Implicit Computational Complexity makes two aspects implicit, by manipulating
programming languages rather than models of com-putation, and by internalizing
the bounds rather than using external measure. We survey how automata theory
contributed to complexity with a machine-dependant with implicit bounds model

Aubert, Clément

English

arXiv

International audienceImplicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of computation, and by internalizing the bounds rather than using external measure. We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model

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An in-between "implicit" and "explicit" complexity: Automata

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HAL Id: hal-01111737https://hal.archives-ouvertes.fr/hal-01111737v2Submitted on 29 May 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Open licence - etalab|An in-between ”implicit” and ”explicit” complexity:AutomataClément AubertTo cite this version:Clément Aubert. An in-between ”implicit” and ”explicit” complexity: Automata. DICE 2015 -Developments in Implicit Computational Complexity, Apr 2015, Londres, United Kingdom. ￿hal-01111737v2￿An in-between “implicit” and “explicit” complexity: AutomataCLÉMENT AUBERTAbstract. Implicit Computational Complexity (ICC) makes two aspectsimplicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure.We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model.IntroductionThis survey justifies a fine-grained view on the definition of what Implicit Com-putational Complexity (ICC) wants to keep implicit:Machine-dependant Machine-independantExplicit boundsTuring machine,Random access machine,Counter machine, . . .Bounded recursion on notation [9],Bounded arithmetic [6],Bounded linear logic [13], . . .Implicit boundsAutomaton,Auxiliary pushdown machine [10],Boolean circuit, . . .Descriptive complexity [11],Recursion on notation [5],Tiered recurrence [19], . . .It is common to refer to the top-left as the classical, or explicit, approach tocomplexity. The three others techniques keep something implicit. The bottom-rightway is considered as the true implicit complexity, the one that produces all themotivating perspectives: quasi-interpretations, non-size-increasing computation, softlambda-calculus and linear logics, to name a few. But there might be hesitationswhen defining ICC, for neither the “machine-independent” neither the “withoutexplicit bounds” slogans are precise enough.The machine-dependant with implicit bounds variant will be our subject. Itcharacterizes complexity classes by bounding the quality of the resources of a machinerather than its quantity. To our knowledge, automata are the most canonical modelof this enterprise, which dates back to the 70’s: “we have attempted to characterizeseveral tape and time complexity classes of Turing machines in terms of devices whosedefinitions involve only ways in which their infinite memory may be manipulatedand no restrictions are imposed on the amount of memory that they use.” ([16,p. 88])Date: February 4, 2015.This work was partly supported by the ANR-10-BLAN-0213 Logoi, the ANR-14-CE25-0005ELICA and the ANR-11-INSE-0007 REVER.12 C. AUBERTAutomata, characterizations and separation resultsDefinition 1. For k > 1, l > 0, a 2-way non-deterministicfinite automaton withk-heads and l pushdown stacks (2NFA(k, l)) is a tuple M = {S, i,F,A,B,B,C, ,σ}where:• S is the finite set of states, with i ∈ S the initial state and F⊆ S the set ofaccepting states;• A is the input alphabet, B is the stack alphabet;• B and C are the left and right endmarkers, B,C/∈A;•   is the bottom symbol of the stack,   /∈B;From now on, we let A./ (resp. B ) be A∪{B,C} (resp. B∪{ }).