4. Pushdown Automata, Conversion of CFG to PDA and Reverse Conversion

Name: 4. Pushdown Automata, Conversion of CFG to PDA and Reverse Conversion
Uploaded: 2021-10-06T12:44:33.000Z
Duration: 2 h 18 min 23 s

Introduction

The lecturer introduces the topic of context-free grammars and explains how they relate to regular languages and finite automata.

The lecture is about context-free grammars.

Regular languages are described by finite automata and regular expressions.

Finite automata are a weak model of computation with limited memory.

Context-free grammars are more powerful models that can do things finite automata cannot.

Recap of Previous Lectures

The lecturer recaps what was covered in previous lectures on regular languages, finite automata, and regular expressions.

Previous lectures covered regular languages, finite automata, and regular expressions.

Conversion between these models was discussed.

Certain languages were shown to be not regular.

Finite automata have limited capabilities compared to general-purpose computers.

Introduction to Context-Free Grammars

The lecturer introduces context-free grammars as a more powerful model than finite automata for describing languages.

Context-free grammars are introduced as a more powerful model than finite automata for describing languages.

Context-free grammars have variables, terminals, and rules.

Variables appear on the left-hand side of rules while terminals appear in the grammar itself.

Limitations of Context-Free Grammars

The lecturer discusses the limitations of context-free grammars despite their increased power over finite automata.

Some things can be done with context-free grammars that cannot be done with finite automata but they still have limitations.

Formal Definition of Context-Free Grammars

The lecturer discusses the formal definition of context-free grammars using variables, terminals, and rules.

A formal definition for context-free grammar is presented using variables, terminals, and rules.

The starting variable is designated to get the computation going.

Context-Free Languages

The lecturer introduces context-free languages as the counterpart to regular languages for finite automata and regular expressions.

Context-free languages are introduced as the counterpart to regular languages for finite automata and regular expressions.

The language of a grammar consists of strings whose alphabet are the terminal symbols.

Push Down Automata

The lecturer introduces push down automata as an automaton-based model that is equivalent in power to context-free grammars.

Push down automata are introduced as an automaton-based model that is equivalent in power to context-free grammars.

Conversion between context-free grammars and push down automata will be discussed.

Context-Free Languages

In this section, the speaker introduces context-free languages and how they relate to regular languages.

Introduction to Context-Free Languages

A context-free language is a type of language that can be generated by a context-free grammar.

The speaker provides an example of a parse tree and resulting string generated from a context-free grammar.

The language of the grammar is all strings that look like runs of zeros followed by runs of ones.

Formal Definition of Context-Free Grammar

A context-free grammar is defined as a four-tuple consisting of variables, terminal symbols, rules, and start variable.

Rules are always in the form of a variable with an arrow to a string of variables and terminals.

The standard notation for processing and producing strings in the grammar is u yields v if it can go from u to v with one substitution step.

U goes to v if there are multiple one-step moves that take you from u to v. This sequence is called derivation.

Language of the Grammar

The language of the grammar is the set of all strings that can be generated starting at the starting variable.

Valid Grammars

In this section, the speaker presents two examples and asks which one(s) are valid grammars.

Identifying Valid Grammars

The speaker presents two examples and asks which one(s) are valid grammars.

Converged Results

The speaker shares the results of a test.

The correct answer is b.

C1 is not a legit context-free grammar because it has things besides a single variable on the left-hand side.

C2 is a valid context-free grammar, but it can't generate any strings of only terminals.

Context-Free Grammar

The speaker explains what context-free grammar is and provides an example.

A string u derives itself according to the definition given by the speaker.

The example provided by the speaker generates arithmetical expressions involving pluses and times.

There are six rules in this example, each line representing two rules.

Variables are symbols that appear on the left-hand side, while terminal symbols are symbols of the language being generated.

Generating Strings

The speaker demonstrates how to use the grammar to generate strings.

A plus a times a is an example of generating strings using this grammar.

As substitutions are made, resulting strings evolve accordingly.

Parentheses are just terminal symbols in this example and do not play any special role besides that.

This might be part of a programming language that you're working with, according to the speaker's explanation at the end.

Introduction to Programming Language

In this section, the speaker introduces the concept of programming languages and how they are interpreted by compilers or interpreters. The meaning of a program is embedded within the structure of the parse tree.

Interpreting Programming Languages

A programming language is interpreted by a compiler or interpreter.

The first step in interpreting a program is to figure out its meaning, which is embedded within the structure of the parse tree.

The grammar used to create the parse tree can affect how meaning is assigned to different parts of a program.

The parse tree contains additional information that can be used to guide interpretation.

Ambiguity in Programming Languages

Some grammars may allow for multiple parse trees for the same string, leading to ambiguity in interpretation.

Ambiguity can be undesirable in programming languages because it can lead to multiple meanings for a single piece of code.

However, ambiguity does occur and is not always seen as a bad thing.

Grammar and Linguistics

This section discusses how grammars and linguistic concepts relate to programming languages.

Grammars and Linguistics

Grammars are used in both programming languages and natural human languages.

