## Equational abstractions (2003)

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Venue: | of LNCS |

Citations: | 27 - 12 self |

### BibTeX

@INPROCEEDINGS{Meseguer03equationalabstractions,

author = {José Meseguer and Miguel Palomino and Narciso Martí-oliet},

title = {Equational abstractions},

booktitle = {of LNCS},

year = {2003},

pages = {2--16},

publisher = {Springer}

}

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### Abstract

Abstract. Abstraction reduces the problem of whether an infinite state system satisfies version. The most common abstractions are quotients of the original system. We present a simple method of defining quotient abstractions by means of equations collapsing the set of states. Our method yields the minimal quotient system together with a set of proof obligations that guarantee its executability and can be discharged with tools such as those in the Maude formal environment.

### Citations

751 | Rewrite systems
- Dershowitz, Jouannaud
- 1992
(Show Context)
Citation Context ... that R = (Σ, E ∪ A, R) is executable if: 1. there exists a matching algorithm modulo the equational axioms A; 4 2. the equational theory (Σ, E ∪ A) is (ground) Church-Rosser and terminating modulo A =-=[18]-=-; and 3. the rules R are (ground) coherent [40] relative to the equations E modulo A. Conditions 1 and 2 ensure that TΣ/E∪A,k is a computable set, since each ground term t can be simplified by applyin... |

623 | Model Checking and Abstraction
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- 1994
(Show Context)
Citation Context ...r, as we will do in Section 5, by identifying some states and thereafter working with the corresponding equivalence classes. The use of the adjective “minimal” is appropriate since, as pointed out in =-=[9]-=-, M h min is the most accurate approximation to M that is consistent with h. However, it is not always can be rephrased y iff there exist a and b such that h(a) = x, h(b) = y, and a →M b. This relatio... |

597 | Counterexample-Guided Abstraction Refinement
- Clarke, Grumberg, et al.
(Show Context)
Citation Context ...e simulation of a system M by another M ′ through a surjective function h was defined and the optimal simulation Mh min was identified. The idea of simulating by quotient has been further explored in =-=[4, 5, 2, 10, 12, 7]-=- among others, although the construction in [7] requires a Galois connection instead of just a function. Theorem proving is proposed in [2] to construct the transition relation of the abstract system,... |

473 |
Conditional rewriting logic as a unified model of concurrency
- Meseguer
- 1992
(Show Context)
Citation Context ...ecification level, in which the computational system of interest is specified, and a property specification level, in which the relevant properties are specified. The main interest of rewriting logic =-=[33]-=- is that it provides a very flexible framework for the system-level specification of concurrent systems. Rewriting logic is parameterized by an underlying equational logic. In this paper we will use m... |

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...f A1 ∧ . . . ∧ An, written 4 Note that, as usual in typed logics in which types can be empty, it is necessary to keep track of the variables that can be instantiated in order to avoid inconsistencies =-=[24, 29]-=-. In particular, this situation arises in many-sorted, order-sorted, and membership equational logic [35, 5]. The notation (∀⃗x) is used in our formulas to make such variables explicit and follows typ... |

247 | Abstract interpretation of reactive systems: abstractions preserving ∀CTL
- DAMS, GRUMBERG, et al.
- 1994
(Show Context)
Citation Context ...t have been used in the literature to illustrate other abstraction methods, including a readers/writers system [11], the alternating bit protocol [13, 6, 12], a mutual exclusion protocol discussed in=-=[7]-=-, and the bounded retransmission protocol [1, 2, 6]. The abstractions were obtained simply by adding some equations to the specifications. Only in the last two cases was it necessary to add some extra... |

215 |
A new solution of dijkstra’s concurrent programming problem
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- 1974
(Show Context)
Citation Context ...mplementations, including CafeOBJ [23], ELAN [4], and Maude [11, 12]. We can illustrate rewriting logic specifications by means of an example, namely a simplified version of Lamport’s bakery protocol =-=[28]-=-. This is an infinite state protocol that achieves mutual exclusion between processes by dispensing a number to each process and serving them in sequential order according to the number they hold. A s... |