• σ ⊆ (S× (A./)k× (B )l)× (S×{−1,0,+1}k×{pop,peek,push(b)}l) is thetransition relation, where −1 means to move the head one cell to the left, 0means to keep the head on the current cell and +1 means to move it onecell to the right. Regarding the pushdown stacks, pop means “erase the topsymbol”, peek “do nothing”, and, for all b ∈B, push(b) is “write b on topof the stack”.Given an input n ∈A∗, the automata is initiated in state i, with B nC written onits only tape, all its heads at B and all its stacks containing  . It makes (non-deterministic) transitions according to σ and halt accepting as soon as it reaches astate belonging to F. We impose that the heads cannot move beyond the endmarkersand that the bottom stack symbol   cannot be erased (“popped”).Finally, we denote with 2NFA(k, l) the class of languages recognized by the2NFA(k, l) automaton. Moreover, we let 2NFA(∗, l) = ∪k>12NFA(k, l).Theorem 1. The following table gives the correspondence between automata andlanguages (or predicates):Automata Languagea) 2NFA(1,2) Computableb) 2NFA(∗,1) Polynomial timec) 2NFA(∗,0) Logarithmic spaced) 2NFA(1,1) Context-freee) 2NFA(1,0) RegularProof. We just sketch them, and provide references for the most difficult one. Wealways suppose that n is the input and |n| its size.a) ⊆: Given a computable language, by the Church–Turing thesis, there existsa Turing machine T that decides it. Using some classical theorem, we can alwaysassume that T has a single reading head and a single read-write tape. A 2NFA(1,2)can simulate T by simulating the movement of the read-only head with its head,and the content of the read-write tape with its two pushdown stacks. The first (resp.second) pushdown stack store the content on the left (resp. right) of the read-writehead.a) ⊇: As 2NFA(1,2) are restrictions of Turing Machines, their simulation isobvious.b) ⊆: This part amounts to designing an equivalent Turing machine whosemovements of heads follow a regular pattern. That permits to seamlessly simulatethe content of the read-write tape with a pushdown stack. A complete proof [10,An in-between “implicit” and “explicit” complexity: Automata 3pp. 9–11] as well as a precise algorithm [27, pp. 238–240] can be found in theliterature.b) ⊇: This way gave birth to memoization. In a nutshell, simulating a 2NFA(∗,1)with a polynomial-time Turing machine cannot amount to simulate step-by-step theautomaton. The reason is that for any automaton, one can design an automatonthat recognizes the same language but runs exponentially slower [1, p. 197]. Thetechnique invented by Alfred V. Aho et. al [1] and made popular by Stephen A. Cookconsists in building a “memoization table” that allows the Turing machine to createshortcuts in the simulation of the automaton, decreasing drastically its computationtime. It is a “clever evaluation strategy, applicable whenever the results of certaincomputations are needed more than once” [2, p. 348]. A nice explanation in thecase of single head automata is in a recent and short article by R. Glück [14].c) ⊆: Let T be a Turing machine deciding the predicate with m× log(|n|) space.Remark that an integer represented by a binary string of length p is no greater than2p, which takes 2p bits in unary. Then, a binary string of length log(|n|) cannot begreater than |n|. So the content of the read-write tape and the position of its headis encoded as distances between B and the read-only heads of the automata. Thatneeds to compute simple operations (modulo, multiplication and remaining of adivision) thanks to an extra head, but m+3 heads [25, pp. 