H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Input Versus Restart Operations as Additional Resources
in Finite Automata
Henning Bordihn
University of Potsdam, Germany
Markus Holzer and Martin Kutrib
University of Giessen, Germany
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Motivation
• Basic machine model in automata theory: Finite State Automata
• Well-known fact: NFA = DFA
• Which additional resources extend the computational power?
• Resources (examples)
–
–
–
–
additional storages (pushdown tape, Turing tape, ...)
two-way, alternation, multiple heads, ...
restart operations
operations on the input word
֒→ forgetting automata, restarting automata
֒→ extended (e.g., input-revolving) finite automata
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Motivation
• Basic machine model in automata theory: Finite State Automata
• Well-known fact: NFA = DFA
• Which additional resources extend the computational power?
• Resources (examples)
–
–
–
–
additional storages (pushdown tape, Turing tape, ...)
two-way, alternation, multiple heads, ...
restart operations
operations on the input word
֒→ forgetting automata, restarting automata
֒→ extended (e.g., input-revolving) finite automata
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Motivation
• Basic machine model in automata theory: Finite State Automata
• Well-known fact: NFA = DFA
• Which additional resources extend the computational power?
• Resources (examples)
–
–
–
–
additional storages (pushdown tape, Turing tape, ...)
two-way, alternation, multiple heads, ...
restart operations
operations on the input word
֒→ forgetting automata, restarting automata
֒→ extended (e.g., input-revolving) finite automata
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Motivation (2)
• Comparing resources of
extended finite automata versus forgetting and restarting automata
• Differences:
– inspiration/motivation
– technical details
– many results
• Similarities: (extended/forgetting/restart automata)
– symbols may be consumed or passed when read
– unconsumed symbols may be read again (cyclic computations)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Motivation (2)
• Comparing resources of
extended finite automata versus forgetting and restarting automata
• Differences:
– inspiration/motivation
– technical details
– many results
• Similarities: (extended/forgetting/restart automata)
– symbols may be consumed or passed when read
– unconsumed symbols may be read again (cyclic computations)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Motivation (2)
• Comparing resources of
extended finite automata versus forgetting and restarting automata
• Differences:
– inspiration/motivation
– technical details
– many results
• Similarities: (extended/forgetting/restart automata)
– symbols may be consumed or passed when read
– unconsumed symbols may be read again (cyclic computations)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Outline
1. Extended Finite Automata: Definitions and Examples
2. Extended Finite Automata: Properties
3. Forgetting vs. Extended Finite Automata
4. Restarting vs. Extended Finite Automata
5. Some Questions for Future Research
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Uniform Machine Model
[Bordihn, Holzer, Kutrib]
(Nondeterministic) extended finite automaton: (Q, Σ, δ, ∆, q0, F ) :
• Q finite set of states
• Σ finite set of input symbols
• q0 ∈ Q is the initial state
• F ⊆ Q is the set of accepting states
• δ : Q × (Σ ∪ {λ}) → 2Q is the transition function
• ∆ : Q × (Σ ∪ {λ}) → 2Q is the operation function
Ordinary transition: (q, aw) ⊢ (p, w) if p ∈ δ(q, a), a ∈ Σ ∪ {λ}, w ∈ Σ∗
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Uniform Machine Model
[Bordihn, Holzer, Kutrib]
(Nondeterministic) extended finite automaton: (Q, Σ, δ, ∆, q0, F ) :
• Q finite set of states
• Σ finite set of input symbols
• q0 ∈ Q is the initial state
• F ⊆ Q is the set of accepting states
• δ : Q × (Σ ∪ {λ}) → 2Q is the transition function
• ∆ : Q × (Σ ∪ {λ}) → 2Q is the operation function
Ordinary transition: (q, aw) ⊢ (p, w) if p ∈ δ(q, a), a ∈ Σ ∪ {λ}, w ∈ Σ∗
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Uniform Machine Model
[Bordihn, Holzer, Kutrib]
(Nondeterministic) extended finite automaton: (Q, Σ, δ, ∆, q0, F ) :
• Q finite set of states
• Σ finite set of input symbols
