## Decidability of Code Properties (0)

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Venue: | Proc. 4th International Conference Developments in Language Theory, (G. Rozenberg, W. Thomas, Eds.) World Scientific |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Fernau_decidabilityof,

author = {Henning Fernau and Klaus Reinhard and Ludwig Staiger},

title = {Decidability of Code Properties},

booktitle = {Proc. 4th International Conference Developments in Language Theory, (G. Rozenberg, W. Thomas, Eds.) World Scientific},

year = {},

pages = {153--163}

}

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### Abstract

We explore the borderline between decidability and undecidability of the following question: "Let C be a class of codes. Given a machine M of type X, is it decidable whether the language L(M) lies in C or not?" for codes in general, !-codes, codes of finite and bounded deciphering delay, prefix, suffix and bi(pre)fix codes, and for finite automata equipped with different versions of push-down stores and counters.

### Citations

4052 |
Introduction to Automata Theory, Languages and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...ur results in Table 1. 1 A good coverage of the corresponding literature can be found in [4]. 2 H. Fernau, K. Reinhardt, L. Staiger 2 Definitions For basics in automata theory, we refer the reader to =-=[1, 10, 11]-=-. Especially, the notion of (deterministic) push-down automaton, (D)PDA for short, should be known, leading to the language classes (D)CF; if the (D)PDA is only allowed to make one turn of the push-do... |

475 |
Computation: Finite and Infinite Machines
- Minsky
- 1967
(Show Context)
Citation Context ... emptiness problem for PBLIND(n) is equivalent to the emptiness problem for PBLIND by adding an additional blank symbol replacing e-moves. 4 H. Fernau, K. Reinhardt, L. Staiger 2. According to Minsky =-=[20, 21]-=-, cf. also [11, Sec. 7.8], the halting problem for two-counter machines is undecidable, even if one takes D2C machines with only one accepting state whose counters never get below zero and to which is... |

244 | Transductions and Context-Free Languages - Berstel - 1979 |

223 |
Theory of Codes
- Berstel, Perrin
- 1985
(Show Context)
Citation Context ... complexities of the decidable code properties; only the finite and regular code problems have received some attention until now [18, 24], although complexity questions have been explicitly raised in =-=[3]-=-. Let us remark as an example that the prefix code problem is just as hard as the emptiness problem for say (P)BC, since L is empty iff fag [ faagL ; a 2 X is a prefix code, cf. also footnote 6. Moreo... |

170 |
An algorithm for the general Petri net reachability problem
- Mayr
- 1984
(Show Context)
Citation Context ...C, e.g., D1C is the family of deterministic one-counter languages. 2 We briefly recall three non-trivial facts on counter machines: 1. From the decidability of the reachability problem for Petri nets =-=[14, 19], the decidabil-=-ity of the emptiness problem for (partially) blind counter machines results, see [9, Theorem 6] 3 . 2 PB1C is called "restricted one-counter languages" in [1], and 1C is called "iterate... |

95 | Decidability of reachability in vector addition systems
- Kosaraju
- 1982
(Show Context)
Citation Context ...C, e.g., D1C is the family of deterministic one-counter languages. 2 We briefly recall three non-trivial facts on counter machines: 1. From the decidability of the reachability problem for Petri nets =-=[14, 19], the decidabil-=-ity of the emptiness problem for (partially) blind counter machines results, see [9, Theorem 6] 3 . 2 PB1C is called "restricted one-counter languages" in [1], and 1C is called "iterate... |

50 |
Remarks on blind and partially blind one-way multicounter machines
- Greibach
- 1978
(Show Context)
Citation Context ...efix relation in X . 2.1 Counter machines Since definitions of counter automata are not standardized in the literature, we have to make the notions we use precise in this subsection, mostly following =-=[9]-=-. A k-counter machine M = (Q; X; ffi; q 0 ; Q f ; k) consists of a finite set Q of states, a designated initial state q 0 a designated subset Q f of final or accepting states, a finite input alphabet ... |

26 |
Introduction to Formal Language Theory (Addison-Wesley
- Harrison
- 1978
(Show Context)
Citation Context ...ur results in Table 1. 1 A good coverage of the corresponding literature can be found in [4]. 2 H. Fernau, K. Reinhardt, L. Staiger 2 Definitions For basics in automata theory, we refer the reader to =-=[1, 10, 11]-=-. Especially, the notion of (deterministic) push-down automaton, (D)PDA for short, should be known, leading to the language classes (D)CF; if the (D)PDA is only allowed to make one turn of the push-do... |