191–192] are sufficient toperform this simulation. Simplest and the fastest simulations exists [27, pp. 223–225],but the overhead in terms of heads is more important.c) ⊇: The addresses of the heads on the input n takes log(|n|) to we written,hence a log-space Turing machine can easily simulate a 2NFA(∗,0).d) and e): The proofs of those two fundamentals results are beyond the scopeof this small survey. Their original proofs are respectively due to Chomsky [7,Theorem 1, p. 188] and Kleene [18, Theorem 6].In the c) case, taking a deterministic automata 2DFA(∗,0) yields a characteri-zation of deterministic log-space predicate L (whereas we characterized here thenon-deterministic case NL). In the b) case, both deterministic and non-deterministicautomata characterize deterministic polynomial time.Automata can also be restricted to be 1-way (simply remove the −1 instructionfrom the transition relation). In the e) case of regular language (which are equal to lin-ear space), we get in fact 1DFA(1,0) = 2DFA(1,0) = 1NFA(1,0) = 2DFA(1,0).Theorem 2 (Other results of interest).1DFA(k,0) ( 2DFA(k,0)([15, p. 95])1NFA(k,0) ( 2NFA(k,0)([15, p. 95])1DFA(k,0) ( 1NFA(2,0)([28, p. 339])1DFA(k,1) ( 1NFA(k,1)([8])1NFA(k,0)⊆ 2DFA(k,0)([25, p. 190])2NFA(k,1)⊆ 2DFA(4×k,1)([24, p. 214])2NFA(k,0)⊆ 2NFA(2×k,1)([25, p. 189])1DFA(k,1) ( 1NFA(k,1)([8])NL = L([23, pp. 75–76])iff1NFA(2,0)⊆ 2DFA(k,0)4 C. AUBERTTheorem 3. For l < 2, k+1 heads are strictly stronger than k heads.This results holds in all situations, whenever automata are 1- or 2-way, determin-istic or not, with or without pushdown stacks.1 Even if some notes on the techniquesand historic of those results exist [26, Chap. 4, Sect. 5.3, 22, pp. 67–68], no extensivesurvey was found, so the reader has to go for the original research papers [28, p. 338,20, p. 106, 21, p. 383, 8, p. 179, 17, p. 35].ConclusionAutomata theory is a major subject of computer science, very active and providingall kind of extensions and restrictions to automata. It contributed to “implicit”complexity in the sense that no external bound or measure is needed to tame theircomputational power: by specifying their number of ways, heads and stacks, oneknows in advance the computational power of the device.Those “implicit characterizations” provided decisive hints and tools to design two(truly implicit) bounded programming languages [3, 4]. Numerous other inspiringresults remains to be explored and should benefit to ICC.We did not mentioned finite state transducer, which are functional finite automata,but they also provide nice characterizations of functional complexity and beautifulresults (even recently [12], where the main theorem fits in the title). The statusof the input is also of interest: automata can bee feeded with 1-way inputs, trees,etc. That makes the expressive power to be really parametric in the input and themachineReferences[1] A. V. Aho, J. E. Hopcroft, and J. D. Ullman. “Time and Tape Complexity of PushdownAutomaton Languages”. In: Inform. Control 13.3 (1968), pp. 186–206. doi: 10.1016/S0019-9958(68)91087-5.[2] T. Amtoft and J. L. Träff. “Partial Memoization for Obtaining Linear Time Behaviorof a 2DPDA”. In: Theoret. Comput. Sci. 98.2 (1992), pp. 347–356. doi: 10.1016/0304-3975(92)90008-4.[3] C. Aubert, M. Bagnol, P. Pistone, and T. Seiller. “Logic Programming and LogarithmicSpace”. In: APLAS. Ed. by J. Garrigue. Vol. 8858. LNCS. Springer, 2014, pp. 39–57. doi:10.1007/978-3-319-12736-1_3.[4] M. Bagnol. “On the Resolution Semiring”. PhD thesis. Aix-Marseille Université – Institutde Mathématiques de Marseille, 2014.