171 | Maude: specification and programming in rewriting logic - Clavel, Duran, et al. |

142 | Membership algebra as a logical framework for equational specification
- Meseguer
- 1997
(Show Context)
Citation Context ...n characteristics we now review. 4s4.1 Membership Equational Logic Membership equational logic is an expressive version of equational logic. A full account of its syntax and semantics can be found in =-=[5, 34]-=-; here we define the basic notions needed in this paper. The logic’s expressiveness is due to its rich type structure, that supports sorts, subsorts, and operator overloading, and also errors and part... |

137 | Property preserving abstractions for the verification of concurrent systems
- Loiseaux, Graf, et al.
- 1995
(Show Context)
Citation Context ...to the cases presented here we have also dealt successfully with a number of examples that have been used in the literature to illustrate other abstraction methods, including a readers/writers system =-=[31]-=- (see also [12, Chapter 12]), the alternating bit protocol [38, 14, 32], a mutual exclusion protocol discussed in [16], and the bounded retransmission protocol [1, 2, 14], which is included in Appendi... |

104 | Specification and proof in membership equational logic
- Bouhoula, Jouannaud, et al.
- 2000
(Show Context)
Citation Context ...n characteristics we now review. 4s4.1 Membership Equational Logic Membership equational logic is an expressive version of equational logic. A full account of its syntax and semantics can be found in =-=[5, 34]-=-; here we define the basic notions needed in this paper. The logic’s expressiveness is due to its rich type structure, that supports sorts, subsorts, and operator overloading, and also errors and part... |

99 | Computing abstractions of infinite state systems automatically and compositionally
- Bensalem, Lakhnech, et al.
- 1998
(Show Context)
Citation Context ... including a readers/writers system [29] (see also [12, Chapter 12]), the alternating bit protocol [36, 14, 30], a mutual exclusion protocol discussed in [16], and the bounded retransmission protocol =-=[1, 2, 14]-=-, which is included in Appendix A. The abstractions were obtained simply by adding some equations to the specifications. Only in the last two cases was it necessary to add some extra rewrite rules (al... |

86 |
Abstract and model check while you prove
- SAIDI, SHANKAR
- 1999
(Show Context)
Citation Context ...rbitrary theory interpretation H : (Σ, E ∪A) −→ (Σ ′ , E ′′ ), allowing arbitrary transformations on the data representation of states. A particular instance of this seems to be predicate abstraction =-=[14, 6]-=-. Under this approach, the abstract domain is a Boolean algebra over a set of assertions and the abstraction function, typically as part of a Galois connection, is symbolically constructed as the conj... |

73 | Experiments in theorem proving and model checking for protocol verification - Havelund, Shankar - 1996 |

67 | T.E.: Generating finite-state abstractions of reactive systems using decision procedures
- Colón, Uribe
- 1998
(Show Context)
Citation Context ...h a number of examples that have been used in the literature to illustrate other abstraction methods, including a readers/writers system [29] (see also [12, Chapter 12]), the alternating bit protocol =-=[36, 14, 30]-=-, a mutual exclusion protocol discussed in [16], and the bounded retransmission protocol [1, 2, 14], which is included in Appendix A. The abstractions were obtained simply by adding some equations to ... |

59 |
Maude: Specification and programming in rewriting logic. Theoretical Computer Science
- Clavel, Durán, et al.
- 2001
(Show Context)
Citation Context ...d by the quotient simulation. Requirements (1) and (2) are proof obligations that can be discharged by theorem proving methods. Our approach can be mechanized using the rewriting logic language Maude =-=[16]-=- and its associated LTL model checker [17], inductive theorem prover [18], Church-Rosser checker [19], and coherence checker [20]. Our present experience with case studies, involving different abstrac... |