• q0 ∈ Q is the initial state
• F ⊆ Q is the set of accepting states
• δ : Q × (Σ ∪ {λ}) → 2Q is the transition function
• ∆ : Q × (Σ ∪ {λ}) → 2Q is the operation function
Ordinary transition: (q, aw) ⊢ (p, w) if p ∈ δ(q, a), a ∈ Σ ∪ {λ}, w ∈ Σ∗
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Uniform Machine Model
[Bordihn, Holzer, Kutrib]
(Nondeterministic) extended finite automaton: (Q, Σ, δ, ∆, q0, F ) :
• Q finite set of states
• Σ finite set of input symbols
• q0 ∈ Q is the initial state
• F ⊆ Q is the set of accepting states
• δ : Q × (Σ ∪ {λ}) → 2Q is the transition function
• ∆ : Q × (Σ ∪ {λ}) → 2Q is the operation function
Ordinary transition: (q, aw) ⊢ (p, w) if p ∈ δ(q, a), a ∈ Σ ∪ {λ}, w ∈ Σ∗
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Input operations by different interpretations of ∆:
• left-revolving transition: (q, awd) ⊢lr (p, daw), for p ∈ ∆(q, a)
···
a
b
···
c
d
Left revolving
• right-revolving transition: (q, aw) ⊢rr (p, wa), for p ∈ ∆(q, a)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Input operations by different interpretations of ∆:
• left-revolving transition: (q, awd) ⊢lr (p, daw), for p ∈ ∆(q, a)
···
a
b
···
c
d
Left revolving
• right-revolving transition: (q, aw) ⊢rr (p, wa), for p ∈ ∆(q, a)
···
a
b
···
c
d
Right revolving
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
• Bi-revolving finite automaton: (Q, Σ, δ, ∆ℓ, ∆r , q0, F )
where
– (Q, Σ, δ, ∆ℓ, q0, F ) is a left-revolving finite automaton,
– (Q, Σ, δ, ∆r , q0, F ) is a right-revolving finite automaton
• An extended finite automaton A is deterministic if in any configuration there
at most one choice of action.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
• Bi-revolving finite automaton: (Q, Σ, δ, ∆ℓ, ∆r , q0, F )
where
– (Q, Σ, δ, ∆ℓ, q0, F ) is a left-revolving finite automaton,
– (Q, Σ, δ, ∆r , q0, F ) is a right-revolving finite automaton
• An extended finite automaton A is deterministic if in any configuration there
at most one choice of action.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Example
Right-revolving finite automaton ({q0, qa, qb}, {a, b}, δ, ∆, q0, {q0}),
where
δ(q0, a)
=
qb
δ(q0, b)
= qa
δ(qa, a)
= q0
δ(qb, b)
= q0
∆(qa, b) = qa
∆(qb, a) =
qb
Accepted language: { w | w ∈ {a, b}∗, |w|a = |w|b }
Analogously:
{ w | w ∈ {a, b, c}∗, |w|a = |w|b = |w|c } is both rr- and lr-DFA language.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Example
Left-revolving finite automaton ({q0, qa, qb}, {a, b}, δ, ∆, q0, {q0}),
where
δ(q0, a)
=
qb
δ(q0, b)
= qa
δ(qa, a)
= q0
δ(qb, b)
= q0
∆(qa, b) = qa
∆(qb, a) =
qb
Accepted language: { w | w ∈ {a, b}∗, |w|a = |w|b }
Analogously:
{ w | w ∈ {a, b, c}∗, |w|a = |w|b = |w|c } is both rr- and lr-DFA language.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Example
Left-revolving finite automaton ({q0, qa, qb}, {a, b}, δ, ∆, q0, {q0}),
where
δ(q0, a)
=
qb
δ(q0, b)
= qa
δ(qa, a)
= q0
δ(qb, b)
= q0
∆(qa, b) = qa
∆(qb, a) =
qb
Accepted language: { w | w ∈ {a, b}∗, |w|a = |w|b }
Analogously:
{ w | w ∈ {a, b, c}∗, |w|a = |w|b = |w|c } is both rr- and lr-DFA language.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Families of Languages
L (lr-DFA)
:
family of languages accepted by some deterministic
left-revolving finite automaton
L (rr-DFA)
:
family of languages accepted by some deterministic
right-revolving finite automaton
L (bi-DFA)
:
family of languages accepted by some deterministic
bi-revolving finite automaton
L (lr-NFA), L (rr-NFA), L (bi-NFA): the nondeterministic families
• In every deterministic extended finite automaton, one can restrict both δ
and any ∆ to Q × Σ (→ no λ-moves needed)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Families of Languages
L (lr-DFA)
:
family of languages accepted by some deterministic
left-revolving finite automaton
L (rr-DFA)
:
family of languages accepted by some deterministic
right-revolving finite automaton
L (bi-DFA)
:
family of languages accepted by some deterministic
bi-revolving finite automaton
L (lr-NFA), L (rr-NFA), L (bi-NFA): the nondeterministic families
• In every deterministic extended finite automaton, one can restrict both δ
and any ∆ to Q × Σ (→ no λ-moves needed)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Computational Power
• Any of the families L (lr-DFA), L (rr-DFA) and L (bi-DFA) is included in
DTime(n2) and DSpace(n).
