},{"id":"card-8","type":"content","order":8,"title":"Worked example: checking an assignment string","blocks":[{"id":"block-1","content":"Suppose the grammar is ` ::= =`, ` ::= | `, ` ::= | `, ` ::= a | b | c`, and ` ::= 0 | 1 | 2`. To test `ab=102`, start from `` and use the first rule to split the string into an identifier, then `=`, then a number."},{"id":"block-2","content":"Derive the identifier part first: `` $\\Rightarrow$ `` $\\Rightarrow$ `a` $\\Rightarrow$ `a` $\\Rightarrow$ `ab`. Both `a` and `b` are allowed by ``."},{"id":"block-3","content":"Now derive the number: `` $\\Rightarrow$ `` $\\Rightarrow$ `1` $\\Rightarrow$ `1` $\\Rightarrow$ `10` $\\Rightarrow$ `10` $\\Rightarrow$ `102`. Because every symbol is matched in order and nothing is left over, `ab=102` is syntactically valid."}]},{"id":"card-9","type":"mcq","order":9,"options":[{"id":"a","text":"$$a2=10$","isCorrect":false},{"id":"b","text":"$$ab=102$","isCorrect":true},{"id":"c","text":"$$ab==102$","isCorrect":false},{"id":"d","text":"$$ab=103$","isCorrect":false}],"question":"Using the grammar in the worked example, which string is syntactically valid?","explanation":"$$ab=102$ can be derived exactly from the grammar. The most common wrong answer is $a2=10$, but $2$ is not a valid $letter$, so it cannot appear inside the identifier. Check both the identifier rules and the allowed digits.","correctOptionId":"b"},{"id":"card-10","type":"categorization","items":[{"id":"item-1","content":"$$a$","correctCategoryId":"cat-1"},{"id":"item-2","content":"$$=$","correctCategoryId":"cat-1"},{"id":"item-3","content":"$$\\langle \\text{identifier} \\rangle$","correctCategoryId":"cat-2"},{"id":"item-4","content":"$$\\langle \\text{number} \\rangle$","correctCategoryId":"cat-2"},{"id":"item-5","content":"$$::=$","correctCategoryId":"cat-3"},{"id":"item-6","content":"$$|$","correctCategoryId":"cat-3"}],"order":10,"categories":[{"id":"cat-1","name":"Terminal"},{"id":"cat-2","name":"Non-terminal"},{"id":"cat-3","name":"BNF notation"}],"instruction":"Classify each item as a terminal, non-terminal, or BNF notation."},{"id":"card-11","type":"content","order":11,"title":"BNF and syntax diagrams","blocks":[{"id":"block-1","content":"![Syntax diagram for signed integers with a three-way choice for plus, no sign, or minus, followed by one required digit and a loop for more digits](https://assets.mathsgenie.co.uk/lesson-v2-img/inline/872fa243-c3ef-4e58-bd49-00295ae20a95.png)\nA syntax diagram is a visual representation of grammar rules. Here the three-way branch represents $+$, no sign, or $-$. The straight path following it forces at least one $$, while the loop allows for extra digits."},{"id":"block-2","content":"This diagram matches BNF definitions such as $ ::= | $ and $ ::= | $. Recursion in BNF often appears as a loop in a syntax diagram."},{"id":"block-3","content":"To follow a diagram, move from START to END and consume the whole string. So $-305$ can take the minus branch, then a digit, then the loop twice. By contrast, a lone $+$ fails because the path requires a digit after the sign to be valid."}]},{"id":"card-12","type":"fill_blank","order":12,"blanks":[{"id":"need","isMath":false,"correctAnswer":"digit"}],"sentence":"Using the signed integer grammar, the string `+` is invalid because after the sign there must still be at least one {{blank:need}}."},{"id":"card-13","type":"content","order":13,"title":"Why BNF can say more than regular expressions","blocks":[{"id":"block-1","content":"Regular expressions are good for simple patterns, but they cannot generally enforce unlimited matching pairs or nesting. A classic context-free language is $L = \\{a^{n} b^{n} \\mid n \\ge 1\\}$, where the number of `a`s must equal the number of `b`s."},{"id":"block-2","content":"BNF can describe that language with the recursive rule ` ::= ab | ab`. To derive `aaabbb`, use ` => ab => aabb => aaabbb`, where the final `ab` is the base case that stops the recursion."},{"id":"block-3","content":"Each recursive step adds one `a` to the front and one `b` to the end, so the counts stay matched. A regular expression like `a*b*` is too weak because it also accepts strings such as `abb` and `aaaab`."