22 |
One-way stack automata
- Ginsburg, Greibach, et al.
- 1967
(Show Context)
Citation Context ...d copy are the accepting states of M 0 . 2 Obviously, the previous construction is applicable to all deterministic machine models with a decidable emptiness problem, like deterministic stack automata =-=[8]-=-, deterministic set automata [15], etc. 6 Theorem 5.8 below shows that we cannot expect to sharpen the previous theorem even from 0-d.d. decidability to 1-d.d. decidability. For (partially) blind coun... |

20 |
1979] Transductions and context-free languages, volume 4. Teubner Stuttgart
- Berstel
(Show Context)
Citation Context ...ur results in Table 1. 1 A good coverage of the corresponding literature can be found in [4]. 2 H. Fernau, K. Reinhardt, L. Staiger 2 Definitions For basics in automata theory, we refer the reader to =-=[1, 10, 11]-=-. Especially, the notion of (deterministic) push-down automaton, (D)PDA for short, should be known, leading to the language classes (D)CF; if the (D)PDA is only allowed to make one turn of the push-do... |

17 | Algebraische Codierungstheorie; Theorie der sequentiellen Codierungen - Lindner, Staiger - 1977 |

12 |
Recursive unsolvability of Post's problem of "tag" and other topics in theory of turing machines
- Minsky
- 1961
(Show Context)
Citation Context ... emptiness problem for PBLIND(n) is equivalent to the emptiness problem for PBLIND by adding an additional blank symbol replacing e-moves. 4 H. Fernau, K. Reinhardt, L. Staiger 2. According to Minsky =-=[20, 21]-=-, cf. also [11, Sec. 7.8], the halting problem for two-counter machines is undecidable, even if one takes D2C machines with only one accepting state whose counters never get below zero and to which is... |

11 |
Theory of Codes. Pure and Applied Mathematics
- Berstel, Perrin
- 1985
(Show Context)
Citation Context ...the following kind: "Let C be a class of codes. Given a device D (automaton or grammar) of some fixed type, is it decidable whether the language L(D) defined by D is in class C or not?" The =-=monograph [2]-=- contains almost no information concerning this question. Known results are surveyed in [13, Sections 3 and 9] based on [12]. In our paper, we aim to show a distinctively sharp boundary between the au... |

6 |
Transducers and the decidability of independence in free monoids. Theoretical Computer Science 134
- Jürgensen, Salomaa, et al.
- 1994
(Show Context)
Citation Context ... whether the language L(D) defined by D is in class C or not?" The monograph [2] contains almost no information concerning this question. Known results are surveyed in [13, Sections 3 and 9] base=-=d on [12]-=-. In our paper, we aim to show a distinctively sharp boundary between the automaton classes with a decidable or undecidable C-code problem, respectively, for the following code classes C: codes in gen... |

5 |
The space complexity of the unique decipherability problem
- Rytter
- 1986
(Show Context)
Citation Context ...ecomes trivial. It would be nice to know more about the (time or space) complexities of the decidable code properties; only the finite and regular code problems have received some attention until now =-=[18, 24]-=-, although complexity questions have been explicitly raised in [3]. Let us remark as an example that the prefix code problem is just as hard as the emptiness problem for say (P)BC, since L is empty if... |

4 |
Codes and infinite words
- Devolder, Latteux, et al.
- 1994
(Show Context)
Citation Context ...called suffix code if the suffix relation restricted to C is the identity. Decidability of Code Properties 5 A language C is called bifix code iff C is both a prefix and a suffix code. 4 According to =-=[2, 5, 25]-=-, a code C has a deciphering delay ms0 (m-d.d. for short) iff for all w; w 0 2 C the relation w \Delta v 1 \Delta \Delta \Delta v m v w 0 \Delta u where v 1 ; : : : ; v m 2 C and u 2 C implies w = w 0... |

4 |
On infinitary finite length codes
- Staiger
- 1986
(Show Context)
Citation Context ..., we aim to show a distinctively sharp boundary between the automaton classes with a decidable or undecidable C-code problem, respectively, for the following code classes C: codes in general, !-codes =-=[25]-=-, codes of finite and bounded deciphering delay, prefix, suffix and bifix codes, because they form (aside from suffix codes) a natural decreasing chain of code classes. Moreover, according to Berstel ... |

