## Tree Automata with Equality Constraints Modulo Equational Theories (2006)

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Venue: | Proceedings of 3rd International Joint Conference on Automated Reasoning, IJCAR, volume 4130 of Lecture Notes in Artificial Intelligence |

Citations: | 16 - 3 self |

### BibTeX

@INPROCEEDINGS{Jacquemard06treeautomata,

author = {Florent Jacquemard and Michael Rusinowitch and Laurent Vigneron},

title = {Tree Automata with Equality Constraints Modulo Equational Theories},

booktitle = {Proceedings of 3rd International Joint Conference on Automated Reasoning, IJCAR, volume 4130 of Lecture Notes in Artificial Intelligence},

year = {2006},

pages = {557--571},

publisher = {Springer}

}

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### Abstract

Abstract. This paper presents new classes of tree automata combining automata with equality test and automata modulo equational theories. We believe that these classes have a good potential for application in e.g. software verification. These tree automata are obtained by extending the standard Horn clause representations with equational conditions and rewrite systems. We show in particular that a generalized membership problem (extending the emptiness problem) is decidable by proving that the saturation of tree automata presentations with suitable paramodulation strategies terminates. Alternatively our results can be viewed as new decidable classes of first-order formula. 1

### Citations

752 | Rewrite systems - Dershowitz, Jouannaud - 1990 |

278 | Mobile values, new names, and secure communication
- Abadi, Fournet
- 2001
(Show Context)
Citation Context ... as this rule for the elimination of stuttering in lists: cons(x, cons(x, y)) → cons(x, y) are monadic rewrite rules. Sublinear and collapsing rewrite rules permit to describe cryptographic functions =-=[1]-=-, like decryption in a symmetric cryptosystem dec ( enc(x, y), y ) → x (the symbols enc and dec stand for encryption and decryption and the variables x and y correspond respectively to the encrypted p... |

95 |
E.: Logic programs as types for logic programs
- Frühwirth, Shapiro, et al.
- 1991
(Show Context)
Citation Context ...acquemard, Rusinowitch, Vigneron positions of [3] and the reduction automata of [7]. This second class captures in particular languages of terms (ir)reducible by non linear rewrite systems. Following =-=[11]-=-, it is classical to represent tree automata by Horn clause sets. In this setting, a recognized language is defined as a least Herbrand model and it is possible to use classical first-order theorem pr... |

67 | Basic paramodulation
- Bachmair, Ganzinger, et al.
- 1995
(Show Context)
Citation Context ...all 1 ≤ i ≤ n; or 2. f ∼ g and, for some j, we have (s1, . . . , sj−1) = (t1, . . . , tj−1), sj ≻lpo tj and s ≻lpo tk, for all k with j < k ≤ n; or 3. sj ≻lpo t, for some j with 1 ≤ j ≤ m. Then as in =-=[2]-=- we identify a positive literal s = t with the multiset {{s}, {t}}, and a negative literal s ̸= t with the multiset {{s, t}}. Then we extend the ordering ≻lpo (resp. ≻lpo) to literals by taking the tw... |

57 | Rewriting for Cryptographic Protocol Verification
- Genet, Klay
(Show Context)
Citation Context ...rder formula. 1 Introduction Combining tree automata and term rewriting systems (TRS) has been successful in domains like automated theorem proving [6] and verification of infinite state systems e.g. =-=[12,18,16]-=-. A problem with such approaches is to extend the decidability results on tree automata languages to equivalence classes of terms modulo an equational theory. Some authors, e.g. [26,20], have investig... |

42 | overloading is dexptime-complete
- Seidl, Haskell
- 1994
(Show Context)
Citation Context ...uch that for all i ≥ n, tiσ ∈ L(A, Qi)? When t1 = . . . = tn = x (a variable), GIP is equivalent to the problem of non-emptiness of intersection of tree automata, which is known to be EXPTIMEcomplete =-=[22]-=-. An inclusion problem L(A, P) ⊆ L(B, Q) is a particular case of GIP when B belong to a class of TA closed under complementation: in this case, inclusion can be expressed as GIP for P, Q and t1 = t2 =... |

40 |
Equality and Disequality Constraints on Direct Subterms in Tree Automata
- Bogaert, Tison
- 1992
(Show Context)
Citation Context ...raints restricted to brother ⋆ This work has been partially supported by the research projects RNTL PROUVÉ (No 03V360) and ACI–SI SATIN and ROSSIGNOL.2 Jacquemard, Rusinowitch, Vigneron positions of =-=[3]-=- and the reduction automata of [7]. This second class captures in particular languages of terms (ir)reducible by non linear rewrite systems. Following [11], it is classical to represent tree automata ... |