[5] S. J. Bellantoni and S. A. Cook. “A New Recursion-Theoretic Characterization of thePolytime Functions (Extended Abstract)”. In: STOC. Ed. by S. R. Kosaraju, M. Fellows,A. Wigderson, and J. A. Ellis. ACM, 1992, pp. 283–93. doi: 10.1145/129712.129740.[6] S. R. Buss. Bounded Arithmetic. Ed. by Bibliopolis. Vol. 3. Studies in Proof Theory.Lecture Notes. 1986.[7] N. Chomsky. Context-Free Grammars and Pushdown Storage. Quarterly Progress Report65. MIT RLE, Apr. 1962, pp. 187–194.[8] M. Chrobak. “Hierarchies of One-Way Multihead Automata Languages”. In: Theoret.Comput. Sci. 48.3 (1986), pp. 153–181. doi: 10.1016/0304-3975(86)90093-9.[9] A. Cobham. “The intrinsic computational difficulty of functions”. In: Logic, methodologyand philosophy of science: Proceedings of the 1964 international congress held at the Hebrewuniversity of Jerusalem, Israel, from August 26 to September 2, 1964. Ed. by Y. Bar-Hillel.Studies in Logic and the foundations of mathematics. North-Holland Publishing Company,1965, pp. 24–30.1To be fair, the author could not find a proof that this is the case for 1-way non-deterministicfinite automata with one stack.REFERENCES 5[10] S. A. Cook. “Characterizations of Pushdown Machines in Terms of Time-Bounded Comput-ers”. In: J. ACM 18.1 (1971), pp. 4–18. doi: 10.1145/321623.321625.[11] R. Fagin. “Contributions to the Model Theory of Finite Structures”. PhD thesis. Universityof California, Berkeley, 1973.[12] E. Filiot, O. Gauwin, P.-A. Reynier, and F. Servais. “From Two-Way to One-Way FiniteState Transducers”. In: LICS. IEEE Computer Society, 2013, pp. 468–477. doi: 10.1109/LICS.2013.53.[13] J.-Y. Girard, A. Scedrov, and P. J. Scott. “Bounded linear logic: a modular approach topolynomial-time computability”. In: Theoret. Comput. Sci. 97.1 (1992), pp. 1–66. doi:10.1016/0304-3975(92)90386-T.[14] R. Glück. “Simulation of Two-Way Pushdown Automata Revisited”. In: Festschrift forDave Schmidt. Ed. by A. Banerjee, O. Danvy, K.-G. Doh, and J. Hatcliff. Vol. 129. EPTCS.2013, pp. 250–258. doi: 10.4204/EPTCS.129.15.[15] M. Holzer, M. Kutrib, and A. Malcher. “Multi-Head Finite Automata: Characterizations,Concepts and Open Problems”. In: CSP. Ed. by T. Neary, D. Woods, A. K. Seda, andN. Murphy. Vol. 1. EPTCS. 2008, pp. 93–107. doi: 10.4204/EPTCS.1.9.[16] O. H. Ibarra. “Characterizations of Some Tape and Time Complexity Classes of TuringMachines in Terms of Multihead and Auxiliary Stack Automata”. In: J. Comput. Syst. Sci.5.2 (1971), pp. 88–117. doi: 10.1016/S0022-0000(71)80029-6.[17] O. H. Ibarra. “On Two-way Multihead Automata”. In: J. Comput. Syst. Sci. 7.1 (1973),pp. 28–36. doi: 10.1016/S0022-0000(73)80048-0.[18] S. C. Kleene. “Representation of events in nerve nets and finite automata”. In: vol. 34.Annals of Mathematics Studies. Princeton, NJ, USA: Princeton University Press, 1956,pp. 3–42.[19] D. Leivant. “Stratified Functional Programs and Computational Complexity”. In: POPL.Ed. by M. S. Van Deusen and B. Lang. Charleston, South Carolina, USA: ACM Press,1993, pp. 325–333. doi: 10.1145/158511.158659.[20] B. Monien. “Transformational Methods and their Application to Complexity Problems”.In: Acta Inform. 6 (1976), pp. 95–108. doi: 10.1007/BF00263746. See also [21].[21] B. Monien. “Corrigenda: Transformational Methods and Their Application to ComplexityProblems”. In: Acta Inform. 8 (1977), pp. 383–384. doi: 10.1007/BF00271346.[22] B. Monien. “Two-Way Multihead Automata Over a One-Letter Alphabet”. In: Theor.Inform. Appl. 14.1 (1980), pp. 67–82.[23] I. H. Sudborough. “On Tape-Bounded Complexity Classes and Multi-Head Finite Automata”.In: SWAT (FOCS). IEEE Computer Society, 1973, pp. 138–144. doi: 10.1109/SWAT.1973.20.