From this question, I gather that whether unambiguous CF grammar can be parsed in linear time is an open problem. I’d like to know what the major roadblocks to achieve this are. That is, what made the attempts to produce such a parser fail ?

This is a follow-up question to Why is it hard to parse unambiguous context-free grammar in linear time? I know that Parsing Expression Grammars (PEG) can be parsed in linear time using a packrat parser. Those parser use memoization to accomplish the linear time complexity: basically each non-terminal can be “expanded” at most once at […]

Can a CFG over large alphabet describe a language, where each terminal appears even number of times? If yes, would the Chomsky Normal Form be polynomial in |Σ| ? EDIT: What about a language where each terminal appears 0 or 2 times? EDIT: Solving this may give interesting extension to “Restricted read twice BDDs and […]

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on numbers: the successor operation, addition and multiplication. As far as I understand, this is not a theorem in the strict sense (because the […]

What makes a context free grammar ambiguous?

Currently, I am dealing with the calculus of natural deduction by Gentzen. This calculus gives us rules to manipulate so-called sequents. Definition. If $\phi_1,\dots, \phi_n,\phi$ are formulas, then $\phi_1\dots\phi_n\vdash\phi$, often abbreviated by $\Gamma\vdash\phi$, is called a sequent. Can somebody please explain to me the following two inference rules? Introduction of the universal quantifier. $$ \begin{array}{c} […]

I’m trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I was taking a class in formal logic last semester, I found that the most efficient way to do my homework was […]

Currently, I am dealing with the calculus of natural deduction by Gentzen. This calculus gives us rules to manipulate so-called sequents. Definition. If $\Gamma$ is a set of formulas and $\phi$ a formula, then $\Gamma\vdash\phi$ is called a sequent. The rules of this calculus of natural deduction are: Hypothesis. $$ \begin{array}{c} \hline \Gamma\vdash\phi \end{array}\text{, where […]

This is somewhat of a minor point about the incompletness theorem, but I’m always a little unsure: So one proves that there is a formula which is unprovable in the theory of consideration. Okay, at this point one is done. Then, as that unproven sentence contains the claim that this (the unprovable-ness) would happen, one […]

In a logical system, there is assignment of truth values to the sentences in the language, and axioms are assigned the true value. A logical system is a formal system. In a formal system, there is no truth value assignment, but there are still axioms. Does that imply that axioms in a formal system are […]

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