Noam Chomsky was instrumental in setting up many concepts related to grammars and linguistics.

English sentences can often have multiple interpretations due to their ambiguous structure.

Example: Ambiguity in English Sentences

The sentence "The boy saw the girl with the mirror" can have two different interpretations.

Ambiguity in English sentences is often resolved through additional information or context.

Conclusion

This section concludes the lecture on programming languages and their interpretation.

Ambiguity in Programming Languages

Ambiguity can be a problem in programming languages, but it is not always avoidable.

Understanding grammars and linguistic concepts can help programmers create more precise and unambiguous code.

Ambiguous Derivation and Pushdown Automata

In this section, the speaker introduces ambiguous derivation and pushdown automata as an automata counterpart for context-free languages. The speaker also presents a new view of finite automata using a schematic diagram.

Ambiguous Derivation

G2 has a unique parse tree for every string that you generate, while G3 can have multiple parse trees for the same string.

G3 has two interpretations for the same string, making it an ambiguously derived string with an ambiguous grammar.

Pushdown Automata

Pushdown automata is introduced as an automata counterpart for context-free languages.

A schematic diagram is presented to show the individual components of a finite automaton from a more abstract perspective.

The picture of a pushdown automaton is shown to have an extra device attached to it called a stack, which serves as auxiliary storage.

The schematic diagram for pushdown automaton includes the stack as part of its components.

Push Down Automata

In this section, we learn about how push down automata uses an extra memory called the stack or push down stack. We also learn about the special names for writing and reading symbols from the top of the stack.

The Stack

A push down automaton uses a stack to write and read symbols.

Symbols can only be read at the very top of the list of symbols.

When a new symbol is added, other symbols already there get pushed down.

This is similar to a cafeteria's stack of plates where plates are on a spring and move up or down depending on whether you add or remove them.

Pushing and Popping

Writing onto a stack is called a push operation, which adds a new symbol on top of the stack.

Reading and removing from the top of the stack is called a pop operation.

These operations are combined as writing comes with adding while reading comes with removing.

Example

We have an example pushdown automaton for language D, which accepts strings of zeros followed by ones where the number of zeros equals that of ones.

The automaton uses its ability to use stacks to count how many zeros it has by storing them on the stack until it sees a one.

It then reads ones while popping zeros from the stack, matching them off one-to-one with each one seen.

If it enters an accept state alone in the middle somewhere, it doesn't matter; it only counts when at end input string.

Returning to Pushdown Automata

In this section, the speaker discusses pushdown automata and how they differ from finite automata. They explain the six-tuple that defines a pushdown automaton and the transition function.

Definition of Pushdown Automata

A pushdown automaton is defined by a six-tuple.

The six-tuple includes an input alphabet (sigma) and a stack alphabet (gamma).

The transition function tells us how the machine operates, including how it reads input symbols and stack symbols, and what new state it can go into.

Transition Function

The transition function is more complicated than in finite automata.

It tells us how the machine goes from state to state based on its current state, input symbol, and top symbol on the stack.

Epsilons can appear in place of input or stack symbols to indicate that no reading or popping needs to occur for a transition.

Non-Determinism in Pushdown Automata

Pushdown automata can use non-determinism through power sets of possible transitions.

Epsilons also play a role in non-determinism by allowing transitions without reading or popping symbols.

Understanding Epsilon in Push-Down Automaton

In this section, the instructor explains the meaning of epsilon in push-down automaton and how it is used to construct a PDA.

Epsilon in PDA

The epsilon appearing over here means something a little different but very similar.

It means that we won't write anything on the top of the stack that's going to be. We will go to a new state but without doing any writing so we'll leave the stack alone.

If it's in this position, it means we're not going to read anything.

All of those things are valid and legal from the perspective of constructing a push-down automaton.

Non-Deterministic Push-Down Automaton

In this section, the instructor explains non-deterministic push-down automata and how they accept input strings.

Acceptance by Non-Deterministic Machine

A non-deterministic machine accepts an input string if at least one thread ends up in an accept state at the end of the input string.

You can use models like guessing or parallelism to understand non-determinism.

Recognizing Language Using Non-Determinism

In this section, the instructor explains how non-determinism is used to recognize languages using an example of recognizing w w reverse for all possible w's over our alphabet zero one.

Recognizing Language Using Non-Determinism

The language being recognized is w w reverse for all possible w's over our alphabet zero one.

The machine recognizes this language by putting the first half of the string on the stack and then removing it to match it with the second half.

Non-determinism is essential in recognizing this language since we don't know what's coming next as we're getting the symbols.

Understanding Non-Determinism in Push-Down Automata

In this section, the speaker explains how non-determinism is essential for recognizing certain languages in push-down automata. The speaker also clarifies some common questions about non-determinism and stack replication.

Non-Deterministic Guessing

When reading input symbols, it's important to not deterministically guess where the midpoint is.

Instead, read and push input symbols without guessing until reaching a point where you can start reading and popping symbols from the stack.

If at any point the input symbol disagrees with the top of the stack symbol, reject the thread of non-determinism.

