Context-Free Languagess are Closed Under Intersection with Regular Languages
Here we consider the problem of intersecting a CFL and a regular language without using a PDA! It turns out that it is possible to do this directly, with a CFG and a DFA. The idea is simple, but we have to make some modifications first (the CFG has to be in CNF, and the DFA has to be an NFA with exactly 1 final state). Then we do a product-type construction that is similar to the PDA to CFG conversion.
The variables model pairs of states in the NFA (representing starting and destination states), as well as a variable in the original grammar. This "triplet" variable then should generate all strings that drive the NFA from the given start to the destination state, as well as a string that the given variable can generate. At the end, we generate the rules that come out of the intersection by considering the three cases that grammars in CNF can have.
Patreon: / easytheory
Facebook: / easytheory
Twitter: / easytheory
If you like this content, please consider subscribing to my channel: / @easytheory
▶ADDITIONAL QUESTIONS◀
1. What fundamental problems can arise with this method if we did a CFG intersect a CFG here?
2. Is it possible to remove the constraint for the DFA/NFA have exactly 1 final state?
▶SEND ME THEORY QUESTIONS◀
[email protected]
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught over 12 courses at Arizona State University, as well as Colgate University, including several sections of undergraduate theory.
▶ABOUT THIS CHANNEL◀
The theory of computation is perhaps the fundamental
theory of computer science. It sets out to define, mathematically, what
exactly computation is, what is feasible to solve using a computer,
and also what is not possible to solve using a computer.
The main objective is to define a computer mathematically, without the
reliance on real-world computers, hardware or software, or the plethora
of programming languages we have in use today. The notion of a Turing
machine serves this purpose and defines what we believe is the crux of
all computable functions.
This channel is also about weaker forms of computation, concentrating on
two classes: regular languages and context-free languages. These two
models help understand what we can do with restricted
means of computation, and offer a rich theory using which you can
hone your mathematical skills in reasoning with simple machines and
the languages they define.
However, they are not simply there as a weak form of computation--the most attractive aspect of them is that problems formulated on them
are tractable, i.e. we can build efficient algorithms to reason
with objects such as finite automata, context-free grammars and
pushdown automata. For example, we can model a piece of hardware (a circuit)
as a finite-state system and solve whether the circuit satisfies a property
(like whether it performs addition of 16-bit registers correctly).
We can model the syntax of a programming language using a grammar, and
build algorithms that check if a string parses according to this grammar.
On the other hand, most problems that ask properties about Turing machines
are undecidable.
This Youtube channel will help you see and prove that several tasks involving Turing machines are unsolvable---i.e., no computer, no software, can solve it. For example,
you will see that there is no software that can check whether a
C program will halt on a particular input. To prove something is possible is, of course, challenging.
But to show something is impossible is rare in computer
science, and very humbling.