[24] I. H. Sudborough. “Separating Tape Bounded Auxiliary Pushdown Automata Classes”. In:STOC. Ed. by J. E. Hopcroft, E. P. Friedman, and M. A. Harrison. ACM, 1977, pp. 208–217.doi: 10.1145/800105.803410.[25] I. H. Sudborough. “Some Remarks on Multihead Automata”. In: Theor. Inform. Appl. 11.3(1977), pp. 181–195.[26] P. M. B. Vitányi and M. Li. “Kolmogorov complexity and its applications”. In: Handbookof Theoretical Computer Science. volume A: Algorithms and Complexity. Ed. by J. VanLeeuwen. Elsevier, 1990, pp. 187–254.[27] K. W. Wagner and G. Wechsung. Computational Complexity. Vol. 21. Mathematics andits Applications. Springer, 1986.[28] A. C.-C. Yao and R. L. Rivest. “k + 1 Heads Are Better Than k”. In: J. ACM 25.2 (1978),pp. 337–340. doi: 10.1145/322063.322076.AcknowledgementsThe author wish to thank, in no particular order, M. Bagnol, G. Bonfante, A.Durand, P. Jacobé de Naurois, U. dal Lago, P. Pistone, P.-A. Reynier, U. Schöpp,T. Seiller and the TEX-LATEX Stack Exchange community.INRIA, Université Paris-Est, LACL (EA 4219), UPEC, F-94010 Créteil, FranceE-mail address: clement.aubert@lacl.frURL: https://lacl.fr/~caubert/

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Open licence - etalab|
An in-between ”implicit” and ”explicit” complexity:
Automata
Clément Aubert
To cite this version:
Clément Aubert. An in-between ”implicit” and ”explicit” complexity: Automata. DICE 2015 -
Developments in Implicit Computational Complexity, Apr 2015, Londres, United Kingdom. ￿hal-
01111737v2￿
An in-between “implicit” and “explicit” complexity: Automata
CLÉMENT AUBERT
Abstract. Implicit Computational Complexity (ICC) makes two aspects
implicit, by manipulating programming languages rather than models of com-
putation, and by internalizing the bounds rather than using external measure.
We survey how automata theory contributed to complexity with a machine-
dependant with implicit bounds model.
Introduction
This survey justifies a fine-grained view on the definition of what Implicit Com-
putational Complexity (ICC) wants to keep implicit:
Machine-dependant Machine-independant
Explicit bounds
Turing machine,
Random access machine,
Counter machine, . . .
Bounded recursion on notation [9],
Bounded arithmetic [6],
Bounded linear logic [13], . . .
Implicit bounds
Automaton,
Auxiliary pushdown machine [10],
Boolean circuit, . . .
Descriptive complexity [11],
Recursion on notation [5],
Tiered recurrence [19], . . .
It is common to refer to the top-left as the classical, or explicit, approach to
complexity. The three others techniques keep something implicit. The bottom-right
way is considered as the true implicit complexity, the one that produces all the
motivating perspectives: quasi-interpretations, non-size-increasing computation, soft
lambda-calculus and linear logics, to name a few. But there might be hesitations
when defining ICC, for neither the “machine-independent” neither the “without
explicit bounds” slogans are precise enough.
The machine-dependant with implicit bounds variant will be our subject. It
characterizes complexity classes by bounding the quality of the resources of a machine
rather than its quantity. To our knowledge, automata are the most canonical model
of this enterprise, which dates back to the 70’s: “we have attempted to characterize
several tape and time complexity classes of Turing machines in terms of devices whose
definitions involve only ways in which their infinite memory may be manipulated
and no restrictions are imposed on the amount of memory that they use.” ([16,
p. 88])
Date: February 4, 2015.
This work was partly supported by the ANR-10-BLAN-0213 Logoi, the ANR-14-CE25-0005
ELICA and the ANR-11-INSE-0007 REVER.
