CS代考计算机代写 Fortran BU CS 332 – Theory of Computation – INTERVIEW&CODEHELP

BU CS 332 – Theory of Computation
Lecture 6:
• NFAs ‐> Regular expressions • Context‐free grammars
• Pumping lemma for CFLs
Reading:
Mark Bun February 10, 2020
Sipser Ch 1.3, 2.1, 2.3

Regular Expressions – Syntax
A regular expression is defined recursively using the following rules:
1. , , and are regular expressions for every
2. If 􏵶 and 􏶊 are regular expressions, then so are
􏵶 􏶊,􏵶􏶊,and􏵶∗ Examples: (over )
∗∗∗∗
2/10/2020 CS332 ‐ Theory of Computation 2

Regular Expressions – Semantics
1. 2. 3.
for every
6.
􏵶∗ 􏵶∗
the language a regular expression describes
􏵶􏶊􏵶􏶊 􏵶􏶊􏵶􏶊
∗∗ 􏶐􏵳
2/10/2020 CS332 ‐ Theory of Computation 3
Example:

Regular Expressions Describe Regular Languages
Theorem: A language is regular if and only if it is described by a regular expression
Theorem 1: Every regular expression has an equivalent NFA Theorem 2: Every NFA has an equivalent regular expression
2/10/2020 CS332 ‐ Theory of Computation 4

NFA ‐> Regular expression
Theorem 2: Every NFA has an equivalent regex
Proof idea: Simplify NFA by “ripping out” states one at a
time and replacing with regexes
0 01*0 0
1
2/10/2020 CS332 ‐ Theory of Computation 5

Generalized NFAs
• Every transition is labeled by a regex
• One start state with only outgoing transitions
• Only one accept state with only incoming transitions • Start state and accept state are distinct
2/10/2020
CS332 ‐ Theory of Computation
6
∗ 𝑠
𝑎

Generalized NFA Example
2/10/2020
CS332 ‐ Theory of Computation
7
∗ 𝑠
𝑎
𝑠 𝑎

…
NFA ‐> Regular expression
2/10/2020
CS332 ‐ Theory of Computation
8
NFA
GNFA
𝑘 􏵷 2 states 𝑘 􏵷 1 states
𝑘 states
GNFA
GNFA
2 states
Regex

NFA ‐> GNFA
• •
Add a new start state with no incoming arrows. Make a unique accept state with no outgoing arrows.
ε
ε ε
NFA
ε
2/10/2020 CS332 ‐ Theory of Computation 9

GNFA ‐> Regular expression
Idea: While the machine has more than 2 states, rip one out and relabel the arrows with regexes to account for the missing state
1
3
2/10/2020
CS332 ‐ Theory of Computation
10
∗ 1
23

GNFA ‐> Regular expression
Idea: While the machine has more than 2 states, rip one out and relabel the arrows with regexes to account for the
missing state
2 1
1
3
2/10/2020
CS332 ‐ Theory of Computation
13
1
3 23
4

Example
2/10/2020 CS332 ‐ Theory of Computation 14

2/10/2020 CS332 ‐ Theory of Computation 15

2/10/2020 CS332 ‐ Theory of Computation 16

Context‐Free Grammars
2/10/2020 CS332 ‐ Theory of Computation 17

Some History
An abstract model for two distinct problems
Rules for parsing natural languages
2/10/2020 CS332 ‐ Theory of Computation 18

Some History
An abstract model for two distinct problems
Specification of syntax and compilation for programming languages
1977 ACM Turing Award citation (John Backus)
For profound, influential, and lasting contributions to the design of practical high‐ level programming systems, notably through his work on FORTRAN, and for seminal publication of formal procedures for the specification of programming languages.
2/10/2020 CS332 ‐ Theory of Computation 19

Context‐Free Grammar (Informal)
Example Grammar
Derivation
2/10/2020 CS332 ‐ Theory of Computation 20

Context‐Free Grammar (Informal)
Example Grammar 𝐺
𝐸 →𝐸􏵷𝑇 𝐸→𝑇
𝑇 →𝑇􏵻𝐹 𝑇→𝐹
𝐹 →􏶒𝐸􏶓 𝐹→𝑎 𝐹→𝑏
Derivation
𝐿􏶒𝐺􏶓 􏵼
2/10/2020
CS332 ‐ Theory of Computation 21

Socially Awkward Professor Grammar
→ → → LIKE
→ UMM
→ YOU KNOW → ε
→ WHOOPS → SORRY → $#@!
2/10/2020
CS332 ‐ Theory of Computation 22

Context‐Free Grammar (Formal)
A CFG is a 4‐tuple
• • •
is a finite set of variables
is a finite set of terminal symbols (disjoint from
)
,
•
is a finite set of production rules of the form
where and
is the start symbol
∗
Example: where
2/10/2020 CS332 ‐ Theory of Computation
24

Context‐Free Grammar (Formal)
A CFG is a 4‐tuple
= variables
= terminals
(“ yields
= rules = start
• Wesay
the grammar
”)if isaruleof or there exists a
• We say ⇒∗ (“ sequence such that
”) if
derives
• Language of the grammar:
Example: where
2/10/2020 CS332 ‐ Theory of Computation
25
􏵳􏵳
􏵶 􏶊
∗ ⇒∗

CFG Examples
Give context‐free grammars for the following languages 1. The empty language
2. Strings of properly nested parentheses
3. Strings with equal # of ’s and ’s
2/10/2020 CS332 ‐ Theory of Computation 26

Pumping Lemma II: Pump Harder
2/10/2020 CS332 ‐ Theory of Computation 27

Non context‐free languages?
• Could it be the case that every language is context‐free? 2/10/2020 CS332 ‐ Theory of Computation 28

Pumping Lemma for regular languages
Let be a regular language.
Then there exists a “pumping length”
such that where:
For every where , can be split into three parts
1.
2.
3. 𝑖  for all
2/10/2020
CS332 ‐ Theory of Computation
29

Pumping Lemma for context‐free languages
Let be a context‐free language.
Then there exists a “pumping length” such that
For every where , can be split into five parts
where:
1.
2.
3. 𝑖 𝑖  forall
2/10/2020
CS332 ‐ Theory of Computation 30
Example:
∗􏵸

Pumping Lemma as a game
1. YOU pick the language 𝐿 to be proved non context‐free.
2. ADVERSARY picks a possible pumping length 𝑝.
3. YOU pick 𝑤 of length at least 𝑝.
4. ADVERSARY divides 𝑤 into 𝑢, 𝑣, 𝑥, 𝑦, 𝑧, obeying rules of the Pumping Lemma: |𝑣𝑦| 􏶔 0 and |𝑣𝑥𝑦| 􏶕 𝑝.
5. YOU win by finding 𝑖 􏶖 0, for which 𝑢𝑣𝑖𝑥𝑦𝑖𝑧 is not in 𝐿.
If regardless of how the ADVERSARY plays this game, you
can always win, then is non context‐free
2/10/2020 CS332 ‐ Theory of Computation 32

Pumping Lemma example
Claim: 􏵳 Proof: Assume
􏵳 􏵳 is not regular
is regular with pumping length
1. Find
2. Show that
with
cannot be pumped
If =
with | |
, then…
2/10/2020
CS332 ‐ Theory of Computation
33

CS代考计算机代写 Fortran BU CS 332 – Theory of Computation

Published by admin

Leave a Reply Cancel reply