• Any of the families L (lr-NFA), L (rr-NFA) and L (bi-NFA) is included in
NTime(n2) and NSpace(n).
֒→ These inclusions are strict:
• Any unary language accepted by some revolving finite automaton is regular.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Computational Power
• Any of the families L (lr-DFA), L (rr-DFA) and L (bi-DFA) is included in
DTime(n2) and DSpace(n).
• Any of the families L (lr-NFA), L (rr-NFA) and L (bi-NFA) is included in
NTime(n2) and NSpace(n).
֒→ These inclusions are strict:
• Any unary language accepted by some hybrid extended finite automaton is
regular.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Inclusion Structure
CSL
L (bi-NFA)
CFL
DCFL
L (lr-NFA)
LIN
L (bi-DFA)
L (rr-NFA)
L (lr-DFA)
L (rr-DFA)
DLIN
REG
(strict inclusions/pairwise incomparabilities)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Closure Properties
L (·)
∪
∩
∼
lr-DFA
rr-DFA
bi-DFA
−
−
−
−
−
−
−
−
−
Operation
∩reg R
·
?
−
?
−
−
+
−
−
−
∗
h−1
hλ
−
−
−
−
−
−
−
−
−
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Forgetting Automata: Another Kind of Input Manipulation
[von Braunmühl, Verbeek], [Jančar, Mráz, Plátek]
• finite control (with an initial and accepting states)
• input word bounded by left and right delimiters on a flexible tape
• may perform (some of) the following operations
M VR , M VL
—
moving the head one cell to the right (left)
ERR, ERL
—
erasing (rewriting the scanned symbol with blank)
and moving the head one cell to the right (left)
DLR, DLL
—
deleting the scanned symbol and moving the head
one cell to the right (left)
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Relations Between the Different Input Operations
transition of extended automata
operation in forgetting automata
ordinary transition
DLR
M VL
M VR
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Relations Between the Different Input Operations
transition of extended automata
operation in forgetting automata
ordinary transition
DLR
···
a
b
···
c
d
M VL
c
d
M VR
Left revolving
···
a
b
···
Right revolving
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Relations Between Automata Classes
• Fact: The language L = { bmambanbn | m, n ≥ 1 } is accepted by a forgetting
automaton with DLR, M VL, M VR operations, but it does not belong to
L (bi-NFA).
• But: {$} · L ∈ L (bi-DFA).
֒→ What if we consider only languages with a (left) delimiter?
• L d(bi-NFA) = L (M VR, M VL, DLR) = L (M V, DL).
• L d(bi-DFA) = Ldet(M VR, M VL, DLR) = Ldet(M V, DL).
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Relations Between Automata Classes
• Fact: The language L = { bmambanbn | m, n ≥ 1 } is accepted by a forgetting
automaton with DLR, M VL, M VR operations, but it does not belong to
L (bi-NFA).
• But: {$} · L ∈ L (bi-DFA).
֒→ What if we consider only languages with a (left) delimiter?
• L d(bi-NFA) = L (M VR, M VL, DLR) = L (M V, DL).
• L d(bi-DFA) = Ldet(M VR, M VL, DLR) = Ldet(M V, DL).
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Relations Between Automata Classes
• Fact: The language L = { bmambanbn | m, n ≥ 1 } is accepted by a forgetting
automaton with DLR, M VL, M VR operations, but it does not belong to
L (bi-NFA).
• But: {$} · L ∈ L (bi-DFA).
֒→ What if we consider only languages with a (left) delimiter?
• L d(bi-NFA) = L (M VR, M VL, DLR) = L (M V, DL).
• L d(bi-DFA) = Ldet(M VR, M VL, DLR) = Ldet(M V, DL).
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Technical Detail
• Whenever the revolving automaton passes the delimiter through a revolving
step, the forgetting automaton has to move to the other end of the input.
• Also when simulating an lr- or rr-automaton, both M VL and M VR seem to
be needed.
֒→ What is the precise relation of lr- and rr- automata with the (9 nondeterministic
and 10 deterministic) classes of forgetting automata?
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Technical Detail
• Whenever the revolving automaton passes the delimiter through a revolving
step, the forgetting automaton has to move to the other end of the input.
• Also when simulating an lr- or rr-automaton, both M VL and M VR seem to
be needed.
֒→ What is the precise relation of lr- and rr- automata with the (9 nondeterministic
and 10 deterministic) classes of forgetting automata?
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
A Technical Detail
• Whenever the revolving automaton passes the delimiter through a revolving
step, the forgetting automaton has to move to the other end of the input.
• Also when simulating an lr- or rr-automaton, both M VL and M VR seem to
be needed.
֒→ What is the precise relation of lr- and rr- automata with the (9 nondeterministic
and 10 deterministic) classes of forgetting automata
?