1
2 C. AUBERT
Automata, characterizations and separation results
Definition 1. For k > 1, l > 0, a 2-way non-deterministicfinite automaton with
k-heads and l pushdown stacks (2NFA(k, l)) is a tuple M = {S, i,F,A,B,B,C, ,σ}
where:
• S is the finite set of states, with i ∈ S the initial state and F⊆ S the set of
accepting states;
• A is the input alphabet, B is the stack alphabet;
• B and C are the left and right endmarkers, B,C/∈A;
•   is the bottom symbol of the stack,   /∈B;
From now on, we let A./ (resp. B ) be A∪{B,C} (resp. B∪{ }).
• σ ⊆ (S× (A./)k× (B )l)× (S×{−1,0,+1}k×{pop,peek,push(b)}l) is the
transition relation, where −1 means to move the head one cell to the left, 0
means to keep the head on the current cell and +1 means to move it one
cell to the right. Regarding the pushdown stacks, pop means “erase the top
symbol”, peek “do nothing”, and, for all b ∈B, push(b) is “write b on top
of the stack”.
Given an input n ∈A∗, the automata is initiated in state i, with B nC written on
its only tape, all its heads at B and all its stacks containing  . It makes (non-
deterministic) transitions according to σ and halt accepting as soon as it reaches a
state belonging to F. We impose that the heads cannot move beyond the endmarkers
and that the bottom stack symbol   cannot be erased (“popped”).
Finally, we denote with 2NFA(k, l) the class of languages recognized by the
2NFA(k, l) automaton. Moreover, we let 2NFA(∗, l) = ∪k>12NFA(k, l).
Theorem 1. The following table gives the correspondence between automata and
languages (or predicates):
Automata Language
a) 2NFA(1,2) Computable
b) 2NFA(∗,1) Polynomial time
c) 2NFA(∗,0) Logarithmic space
d) 2NFA(1,1) Context-free
e) 2NFA(1,0) Regular
Proof. We just sketch them, and provide references for the most difficult one. We
always suppose that n is the input and |n| its size.
a) ⊆: Given a computable language, by the Church–Turing thesis, there exists
a Turing machine T that decides it. Using some classical theorem, we can always
assume that T has a single reading head and a single read-write tape. A 2NFA(1,2)
can simulate T by simulating the movement of the read-only head with its head,
and the content of the read-write tape with its two pushdown stacks. The first (resp.
second) pushdown stack store the content on the left (resp. right) of the read-write
head.
a) ⊇: As 2NFA(1,2) are restrictions of Turing Machines, their simulation is
obvious.
b) ⊆: This part amounts to designing an equivalent Turing machine whose
movements of heads follow a regular pattern. That permits to seamlessly simulate
the content of the read-write tape with a pushdown stack. A complete proof [10,
An in-between “implicit” and “explicit” complexity: Automata 3
pp. 9–11] as well as a precise algorithm [27, pp. 238–240] can be found in the
literature.
b) ⊇: This way gave birth to memoization. In a nutshell, simulating a 2NFA(∗,1)
with a polynomial-time Turing machine cannot amount to simulate step-by-step the
automaton. The reason is that for any automaton, one can design an automaton
that recognizes the same language but runs exponentially slower [1, p. 197]. The
technique invented by Alfred V. Aho et. al [1] and made popular by Stephen A. Cook
consists in building a “memoization table” that allows the Turing machine to create
shortcuts in the simulation of the automaton, decreasing drastically its computation
time. It is a “clever evaluation strategy, applicable whenever the results of certain
computations are needed more than once” [2, p. 348]. A nice explanation in the
case of single head automata is in a recent and short article by R. Glück [14].
c) ⊆: Let T be a Turing machine deciding the predicate with m× log(|n|) space.
