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Theory of Computation
Stack Machines
Vladimir Kulyukin
www.vkedco.blogspot.com
Outline
Stack Machines Stack Machine & Non-Determinism Stack Machine Construction Formalization of Stack Machines Stack Machines & Robot Control
Stack Machines
Stack Machine
FiniteControl
Input Tape
Stack
Stack Machine Move
read pop push
a C efC
If the current input symbol is a and the symbol on top of the stack is c, pop C off the stack, push efC onto the stack, and advance to the next symbol (move right). Reading a symbol means seeing it in the input and advancing to the next symbol.
Stack Machine Move: Example
read pop push
a c b
a cd
Input Tape Stacka
Stack Machine Move: Example
read pop push
a c b
a cd
Input Tape Stacka
Read a; Move right
Stack Machine Move: Example
read pop push
a c b
ad
Input Tape Stacka
Pop c off the Stack
Stack Machine Move: Example
read pop push
a c b
a bd
Input Tape Stacka
Push b on to the Stack
Stack Machines &
Non-Determinism
Stack Machine & Non-Determinism
A stack machine is like NFA, because it may have multiple sequences of moves on the same input
A stack machine accepts if and only if the input has been read and the stack is empty
Acceptance can also be defined in terms of final states
Non-Deterministic Stack Machine: Example
read pop push
ε S ab
a S ef
a S ε
Non-Deterministic Stack Machine: Example
Suppose the input is a The initial stack contains S There are three possible move sequences: Sequence 1: no input is read, pop S, push ab Sequence 2: read a, pop S, push ef Sequence 3: read a, pop S, push ε
Stack Machine Construction
Construct a stack machine for {anbn}
Sample Problem
Stack Machine Construction: Basic Insight
If a is read, pop off S, push 1, and push S In the middle of the string, pop off S If b is read, pop 1 off the stack If the stack is empty, that means we have read some number of
a’s (pushing) followed by the same number of b’s (popping)
Stack Machine Construction
read pop push
a S S1
ε S ε
b 1 ε
How Stack Machines Work: Example
Sa a b b b
Read a; Pop S; Push S1 (i.e., Push 1, then Push S); Move Right
a
Stack
a a a b b b S1
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
Read a; Pop S; Push S1 (i.e., Push 1, then Push S); Move Right
a
Stack
a a a b b b S11
S1
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
Read a; Pop S; Push S1 (i.e., Push 1, then Push S); Move Right
a
Stack
a a a b b bS111
S11
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
Read ε; Pop S; Push ε ; Stay in place (the machine has not read anything)
a
Stack
a a a b b b 111
S111
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
Read b ; Pop 1; Push ε ; Move right
a
Stack
a a a b b b
11
111
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
Read b ; Pop 1; Push ε ; Move right
a
Stack
a a a b b b
1
11
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
Read b ; Pop 1; Push ε ; Move right
a
Stack
a a a b b b
1
Stack
Before Move
After Move
How Stack Machines Work: Example
a a b b b
End of input is reached; Stack is empty; Accept
a
Stack
Formalization of Stack Machines
Formal Definition
tic.determinis-non is because strings,
ofset a isoutput thei.e., , ofset power the
is where,
symbol;alphabet initial theis
alphabet;input theis
alphabet;stack theis
where,S,,, tuple-4 a is machinestack A
*
**
M
Γ
ΓPΓPΓ
Γ S
Γ
ΓM
Example
read pop push
ε S ab
a S ef
a S ε
efSaefSa
abSabS
push or push , pop , read- ,,
push , pop , read- ,
Instantaneous Description of Stack Machine
stack theof on top is ofcharacter
leftmost thecontents,stack current theis and
input theofpart unread theis where,*
*
y
y
xyx
Stack Machine Derivation
Stack Machine Derivation
→* is a relation on instantaneous descriptions such thatif A & B are two instantaneous descriptions, then A →* B iff there is sequence of zero or more → transitions that starts at A and ends at B.
Acceptance by the Empty Stack
L(M) = {x is in Σ* | (x, S) →* (ε, ε)}
CFGs → Stack Machines:
Construction of Stack Machines from
Context-Free Grammars
CFG → Stack Machine
nothing.push , pop , read ;, , allfor .2
. of side hand-right the
push , pop , read ;|, , allfor 1.
:follows as is where,,,
:Sketch Proof
.such that machinestack
a is thereCFG, a is ,,, If :Theorem
aaaaa
xv
vPxvxvVv
S,VM
MLGLM
PSV G
CFG → Stack Machine
S → aSaS → bSbS → ε
read pop push
ε S aSa
ε S bSb
ε S ε
a a ε
b b ε
CFG Stack Machine
Stack Machines → CFGs:
Construction of Context-Free Grammars
fromStack Machines
Stack Machine CFG→
.in production a is |...|Then . where
,,...,,n transitioa has If :Insight Basic
.
such that ,,, CFG a is then there
machine,stack a is ,,, If .
that)generality of loss (w/o Assume :Theorem
1
1
PttA
ttAM
MLGL
PSΓG
SΓ M Γ
n
n
Example: Stack Machine CFG
S a0S
Stack MachineREAD POP PUSH
a S 0S
a 0 00
b 0 ɛ
b S 1S
b 1 11
a 1 ɛ
ɛ S ɛ
CFG Productions
0 a00
0 b
S b1S
1 b11
1 a
S ɛ
Stack Machines & Robot Control
Stack Machines for Robots Stack machines can be implemented in various robot control
mechanisms Behaviors can be described as context-free grammars The robot is constructed to parse the signals it receives from
the world CFGs are more powerful than FAs in that they allow the robot
to perceive signal dependencies but they do require memory (i.e., stack)
Robots & Stack Machines
Recall the line following problem for this robot. A control mechanism can be constructed so that the robot pushes and pulls its states onto and off the stack and executes specific behaviors in the following loop:
read (sense X); pop Y (Y is a memory state); execute Z
References & Reading Suggestions
Davis, Weyuker, Sigal. Computability, Complexity, and Languages, 2nd Edition, Academic Press
Brooks Webber. Formal Language: A Practical Introduction, Franklin, Beedle & Associates, Inc
Hopcroft & Ullman. Introduction to Automata Theory, Languages, and Computation, Narosa Publishing House
Moll, Arbib, & Kfoury. An Introduction to Formal Language Theory
