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COMP4141 Theory of Computation

创建日期：2024-04-29 16:35:56

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标签： COMP4141

Theory of Computation

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COMP4141

Theory of Computation
Lecture 7: Algorithms for CFLs
Administrivia
Assignment 1 now overdue
Assignment 2 available: due Monday, 21 March at 12 noon
(AEDST)
2
Outline
Decision Problems
Algorithms for Regular Languages
Algorithms for Context-Free Languages
3
Common questions about languages
We have seen a number of ways of representing languages in a
finite way (automata, regular expresions, grammars). This lets us
address several questions algorithmically:
Emptiness: Given a representation, is the associated language
empty?
Universality: Given a representation, is the associated
language universal?
Word/Acceptance problem: Given a word w and a
representation of a language L, is w ∈ L?
Language equivalence: Given representations of two
languages, L1 and L2, is L1 = L2?
Language inclusion: Given representations of two languages,
L1 and L2, is L1 ⊆ L2?
4
Decision Problems
When we can phrase a question we want answered as a Yes/No
question this is known as a Decision Problem.
For example:
NFA Emptiness
Input: An NFA N
Question: Is L(N) = ∅
5
Outline
Decision Problems
Algorithms for Regular Languages
Algorithms for Context-Free Languages
6
Deciding emptiness of a regular language
How to decide emptiness of a regular language L depends on its
representation, of which we’ve met a few.
NFA Emptiness
Input: An NFA N
Question: Is L(N) = ∅?
RegExp Emptiness
Input: A regular expression R
Question: Is L(R) = ∅?
7
NFA Emptiness
When L is given as L(A) of an NFA (Q,Σ, δ, q0,F ) (or DFA,
-NFA) is an exercise in graph reachability: Is there a final state
that can be reached from the initial state?
This can be done by a depth-first search in time linear in the
number of edges and vertices. (But note there could be |Q|2
edges.)
8
RegExp Emptiness
When L is given as L(R) of a regular expression R then we can
abstract inductively as follows:
Base: L(∅) = ∅, L() 6= ∅, and L(a) 6= ∅.
Induction:
L(R1 ∪ R2) = ∅ iff L(R1) = L(R2) = ∅.
L(R1 · R2) = ∅ iff L(R1) = ∅ or L(R2) = ∅.
L(R∗1 ) 6= ∅.
L((R1)) = ∅ iff L(R1) = ∅.
This implies that L(R) = ∅ can be decided in O(|R|) time.
9
The word problem for regular languages
Like the emptiness problem, the word problem depends on the
representation.
DFA Acceptance
Input: A DFA D and a word w
Question: Is w ∈ L(D)?
NFA Acceptance
Input: An NFA N and a word w
Question: Is w ∈ L(N)?
10
Word problems
DFA: easy, just feed the word to the automaton.
NFA: marking algorithm:
On input w = a1 . . . a|w |
1 Set of marks M := {q0}.
2 For i = 1 to |w | do
M :=
⋃
q∈M δ(q, ai )
3 Return whether F ∩M 6= ∅.
Others: translate to an equivalent DFA.
11
The inclusion problem for regular languages
DFA Language Inclusion
Input: Two DFAs A1 and A2
Question: Is L(A1) ⊆ L(A2)?
Algorithm:
1 Construct a DFA recognising L(A1) ∩ L(A2)
2 Check for emptiness
Other represntations: Translate to equivalent DFAs.
12
The inclusion problem for regular languages
DFA Language Inclusion
Input: Two DFAs A1 and A2
Question: Is L(A1) ⊆ L(A2)?
Algorithm:
1 Construct a DFA recognising L(A1) ∩ L(A2)
2 Check for emptiness
Other represntations: Translate to equivalent DFAs.
12
The inclusion problem for regular languages
DFA Language Inclusion
Input: Two DFAs A1 and A2
Question: Is L(A1) ⊆ L(A2)?
Algorithm:
1 Construct a DFA recognising L(A1) ∩ L(A2)
2 Check for emptiness
Other represntations: Translate to equivalent DFAs.
12
The equivalence problem for regular languages
DFA Language Equivalence
Input: Two DFAs A1 and A2
Question: Is L(A1) = L(A2)?
Algorithm:
1 Check if L(A1) ⊆ L(A2)
2 Check if L(A2) ⊆ L(A1)
Other representations: translate to equivalent DFAs.
13