53 | ELAN from a rewriting logic point of view
- Borovansky, Kirchner, et al.
(Show Context)
Citation Context ...12]. Under reasonable assumptions about E and R, rewrite theories are executable (more on this below). Indeed, there are several rewriting logic language implementations, including CafeOBJ [23], ELAN =-=[4]-=-, and Maude [11, 12]. We can illustrate rewriting logic specifications by means of an example, namely a simplified version of Lamport’s bakery protocol [28]. This is an infinite state protocol that ac... |

51 | The Maude LTL Model Checker
- Eker, Meseguer, et al.
- 2002
(Show Context)
Citation Context ... (1) and (2) are proof obligations that can be discharged by theorem proving methods. Our approach can be mechanized using the rewriting logic language Maude [16] and its associated LTL model checker =-=[17]-=-, inductive theorem prover [18], Church-Rosser checker [19], and coherence checker [20]. Our present experience with case studies, involving different abstractions discussed in the literature, suggest... |

50 |
Orna Grumberg, Somesh Jha, Yuan Lu, and Helmut Veith. Counterexample-guided abstraction refinement
- Clarke
- 2000
(Show Context)
Citation Context ...simulation of a system M by another M ′ through a surjective function h was defined and the optimal simulation Mh was identified. The idea of simulating by a quotient has min been further explored in =-=[10, 8, 2, 27, 30, 16]-=- among others, although the construction in [16] requires a Galois connection instead of just a function. Theorem proving is proposed in [2] to construct the transition relation of the abstract system... |

49 | Conditional Rewriting Logic as a Uni Model of Concurrency - Meseguer - 1992 |

45 | Combining model checking and deduction for I/O-automata. In Brinksma et al Ed (eds) Tools and algorithms for the construction and analysis of systems
- Müller, Nipkow
- 1995
(Show Context)
Citation Context ... also dealt successfully with a number of examples that have been used in the literature to illustrate other abstraction methods, including a readers/writers system [11], the alternating bit protocol =-=[13, 6, 12]-=-, a mutual exclusion protocol discussed in[7], and the bounded retransmission protocol [1, 2, 6]. The abstractions were obtained simply by adding some equations to the specifications. Only in the las... |

42 |
A characterisation of computable data types by means of a finite equational specification method
- Bergstra, Tucker
- 1980
(Show Context)
Citation Context ...owed in the implementation of rewriting logic in Maude under appropriate Church-Rosser and termination assumptions, but not in rewriting logic itself. However, by a metatheorem of Bergstra and Tucker =-=[3]-=-, under the conditions of the proposition it is always possible to define such inequality in an equational way. The reason not to do it here is because it is more convenient and concise to express the... |

42 | The bounded retransmission protocol must be on time - D’Argenio, Katoen, et al. - 1997 |

39 |
Mechanical Verification of Reactive Systems
- Manolios
- 2001
(Show Context)
Citation Context ...h a number of examples that have been used in the literature to illustrate other abstraction methods, including a readers/writers system [29] (see also [12, Chapter 12]), the alternating bit protocol =-=[36, 14, 30]-=-, a mutual exclusion protocol discussed in [16], and the bounded retransmission protocol [1, 2, 14], which is included in Appendix A. The abstractions were obtained simply by adding some equations to ... |

36 | A.: Symbolic verification of lossy channel systems: Application to the bounded retransmission protocol. In: Tools and Algorithms for Construction and Analysis of Systems
- Abdulla, Annichini, et al.
- 1999
(Show Context)
Citation Context ... including a readers/writers system [29] (see also [12, Chapter 12]), the alternating bit protocol [36, 14, 30], a mutual exclusion protocol discussed in [16], and the bounded retransmission protocol =-=[1, 2, 14]-=-, which is included in Appendix A. The abstractions were obtained simply by adding some equations to the specifications. Only in the last two cases was it necessary to add some extra rewrite rules (al... |

35 | Generalized rewrite theories
- Bruni, Meseguer
(Show Context)
Citation Context ... (resp. crl) for rules (resp. conditional rules) that optionally can be labeled. Rewriting logic then has the inference rules in Figure 2 to infer all the possible concurrent computations in a system =-=[33, 7]-=-, in the sense that, given two states [u], [v] ∈ TΣ/E,k, we can reach [v] from [u] by some possibly complex concurrent computation iff we can prove u −→ v in the logic; we denote this provability by R... |