}]},{"id":"card-14","type":"mcq","order":14,"options":[{"id":"a","text":"Because it allows any number of $a$s followed by any number of $b$s, so it also accepts strings like $abb$","isCorrect":true},{"id":"b","text":"Because $*$ means the $a$s and $b$s must repeat the same number of times","isCorrect":false},{"id":"c","text":"Because it forbids the string $ab$","isCorrect":false},{"id":"d","text":"Because regular expressions can only describe alphabets of size 1","isCorrect":false}],"question":"[Stretch] Why is the regular expression $a^{*}b^{*}$ not enough to describe $L = \\{a^{n} b^{n} \\mid n \\ge 1\\}$?","explanation":"Option A is correct because $a^{*}b^{*}$ has no memory of how many $a$s were used before the $b$s begin. The most common wrong choice is B, but $*$ only means repeat the preceding pattern zero or more times, and it does not force two different parts of a pattern to repeat equally often.","correctOptionId":"a"},{"id":"card-15","type":"flashcard","cards":[{"id":"fc-1","back":"A language that can be generated by a context-free grammar.","front":"Context-free language"},{"id":"fc-2","back":"A sequence of production-rule applications that turns the start symbol into a string of terminals.","front":"Derivation"},{"id":"fc-3","back":"Defining a rule partly in terms of itself, which allows repetition or nesting.","front":"Recursion"},{"id":"fc-4","back":"A visual form of grammar rules using paths, branches, and loops.","front":"Syntax diagram"}],"order":15},{"id":"card-16","type":"multi-question","order":16,"questions":[{"id":"mq-1","options":[{"id":"a","text":"A terminal","isCorrect":false},{"id":"b","text":"A non-terminal","isCorrect":true},{"id":"c","text":"A parser","isCorrect":false}],"question":"In ` ::= 0 | 1 | 2`, what is ``?","explanation":"Non-terminals are symbols in a grammar that represent sets of strings and can be further expanded, often denoted by angle brackets `< >`."},{"id":"mq-2","options":[{"id":"a","text":"`::=`","isCorrect":false},{"id":"b","text":"`|`","isCorrect":true},{"id":"c","text":"`=`","isCorrect":false}],"question":"Which BNF symbol means 'or'?","explanation":"The vertical bar `|` is used in BNF to separate alternative definitions for a non-terminal, acting as a logical 'or'."},{"id":"mq-3","options":[{"id":"a","text":"`cab`","isCorrect":true},{"id":"b","text":"`c2`","isCorrect":false},{"id":"c","text":"`-ab`","isCorrect":false}],"question":"If ` ::= a | b | c` and ` ::= | `, which string is valid?","explanation":"The grammar allows any sequence of one or more letters (a, b, or c). 'cab' is composed entirely of these letters."},{"id":"mq-4","options":[{"id":"a","text":"`+`","isCorrect":false},{"id":"b","text":"`-305`","isCorrect":true},{"id":"c","text":"`3-05`","isCorrect":false}],"question":"Which string fits the signed-integer grammar?","explanation":"A signed integer requires a sign followed by at least one digit. '-305' satisfies this, while '+' lacks digits and '3-05' has a misplaced sign."},{"id":"mq-5","options":[{"id":"a","text":"` ::= 0 | 1`","isCorrect":false},{"id":"b","text":"` ::= ab | ab`","isCorrect":true},{"id":"c","text":"` ::= + | -`","isCorrect":false}],"question":"Which rule is recursive?","explanation":"A rule is recursive if the non-terminal being defined (on the left side) also appears on the right side of the definition."},{"id":"mq-6","options":[{"id":"a","text":"` ::= 0 | 1`","isCorrect":false},{"id":"b","text":"` ::= 0 | 1` and ` ::= | `","isCorrect":true},{"id":"c","text":"` ::= 01`","isCorrect":false}],"question":"Which BNF can generate one or more binary digits?","explanation":"To generate 'one or more' items, you need a base case (a single bit) and a recursive case that allows appending more bits."},{"id":"mq-7","options":[{"id":"a","text":"`aabb`","isCorrect":false},{"id":"b","text":"`aaabbb`","isCorrect":false},{"id":"c","text":"`aab`","isCorrect":true}],"question":"Which string is not in $L = \\{a^{n} b^{n} \\mid n \\ge 1\\}$?","explanation":"The language $L = \\{a^{n} b^{n} \\mid n \\ge 1\\}$ requires an equal number of $a$'s and $b$'s. 'aab' has two $a$'s but only one $b$."}]}],"isPublished":true,"createdAt":"2026-07-06T07:29:47.105Z","updatedAt":"2026-07-06T07:30:37.218Z"},"lessonHref":"/as-level/computer-science/aqa/as-context-free-languages/lessons","lessonTitle":"Context-free languages","subjectId":"1bd3bb61-fba3-4297-adae-98adb356b835","subjectName":"Computer Science","subjectSlug":"computer-science","topicSidebar":"$L37","previewCardCount":4,"topicSlug":"context-free-languages","topicUrl":"/as-level/computer-science/aqa/as-context-free-languages"}],"$L38","$L39"]}]

Context-free languages Lesson

Context-free languages Lesson