Remark that an integer represented by a binary string of length p is no greater than
2p, which takes 2p bits in unary. Then, a binary string of length log(|n|) cannot be
greater than |n|. So the content of the read-write tape and the position of its head
is encoded as distances between B and the read-only heads of the automata. That
needs to compute simple operations (modulo, multiplication and remaining of a
division) thanks to an extra head, but m+3 heads [25, pp. 191–192] are sufficient to
perform this simulation. Simplest and the fastest simulations exists [27, pp. 223–225],
but the overhead in terms of heads is more important.
c) ⊇: The addresses of the heads on the input n takes log(|n|) to we written,
hence a log-space Turing machine can easily simulate a 2NFA(∗,0).
d) and e): The proofs of those two fundamentals results are beyond the scope
of this small survey. Their original proofs are respectively due to Chomsky [7,
Theorem 1, p. 188] and Kleene [18, Theorem 6].

In the c) case, taking a deterministic automata 2DFA(∗,0) yields a characteri-
zation of deterministic log-space predicate L (whereas we characterized here the
non-deterministic case NL). In the b) case, both deterministic and non-deterministic
automata characterize deterministic polynomial time.
Automata can also be restricted to be 1-way (simply remove the −1 instruction
from the transition relation). In the e) case of regular language (which are equal to lin-
ear space), we get in fact 1DFA(1,0) = 2DFA(1,0) = 1NFA(1,0) = 2DFA(1,0).
Theorem 2 (Other results of interest).
1DFA(k,0) ( 2DFA(k,0)
([15, p. 95])
1NFA(k,0) ( 2NFA(k,0)
([15, p. 95])
1DFA(k,0) ( 1NFA(2,0)
([28, p. 339])
1DFA(k,1) ( 1NFA(k,1)([8])
1NFA(k,0)⊆ 2DFA(k,0)
([25, p. 190])
2NFA(k,1)⊆ 2DFA(4×k,1)
([24, p. 214])
2NFA(k,0)⊆ 2NFA(2×k,1)
([25, p. 189])
1DFA(k,1) ( 1NFA(k,1)([8])
NL = L
([23, pp. 75–76])
iff
1NFA(2,0)⊆ 2DFA(k,0)
4 C. AUBERT
Theorem 3. For l < 2, k+1 heads are strictly stronger than k heads.
This results holds in all situations, whenever automata are 1- or 2-way, determin-
istic or not, with or without pushdown stacks.1 Even if some notes on the techniques
and historic of those results exist [26, Chap. 4, Sect. 5.3, 22, pp. 67–68], no extensive
survey was found, so the reader has to go for the original research papers [28, p. 338,
20, p. 106, 21, p. 383, 8, p. 179, 17, p. 35].
Conclusion
Automata theory is a major subject of computer science, very active and providing
all kind of extensions and restrictions to automata. It contributed to “implicit”
complexity in the sense that no external bound or measure is needed to tame their
computational power: by specifying their number of ways, heads and stacks, one
knows in advance the computational power of the device.
Those “implicit characterizations” provided decisive hints and tools to design two
(truly implicit) bounded programming languages [3, 4]. Numerous other inspiring
results remains to be explored and should benefit to ICC.
We did not mentioned finite state transducer, which are functional finite automata,
but they also provide nice characterizations of functional complexity and beautiful
results (even recently [12], where the main theorem fits in the title). The status
of the input is also of interest: automata can bee feeded with 1-way inputs, trees,
etc. That makes the expressive power to be really parametric in the input and the
machine
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Acknowledgements
The author wish to thank, in no particular order, M. Bagnol, G. Bonfante, A.
Durand, P. Jacobé de Naurois, U. dal Lago, P. Pistone, P.-A. Reynier, U. Schöpp,
T. Seiller and the TEX-LATEX Stack Exchange community.
INRIA, Université Paris-Est, LACL (EA 4219), UPEC, F-94010 Créteil, France
E-mail address: clement.aubert@lacl.fr
URL: https://lacl.fr/~caubert/


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