32 | Control and data abstraction: The cornerstones of practical formal verification - Kesten, Pnueli - 2000 |

30 |
J.-P.: Rewrite systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...hat R = (Σ, E ∪A, R) is executable if: (1) there exists a matching algorithm modulo the equational axioms A 5 ; (2) the equational theory (Σ, E ∪ A) is (ground) Church-Rosser and terminating modulo A =-=[27]-=-; and (3) the rules R are (ground) coherent [28] relative to the equations E modulo A. Conditions (1–2) ensure that T Σ/E∪A,k is a computable set, since each ground term t can be simplified by applyin... |

27 |
Orna Grumberg. Abstract interpretation of reactive systems
- Dams, Gerth
- 1997
(Show Context)
Citation Context ...ature to illustrate other abstraction methods, including a readers/writers system [29] (see also [12, Chapter 12]), the alternating bit protocol [36, 14, 30], a mutual exclusion protocol discussed in =-=[16]-=-, and the bounded retransmission protocol [1, 2, 14], which is included in Appendix A. The abstractions were obtained simply by adding some equations to the specifications. Only in the last two cases ... |

20 | Counterexample-guided abstraction re - Clarke, Grumberg, et al. |

17 |
Equational rules for rewriting logic
- Viry
- 2002
(Show Context)
Citation Context ... exists a matching algorithm modulo the equational axioms A 5 ; (2) the equational theory (Σ, E ∪ A) is (ground) Church-Rosser and terminating modulo A [27]; and (3) the rules R are (ground) coherent =-=[28]-=- relative to the equations E modulo A. Conditions (1–2) ensure that T Σ/E∪A,k is a computable set, since each ground term t can be simplified by applying the equations E from left to right modulo A to... |

16 | Property preserving abstractions for the veri of concurrent systems - Loiseaux, Graf, et al. - 1995 |

15 | Computing abstractions of in state systems compositionally and automatically - Bensalem, Lakhnech, et al. - 1998 |

13 | Generating abstractions of reactive systems using decision procedures - Colon, Uribe - 1998 |

12 | Theoroidal maps as algebraic simulations
- Martí-Oliet, Meseguer, et al.
- 2004
(Show Context)
Citation Context ... more challenging range of examples. Also, the method itself can be generalized along several directions. For example, the equational theory extension (Σ, E ∪ A) ⊆ (Σ, E ∪ A ∪ E ′ ) is generalized in =-=[32]-=- to an arbitrary theory interpretation H : (Σ, E ∪ A) −→ (Σ ′ , E ′′ ), allowing arbitrary transformations on the data representation of states. A particular instance of this is predicate abstraction ... |

12 |
and José Meseguer. Completeness of many-sorted equational logic
- Goguen
- 1982
(Show Context)
Citation Context ...f A1 ∧ . . . ∧ An, written 4 Note that, as usual in typed logics in which types can be empty, it is necessary to keep track of the variables that can be instantiated in order to avoid inconsistencies =-=[24, 29]-=-. In particular, this situation arises in many-sorted, order-sorted, and membership equational logic [35, 5]. The notation (∀⃗x) is used in our formulas to make such variables explicit and follows typ... |

11 |
Saddek Bensalem. Property preserving abstractions for the verification of concurrent systems
- Loiseaux, Graf, et al.
- 1995
(Show Context)
Citation Context ...to the cases presented here we have also dealt successfully with a number of examples that have been used in the literature to illustrate other abstraction methods, including a readers/writers system =-=[29]-=- (see also [12, Chapter 12]), the alternating bit protocol [36, 14, 30], a mutual exclusion protocol discussed in [16], and the bounded retransmission protocol [1, 2, 14], which is included in Appendi... |

