There are grammars we convert to unambiguous by using left recursion. Are there grammars that cannot be converted to unambiguous grammars?
There are unambiguous context-free grammars for most practical languages (ignoring context-sensitive features such as variable declarations, whitespace sensitivity, etc.).
But there is no algorithm which can find an unambiguous grammar given an ambiguous grammar. Furthermore, there is not even an algorithm which can tell you for certain whether a given grammar is ambiguous. These are both undecidable problems.
And, to answer your question, yes there are context-free languages for which there is no unambiguous grammar. Such languages are said to be inherently ambiguous.
Related
I'm looking for the mathematical theory which deals with describing formal languages (set of strings) in general and not just grammar hierarchies.
Grammars give you the algorithm that lists all possible strings in the language. You could specify the algorithm any other way, but grammars are a concise and well-accepted format to do so.
Another way is to list every string that belongs to the language -- this will only work if the set of strings in the language is small (and definitely not when the set is infinite).
Regular expressions are a formalism for describing a set of languages, for instance. Although there are algorithms for transforming regular grammars and expressions in both ways, they are still two different theories. Also, automata (as a plural of automaton) can help you describe languages, not just DFA and NFA which describe the same set as regular languages, but 2DFA, stack automata. For example, a two-stacks automata is as powerful as a Turing machine. Finally, Turing machines itself are a formalism for languages. For any Turing machine, the set of all string on which the given Turing machine stops on a finite number of steps is a formally defined language.
My book gives similar but slightly different explanations of regular grammar and regular language.
I doubt it's wrong, is a regular language the same thing of a regular grammar?
The definition of my book is:
A grammar is regular if all the productions are V-> aW or V->Wa with V,W non terminal or terminal symbols, "a" terminal symbol.W can also be empty or be the same of V.
Regular grammars and regular languages are two different terms:
A language is a (possibly infinite) set of valid sequences of terminal symbols.
A grammar defines which are the valid sequences.
The same language could be represented with different class of grammars (regular, context free, etc.). A language is said to be regular if it can be represented with a regular grammar. On the othet hand, a regular grammar always defines a regular language. What you have posted is the definition of the regular grammar.
See this Wikipedia post for further information.
A formal grammar is a set of rules, whereas a formal language is a set of strings.
A regular grammar is a formal grammar that describes a regular language.
According to Wikipedia:
[T]he left regular grammars generate exactly all regular languages. The right regular grammars describe the reverses of all such languages, that is, exactly the regular languages as well.
If mixing of left-regular and right-regular rules is allowed, we still have a linear grammar, but not necessarily a regular one.
In the above, left-regular rules are rules of the form V->Wa (right-regular, of the form V->aW).
I think if I explain the difference between a language and grammar, your queries will automatically get resolved.
A language is a set of strings over some set of alphabets satisfying certain rules encoded as grammars, while
Grammars are used to generate languages.
So basically grammars denote the syntactical rules of a string and the set of strings that can be generated with the start symbol of the grammar is called the Language of the grammar
I'm trying to find a plain (i.e. non-formal) explanation of the 4 levels of formal grammars (unrestricted, context-sensitive, context-free, regular) as set out by Chomsky.
It's been an age since I studied formal grammars, and the various definitions are now confusing for me to visualize. To be clear, I'm not looking for the formal definitions you'll find everywhere (e.g. here and here -- I can google as well as anyone else), or really even formal definitions of any sort. Instead, what I was hoping to find was clean and simple explanations that don't sacrifice clarity for the sake of completeness.
Maybe you get a better understanding if you remember the automata generating these languages.
Regular languages are generated by regular automata. They have only have a finit knowledge of the past (their compute memory has limits) so everytime you have a language with suffixes depending on prefixes (palindrome language) this can not be done with regular languages.
Context-free languages are generated by nondeterministic pushdown automata. They have a kind of knowledge of the past (the stack, which is not limited in contrast to regular automata) but a stack can only be viewed from top so you don't have complete knowledge of the past.
Context-sensitive languages are generated by linear-bound non-deterministic turing machines. They know the past and can deal with different contexts because they are non-deterministic and can access all the past at every time.
Unrestricted languages are generated by Turing machines. According to the Church-Turing-Thesis turing machines are able to calculate everything you can imagine (which means everything decidable).
As for regular languages, there are many equivalent characterizations. They give many different ways of looking at regular languages. It is hard to give a "plain English" definition, and if you find it hard to understand any of the characterizations of regular languages, it is unlikely that a "plain English" explanation will help. One thing to note from the definitions and various closure properties is that regular languages embody the notion of "finiteness" somehow. But this is again hard to appreciate without better familiarity with regular languages.
Do you find the notion of a finite automaton to be not simple and clean?
Let me mention some of the many equivalent characterizations (at least for other readers) :
Languages accepted by deterministic finite automata
Languages accepted by nondeterministic finite automata
Languages accepted by alternating finite automata
Languages accepted by two-way deterministic finite automata
Languages generated by left-linear grammars
Languages generated by right-linear grammars
Languages generated by regular expressions.
A union of some equivalence classes of a right-congruence of finite index.
A union of some equivalence classes of a congruence of finite index.
The inverse image under a monoid homomorphism of a subset of a finite monoid.
Languages expressible in monadic second order logic over words.
Regular: These languages answer yes/no with finite automata
Context free: These languages when given input word ( using state machiene and stack ) we can always answer yes/no if it is member of the language
Context sensitive: As long as production in grammar never shrinks ( α -> β ) we can answer yes/no (using state machiene and chunk of memory that is linear in size with input)
Recursively ennumerable: It can answer yes but in case of no it will go into infinite loop
see this video for full explanation.
I've understood how lexical analysis works,
but no idea how the syntactic analysis is done,
though in principle they two should similar(The only difference lies in the
type of their input symbols, characters or tokens.) ,
but the generated parser code is greatly different.
Especially the yy_action,yy_lookahead,there's no such thing in lexical analysis...
The grammars used to generate lexical analyzers generally are regular grammars, while the grammars used to generated syntatic analyzers generally are context-free grammars. Although they might look the same at the surface, they have very different characteristics and capabilities. Regular grammars can be recognized by deterministic finite automatons, which are relatively simple to construct and make fast. Context-free grammars are more challenging to build a recognizer for and usually a parser generator tool will construct a parser for only a subset of context-free grammars. For example, yacc constructs parsers for context-free grammars that are also LALR(1) grammars using push-down automata.
For more information on parsing, I would highly recommend Parsing Techniques, which walks through all the nuances of parsing in excruciating (but well described!) detail.
Is there a tool for converting an ANTLR grammar to a Bison grmmar?
I doubt it. Since ANTLR supports a broader class of grammars than Bison, it's only even possible for a subset of ANTLR grammars. At least from what I've seen, relatively few ANTLR grammars fit in the subset that could be directly converted to Bison.