37 | Splitting without Backtracking
- Riazanov, Voronkov
- 2001
(Show Context)
Citation Context ...ve equational clauses are applied as in [2].Tree automata with equality constraints modulo equational theories 7 Splitting. We shall use ε-splitting [13], a variant of splitting without backtracking =-=[21]-=-. B, Γ ⇒ H�θ� εsplit B ⇒ qB�θ� qB, Γ ⇒ H�θ� where the literals of Γ ∪H are not equational, Bθ is an ε-block, i.e. a set of literals of the form Q1(x), . . . , Qn(x), with Q1, . . .,Qn ∈ P, x is a vari... |

25 |
The authentication protocol
- Bull, Otway
- 1997
(Show Context)
Citation Context ...esenting B (with i = 0, 1 and t = adec(adec(snd(x), inv(pub(B))), pub(fst(x)))): Qi0(x), snd(fst(t)) = xB, fst(fst(t)) = fst(x), ⇒ Qi1(enc(S, snd(t))) Example 7. The recursive authentication protocol =-=[4]-=- ensures the distribution of certified session keys to a group of clients by a server which process recursively an unbounded list of requests. The automated verification of such group protocols has be... |

20 |
Tree Automata Techniques and Applications. http://tata.gforge.inria.fr
- Comon, Dauchet, et al.
- 2007
(Show Context)
Citation Context ...ts can be viewed as new decidable classes of first-order formula. 1 Introduction Combining tree automata and term rewriting systems (TRS) has been successful in domains like automated theorem proving =-=[6]-=- and verification of infinite state systems e.g. [12,18,16]. A problem with such approaches is to extend the decidability results on tree automata languages to equivalence classes of terms modulo an e... |

20 | H.: Normalizable horn clauses, strongly recognizable relations, and spi
- Nielson, Nielson, et al.
- 2002
(Show Context)
Citation Context ...rder formula. 1 Introduction Combining tree automata and term rewriting systems (TRS) has been successful in domains like automated theorem proving [6] and verification of infinite state systems e.g. =-=[12,18,16]-=-. A problem with such approaches is to extend the decidability results on tree automata languages to equivalence classes of terms modulo an equational theory. Some authors, e.g. [26,20], have investig... |

13 |
J.L.: Automata for reduction properties solving
- Dauchet, Caron, et al.
- 1995
(Show Context)
Citation Context ...presents a halting computation of M. ⊓⊔ 6.2 Tree Automata with Equational Constraints We propose here the definition of a new class of tree automata where the constraints are generalized (compared to =-=[7]-=-) to equations between arbitrary terms and where the transitions comply to stronger ordering conditions, based on the ordering ≻ on states, in order to obtain a decidable GIP. We call below test predi... |

10 | Set-based analysis for logic programming and tree automata
- Devienne, Talbot, et al.
- 1997
(Show Context)
Citation Context ...ar and quadratic. GMP for a linear term can be decided by a procedure of the same quadratic time complexity. For a non-linear term,8 Jacquemard, Rusinowitch, Vigneron the problem is EXPTIME-complete =-=[10]-=-. We sketch below a slight variation of a DEXPTIME procedure of [13] in our framework, in order to introduce the principles of the proofs in the next sections. It is based on the function sel1 which s... |

10 | Automata-Based Analysis of Recursive Cryptographic Protocols
- Küsters, Wilke
(Show Context)
Citation Context ...rder formula. 1 Introduction Combining tree automata and term rewriting systems (TRS) has been successful in domains like automated theorem proving [6] and verification of infinite state systems e.g. =-=[12,18,16]-=-. A problem with such approaches is to extend the decidability results on tree automata languages to equivalence classes of terms modulo an equational theory. Some authors, e.g. [26,20], have investig... |

10 | Decidability and Closure Properties of Equational Tree Languages
- Ohsaki, Takai
- 2002
(Show Context)
Citation Context ...ystems e.g. [12,18,16]. A problem with such approaches is to extend the decidability results on tree automata languages to equivalence classes of terms modulo an equational theory. Some authors, e.g. =-=[26,20]-=-, have investigated the problem of emptiness decision for tree automata modulo specific equational theories, e.g. A, AC, ACU. . . Moreover, it is also shown in [20] that emptiness is decidable for any... |

10 | Layered transducing term rewriting system and its recognizability preserving property
- Seki, Takai, et al.
- 2002
(Show Context)
Citation Context ...hown in [20] that emptiness is decidable for any linear equational theory, and results about regularity preservation under rewriting have been established for several general classes of TRS (see e.g. =-=[23]-=- § 2.3). Another important difficulty stems from the non linear variables (variables with multiple occurrences) in the rewrite rules, which impose in general some over-approximations of the rewrite re... |