3 |
IIFS and codes
- Fernau
- 1992
(Show Context)
Citation Context ...very beginning of the theory of codes". 1 Furthermore, these code classes seem to be important for applications like the computation of Hausdorff dimension of language-defined fractals as propose=-=d in [6, 7, 26]-=-. The paper is structured as follows: in the next section, we present the definitions necessary for the understanding of this paper. In Section 3, so-called C-chains are introduced as a basis for seve... |

3 |
Prioritatszahlerautomaten und die Synchronisation von Halbspursprachen
- Reinhardt
- 1994
(Show Context)
Citation Context ...code. Proof. The code C constructed in the proof of Theorem 5.2 is an !-code iff C has finite deciphering delay iff C has bounded deciphering delay. 2 7 Details work like in the proof of Lemma 5.1 in =-=[22]-=-. The counter is used to check a multiplication by 2 or 3 simulating an increment by reading jj or jjj in the second configuration for every letter j in the first configuration and vice versa for a di... |

3 | Codes, simplifying words, and open set condition - Staiger - 1996 |

3 | On infinitary finite length codes. RAIRO Informatique th'eorique et - Staiger |

2 |
Parallel complexity of the regular code problem
- Litow
- 1990
(Show Context)
Citation Context ...ecomes trivial. It would be nice to know more about the (time or space) complexities of the decidable code properties; only the finite and regular code problems have received some attention until now =-=[18, 24]-=-, although complexity questions have been explicitly raised in [3]. Let us remark as an example that the prefix code problem is just as hard as the emptiness problem for say (P)BC, since L is empty if... |

1 |
Automata and codes with bounded deciphering delay
- Bruy`ere
- 1992
(Show Context)
Citation Context ...ecidability results are collected, while Section 5 contains the undecidability results. Finally, we summarize our results in Table 1. 1 A good coverage of the corresponding literature can be found in =-=[4]-=-. 2 H. Fernau, K. Reinhardt, L. Staiger 2 Definitions For basics in automata theory, we refer the reader to [1, 10, 11]. Especially, the notion of (deterministic) push-down automaton, (D)PDA for short... |

1 |
IFS and control languages
- Fernau, Staiger
- 1998
(Show Context)
Citation Context ...very beginning of the theory of codes". 1 Furthermore, these code classes seem to be important for applications like the computation of Hausdorff dimension of language-defined fractals as propose=-=d in [6, 7, 26]-=-. The paper is structured as follows: in the next section, we present the definitions necessary for the understanding of this paper. In Section 3, so-called C-chains are introduced as a basis for seve... |

1 | Set automata
- Lange, Reinhardt
- 1996
(Show Context)
Citation Context ...of M 0 . 2 Obviously, the previous construction is applicable to all deterministic machine models with a decidable emptiness problem, like deterministic stack automata [8], deterministic set automata =-=[15]-=-, etc. 6 Theorem 5.8 below shows that we cannot expect to sharpen the previous theorem even from 0-d.d. decidability to 1-d.d. decidability. For (partially) blind counters, however, we obtain an even ... |

1 |
Some properties of coding and self-adjusting automata for decoding messages (in Russian
- Levenshtejn
- 1964
(Show Context)
Citation Context ... \Delta v 0 m v w 0 \Delta v 0 1 \Delta \Delta \Delta v 0 j \Gamma1 @ w \Delta v 1 \Delta \Delta \Delta v m , whence again w 0 = w. 2 3 C-Chains In this section, following an idea of Levenshtejn (cf. =-=[16]-=- and also [17, Section 2.2.1] or [25]), we introduce a relation useful in the study of codes. For a subset C ` X + we define Levenshtejn's relation OE on X as follows. w OE 1 v :, wv 2 C ; 4 Such code... |

1 |
Codes and automata
- Restivo
- 1988
(Show Context)
Citation Context ...which it is still an open question to determine the borderline between the decidability and undecidability of the corresponding code class problems. Associated decidability questions are discussed in =-=[23]-=-. 14 H. Fernau, K. Reinhardt, L. Staiger Acknowledgment: We gratefully acknowledge discussions with our colleagues U. Hertrampf, P. Kadau and R. Stiebe and the useful comments of the referees. Referen... |

1 | Codes and in words - Devolder, Latteux, et al. - 1994 |

1 | Recursive unsolvability of Post's problem of \tag" and other topics in theory of Turing machines - Minsky - 1961 |

1 | On in length codes. RAIRO Informatique theorique et Applications/Theoretical Informatics and Applications - Staiger - 1986 |