Stack Replication

Every time there's a fork in non-determinism, the entire machine replicates its current state including its position, stack contents, etc.

The two branches of a fork operate independently until one accepts and raises a flag for communication.

Testing for Empty Stack

There's no primitive for testing if the stack is empty in push-down automata.

However, you can mark the bottom of the stack with a special symbol like "$" to simulate testing for an empty stack.

Testing for Empty Stack

The speaker explains that it is possible to test for an empty stack even if there is no primitive for it. This assumption can be used when writing homework sets.

Converting Grammars to Push Down Automata

The speaker claims that push down automata and context-free grammars are equivalent, and they will prove this equivalence in one direction.

To convert a grammar into a push down automaton, the speaker suggests starting with the start string and trying substitutions until the input string is derived.

Non-determinism comes into play when choosing which substitutions to make among many possibilities.

The machine writes intermediate results on the stack, but there is an important subtlety to consider.

Conclusion

The push down automaton starts with the starting variable and guesses substitutions while keeping intermediate results on the stack. When all substitutions have been made, it compares the result with the input string to see if they match.

Introduction to Pushdown Automata

In this section, the speaker introduces the concept of pushdown automata and explains how they operate.

Operating a Pushdown Automaton

The pushdown automaton operates by writing the starting variable on the stack and then doing substitutions as it goes along.

Substitutions are made one after another until there are no variables left, and then the resulting string is compared with the input.

However, doing substitutions at the very top of the stack is cheating because pushdown automata do not have access deep down within the stack.

Dealing with Access Below Top of Stack

To deal with this problem, substitutions will only be made at the top of the stack where variables can be accessed.

If a terminal symbol is sitting at the top blocking access to variables below, it will be matched with an input symbol until a variable appears on top for substitution.

This approach allows for substitutions to rise up to the top without ever needing to dig down into the interior of the stack.

Conclusion

By making substitutions only at the top of the stack, pushdown automata can effectively handle access below top of stack without cheating.

Construction of a Pushdown Automaton

In this section, the speaker explains how to construct a pushdown automaton.

Steps for constructing a pushdown automaton

Push the start symbol on the stack.

If the top of the stack is a variable, replace it with a corresponding right-hand side doing a non-deterministic choice among the various possibilities.

If it's a terminal, pop it and match it with the next input symbol.

If the stack is empty, accept.

Example of constructing a pushdown automaton

Start with e on top of the stack.

Substitute e with e plus t at the top.

Substitute t with f at the top.

When an 'a' appears at the top, match it with the next input symbol and remove it from the stack.

The remaining symbols are plus and t. Match plus with its corresponding input symbol and remove it from the stack.

Now only t remains on top of the stack, allowing for another substitution.

Context-Free Languages

In this section, context-free languages are defined and their relationship to pushdown automata is explained.

Definition of context-free languages

A language is context-free if and only if some push-down automaton recognizes it.

Proving context-free languages

To prove that a language is context-free, we need to show that there exists some push-down automaton that recognizes it.

To prove that every regular language is also context-free:

Every regular language can be recognized by either DFA or NFA.

DFA or NFA can be thought of as a push-down automaton that never uses its stack.

Pushdown automata are equivalent to context-free grammars, so every regular language is also context-free.

Recap

In this section, the speaker provides a quick recap of the topics covered in the class.

Regular and context-free languages

Regular languages and context-free languages were defined.

The recognizer form for regular languages is either DFA or NFA, while for context-free languages it is pushdown automata.

The generator form for regular languages is regular expressions, while for context-free languages it is context-free grammars.

Relationship between regular and context-free languages

Regular languages are a proper subset of context-free languages.

Some context-free languages are not regular.

Introduction to Turing Machines

In this section, the speaker introduces the concept of random access memory and the Turing Machine as a computational model that will be used throughout the term. The weaker models were introduced as a prelude to the more general purpose computational model.

Random Access Memory

Random Access Memory (RAM) is used in the next model called the Turing Machine.

The Turing Machine will be used throughout the term as a computational model.

Weaker Models

Weaker models were introduced as a prelude to the more general purpose computational model.

For weaker models, you can fully analyze them in a way that you cannot for Turing Machines.

You can determine properties of languages for weaker models that you cannot for more general models.

Why Use Stacks?

The reason why stacks are used instead of other data structures like queues is because it is exactly what you need to get to correspondence with context-free grammars.

If you use some other storage like queues instead of stacks, you get a very different outcome.

Non-Deterministic Pushdown Automata

In this section, non-determinism and determinism are compared for finite automata and pushdown automata. It is shown that non-determinism can be eliminated for finite automata but not for pushdown automata.

Nondeterministic vs Deterministic Pushdown Automata

Non-determinism can be eliminated for finite automata so NFA's and DFA's are equivalent.

However, they are not equivalent for pushdown automata.

There are certain languages that can only be done with non-deterministic push down automata and cannot be done with deterministic push down automata.

Conclusion

The lecture concludes by mentioning an example language "w w reverse" that requires non-determinism in order for the machine to be able to guess where the middle is. The speaker thanks the audience and ends the lecture.