10 |
Selecting theories and recursive protocols., in
- Truderung
(Show Context)
Citation Context ...asic ordered paramodulation terminates. ⊓⊔ Note that the expanded form of the above clause Q1(ℓ1), . . . , Qn(ℓn) ⇒ Q(y) is related to the push clauses of two-ways automata [26] or selecting theories =-=[25]-=-. We will come back to this remark in Example 7 showing how the approach for protocol verification of this last paper can be carry on by TACE. Corollary 2. GIP is decidable for TAE modulo a ≻-converge... |

9 |
Deciding H1 by resolution
- Goubault-Larrecq
- 2005
(Show Context)
Citation Context ...use sets. In this setting, a recognized language is defined as a least Herbrand model and it is possible to use classical first-order theorem proving techniques in order to establish decision results =-=[26,13]-=-. In this paper, we follow this approach in order to unify the two problems mentioned above: we show how techniques of basic ordered paramodulation with selection and a variant of splitting without ba... |

9 | Unification in Extensions of Shallow Equational Theories
- Jacquemard, Meyer, et al.
- 1998
(Show Context)
Citation Context ...resented as follows in the paper: E = ∅ E TA Section 4 Section 5 TAC Section 6 Section 7 The last result (lower right corner of the table) is to our knowledge one of the first decision results (after =-=[14]-=-) concerning tree automata with equality constraints modulo equational theories. We show that emptiness is undecidable for TA extended with non-linear facts, even with only one state. Unlike stated in... |

8 |
On the relative soundness of the free algebra model for public key encryp
- Lynch, Meadows
- 2005
(Show Context)
Citation Context ...used to characterize the behaviour of security protocols running in an insecure environment, following a model with explicit destructors [1] specified with the rewrite rules of Example 2. It is known =-=[17]-=- that such model with rewrite rules is more expressive than a standard model of cryptosystems based on free algebras. For instance, the attack mentioned in Example 6 cannot be captured by free algebra... |

5 |
Tree automata and term rewrite systems
- Tison
- 2000
(Show Context)
Citation Context ...ee automata B such that L(B, Q) is the complement of L(B, Q) in T (F). The General Membership Problem (GMP, [15]) is the particular case of GIP where n = 1. This problem was shown EXPTIME-complete in =-=[24]-=- for standard tree automata. When t1 is a ground term, GMP is equivalent to a membership problem for A: t ∈ L(A, Q)? When t is a variable, GMP is equivalent to a non-emptiness problem for A: L(A, Q) ̸... |

4 |
Rewrite systems, chapter Handbook
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...ion). A subterm of t at position p is denoted t|p, and the replacement in t of the subterm at position p by u denoted t[u]p. Rewriting. We assume standard definitions and notations for term rewriting =-=[9]-=-. A term rewriting system (TRS) is a finite set of rewrite rules ℓ → r, where ℓ ∈ T (F, X) and r ∈ T (F,vars(ℓ)). A term t ∈ T (F, X) rewrites to s by a TRS R, denoted by t →R s, if there is a rewrite... |

1 |
Paramodulation-Based Theorem Proving, chapter Handbook of Automated Reasoning
- Nieuwenhuis, Rubio
- 2001
(Show Context)
Citation Context ...ty clause) calculus for Horn clauses called basic ordered 4 We shall sometimes underline literals to indicate that they are selected.6 Jacquemard, Rusinowitch, Vigneron paramodulation with selection =-=[2, 19]-=-. Γ ⇒ ℓ = r �θ� Γ ′ ⇒ u[ℓ ′ ]p = v �θ ′ � RP Γ, Γ ′ ⇒ u[x]p = v �θ, θ ′ , ℓ ′ = ℓ, x = r� if x is fresh, and (i) ℓ ′ /∈ X, (ii) no literal is selected in Γ and Γ ′, (iii) and (v) hold. Γ ⇒ ℓ = r �θ� Γ... |

1 |
Two-Way Equational Tree Automata
- Verma
- 2003
(Show Context)
Citation Context ...ystems e.g. [12,18,16]. A problem with such approaches is to extend the decidability results on tree automata languages to equivalence classes of terms modulo an equational theory. Some authors, e.g. =-=[26,20]-=-, have investigated the problem of emptiness decision for tree automata modulo specific equational theories, e.g. A, AC, ACU. . . Moreover, it is also shown in [20] that emptiness is decidable for any... |

1 | Paramodulation-Based Theorem Proving, Elsevier Science and - Nieuwenhuis, Rubio |