The equivalence problem for regular languages
DFA Language Equivalence
Input: Two DFAs A1 and A2
Question: Is L(A1) = L(A2)?
Algorithm:
1 Check if L(A1) ⊆ L(A2)
2 Check if L(A2) ⊆ L(A1)
Other representations: translate to equivalent DFAs.
13
The equivalence problem for regular languages
DFA Language Equivalence
Input: Two DFAs A1 and A2
Question: Is L(A1) = L(A2)?
Algorithm:
1 Check if L(A1) ⊆ L(A2)
2 Check if L(A2) ⊆ L(A1)
Other representations: translate to equivalent DFAs.
13
Outline
Decision Problems
Algorithms for Regular Languages
Algorithms for Context-Free Languages
14
Emptiness of CFLs
CFG Emptiness
Input: A CFG G = (V ,Σ,P,S)
Question: Is L(G ) = ∅?
The emptiness problem for CFGs can be decided as follows:
1 Mark the terminals and , as generating
2 Mark as generating all those non-terminals that have a
production with only generating symbols in the RHS.
3 Repeat step 2 until nothing new is marked generating.
4 Check whether the start symbol is marked as generating.
15
Emptiness of CFLs
CFG Emptiness
Input: A CFG G = (V ,Σ,P,S)
Question: Is L(G ) = ∅?
The emptiness problem for CFGs can be decided as follows:
1 Mark the terminals and , as generating
2 Mark as generating all those non-terminals that have a
production with only generating symbols in the RHS.
3 Repeat step 2 until nothing new is marked generating.
4 Check whether the start symbol is marked as generating.
15
Chomsky normal form
For many of the remaining questions, it is convenient to introduce
a particularly simple class of CFGs that is still as powerful as CFGs
in general.
Definition
A context free (type 2) grammar is in Chomsky normal form if
every production is of one of the forms
S → , or
A→ BC where B and C are not S , or
A→ a.
Theorem
Any CFL is generated by a CFG in Chomsky normal form.
16
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals
Step 0: Define an equivalent CFG G0 = (V ∪ {S0},Σ,P0,S0) with a fresh start
variable S0 and the production S0 → S . We also remove all productions of
the form A → and patch this up by introducing new productions for every
occurrence of A in a body with that occurrence removed.
(E.g., A→ | aAbA becomes A→ ab | abA | aAb | aAbA.)
Productions B → A are replaced by B → unless that is one we had removed
earlier.
Repeat until occurs at most in S0 → .
17
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals
Step 1: Define an equivalent CFG G1 = (V1,Σ,P1,S0) with fresh non-terminals
for terminals V1 = V ∪ {S0} ∪ { Xa | a ∈ Σ }. Here P1 is derived from P0 by
replacing all occurrences of a terminal a ∈ Σ in the body of a production by the
corresponding non-terminal Xa and then adding productions Xa → a.
17
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals
Step 2: Define an equivalent CFG G2 = (V1,Σ,P2, S0). To generate P2 from
P1, eliminate productions of the form A→ B by
1 determining all derivations A1 ⇒G1 . . .⇒G1 Ak ⇒G1 α /∈ V1 not
containing repetitions of non-terminals,
2 drop all productions A→ B, and
3 introduce A1 → α for each of the derivations determined before.
17
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals
Step 3: Define an equivalent CFG G3 = (V3,Σ,P3, S0). To generate P3 from
P2, replace all productions A→ B1 . . .Bn with n > 2 by A→ B1C1, C1 → B2C2,
. . . , Cn−2 → Bn−1Bn for fresh non-terminals Ci . (These are distinct for each
such production.)
17
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals
S → ASA | aB
A → B | S
B → b |
17
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals
S0 → S
S → ASA | aB | a
A → B | S | b |
B → b
17
Constructing the Chomsky normal form
Let G = (V ,Σ,P, S) be a CFG.
Step 0 Remove S and from RHS
Step 1 Remove terminals from RHS
Step 2 Remove singleton rules (A→ B)
Step 3 Reduce RHS to pairs of non-terminals

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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