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Restarting Versus Extended Finite Automata
1. The standard model of restarting automata:
• R- and RR-automata can be simulated by nondeterministic
right-revolving (hence, bi-revolving) automata with delimiter.
• In the deterministic case: bi-revolving seems to be needed.
• RL-automata are simulated by bi-revolving automata.
?
• Conversely
– After restart-operations, the last state is lost.
– After restart-operations, the position of the last rewriting is lost.
– Length-reduction is required in every cycle.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Restarting Versus Extended Finite Automata
1. The standard model of restarting automata:
• R- and RR-automata can be simulated by nondeterministic
right-revolving (hence, bi-revolving) automata with delimiter.
• In the deterministic case: bi-revolving seems to be needed.
• RL-automata are simulated by bi-revolving automata.
?
• Conversely
– After restart-operations, the last state is lost.
– After restart-operations, the position of the last rewriting is lost.
– Length-reduction is required in every cycle.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Restarting Versus Extended Finite Automata
1. The standard model of restarting automata:
• R- and RR-automata can be simulated by nondeterministic
right-revolving (hence, bi-revolving) automata with delimiter.
• In the deterministic case: bi-revolving seems to be needed.
• RL-automata are simulated by bi-revolving automata.
?
• Conversely
– After restart-operations, the last state is lost.
– After restart-operations, the position of the last rewriting is lost.
– Length-reduction is required in every cycle.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Restarting Versus Extended Finite Automata
1. The standard model of restarting automata:
• R- and RR-automata can be simulated by nondeterministic
right-revolving (hence, bi-revolving) automata with delimiter.
• In the deterministic case: bi-revolving seems to be needed.
• RL-automata are simulated by bi-revolving automata.
?
• Conversely
– After restart-operations, the last state is lost.
– After restart-operations, the position of the last rewriting is lost.
– Length-reduction is required in every cycle.
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
2. More involved models: For simulating revolving automata is needed:
• non-forgetting restarting automata
• freely rewriting restarting automata
• length-preserving (instead of length-reducing) restarting automata
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Questions, Questions, Questions
• Comparing the computational powers of the variants of forgetting, restarting
and extended finite automata
– several different input operations, hybrid extended finite automata,
֒→ with and without delimiter
– 19 (potentially pairwise different) classes of forgetting automata
– forgetting vs. non-forgetting restarting automata
– taking concepts of determinism and monotonicity into consideration, ...
• Comparing the efficiency with respect to other resources when describing
languages of a suitable family
• Investigating the classes of extended finite automata with delimiter for
themselves
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Questions, Questions, Questions
• Comparing the computational powers of the variants of forgetting, restarting
and extended finite automata
– several different input operations, hybrid extended finite automata,
֒→ with and without delimiter
– 19 (potentially pairwise different) classes of forgetting automata
– forgetting vs. non-forgetting restarting automata
– taking concepts of determinism and monotonicity into consideration, ...
• Comparing the efficiency with respect to other resources when describing
languages of a suitable family
• Investigating the classes of extended finite automata with delimiter for
themselves
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Questions, Questions, Questions
• Comparing the computational powers of the variants of forgetting, restarting
and extended finite automata
– several different input operations, hybrid extended finite automata,
֒→ with and without delimiter
– 19 (potentially pairwise different) classes of forgetting automata
– forgetting vs. non-forgetting restarting automata
– taking concepts of determinism and monotonicity into consideration, ...
• Comparing the efficiency with respect to other resources when describing
languages of a suitable family
• Investigating the classes of extended finite automata with delimiter for
themselves
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Questions, Questions, Questions
• Comparing the computational powers of the variants of forgetting, restarting
and extended finite automata
– several different input operations, hybrid extended finite automata,
֒→ with and without delimiter
– 19 (potentially pairwise different) classes of forgetting automata
– forgetting vs. non-forgetting restarting automata
– taking concepts of determinism and monotonicity into consideration, ...
• Comparing the efficiency with respect to other resources when describing
languages of a suitable family
• Investigating the classes of extended finite automata with delimiter for
themselves
ABCD Workshop 2009
H. Bordihn, M. Holzer, M. Kutrib
Input Versus Restart Operations
Questions, Questions, Questions
• Comparing the computational powers of the variants of forgetting, restarting
and extended finite automata
– several different input operations, hybrid extended finite automata
֒→ with and without delimiter
– 19 (potentially pairwise different) classes of forgetting automata
– forgetting vs. non-forgetting restarting automata
– taking concepts of determinism and monotonicity into consideration, ...
• Comparing the efficiency with respect to other resources when describing
languages of a suitable family
• Investigating the classes of extended finite automata with delimiter for
themselves
ABCD Workshop 2009