10 | A Church-Rosser checker tool for Maude equational specifications. http://maude.cs.uiuc.edu/tools
- Durán, Meseguer
- 2000
(Show Context)
Citation Context ...y theorem proving methods. Our approach can be mechanized using the rewriting logic language Maude [16] and its associated LTL model checker [17], inductive theorem prover [18], Church-Rosser checker =-=[19]-=-, and coherence checker [20]. Our present experience with case studies, involving different abstractions discussed in the literature, suggests a fairly wide applicability for this method. After summar... |

9 |
The ITP Tool
- Clavel
- 2001
(Show Context)
Citation Context ...ons that can be discharged by theorem proving methods. Our approach can be mechanized using the rewriting logic language Maude [16] and its associated LTL model checker [17], inductive theorem prover =-=[18]-=-, Church-Rosser checker [19], and coherence checker [20]. Our present experience with case studies, involving different abstractions discussed in the literature, suggests a fairly wide applicability f... |

9 | Coherence Checker and Completion Tools for Maude Specifications
- Durán
- 2000
(Show Context)
Citation Context ...ur approach can be mechanized using the rewriting logic language Maude [16] and its associated LTL model checker [17], inductive theorem prover [18], Church-Rosser checker [19], and coherence checker =-=[20]-=-. Our present experience with case studies, involving different abstractions discussed in the literature, suggests a fairly wide applicability for this method. After summarizing LTL prerequisites (Sec... |

9 |
ELAN from a rewriting logic point of view. Theor
- Borovansk´y, Kirchner, et al.
(Show Context)
Citation Context ...e conjunction must hold for a substitution instance θ before we are allowed to rewrite θ(t) to θ(t ′ ). We also assume that vars(t ′ ) ∪ vars(cond) ⊆ vars(t).language implementations, including ELAN =-=[24]-=-, CafeOBJ [25], and Maude [16]. We can illustrate rewriting logic specifications by means of a simple example, namely Lamport’s bakery protocol [26]. This is an infinite state protocol that achieves m... |

8 | Termination Checker and Knuth-Bendix Completion Tools for Maude Equational Specifications
- Durán
- 2000
(Show Context)
Citation Context ...n always try to check such a property with a tool such as Maude’s Church-Rosser checker [19]; if the check fails, we can try to complete the equations with a Knuth-Bendix completion tool, for example =-=[29, 30]-=-, to get a theory (Σ, E ′′ ∪ A) equivalent to(Σ, E ∪ A ∪ E ′ ) for which the first question has an affirmative answer. Likewise, we can try to check whether the rules R are ground coherent relative t... |

7 |
Orna Grumberg. Characterizing finite Kripke structures in propositional temporal logic
- Browne, Clarke
- 1988
(Show Context)
Citation Context ...dered, yielding another increase in generality when relating systems. Yet another direction along which our methods can be generalized is considering stuttering notions of simulation and bisimulation =-=[6, 37, 30]-=- allowing changes in the atomicity levels of transitions when relating systems. All these extensions, together with the more general representations of simulations in rewriting logic by means of equat... |

5 | Symbolic Veri of Lossy Channel Systems: Application to the Bounded Retransmission Protocol - Abdulla, Annichini, et al. - 1999 |

4 |
José Meseguer, and Ambarish Sridharanarayanan. The maude ltl model checker
- Eker
- 2002
(Show Context)
Citation Context ... 1 and 2 are proof obligations that can be discharged by theorem proving methods. Our approach can be mechanized using the rewriting logic language Maude [11, 12] and its associated LTL model checker =-=[22]-=-, inductive theorem prover [13], Church-Rosser checker [19], termination tool [21], coherence checker [20], and sufficient completeness checker [25]. Our present experience with case studies, involvin... |

3 |
Hitoshi Ohsaki. A sufficient completeness checker for linear order-sorted specifications modulo axioms
- Hendrix, Meseguer
- 2006
(Show Context)
Citation Context ...Maude [11, 12] and its associated LTL model checker [22], inductive theorem prover [13], Church-Rosser checker [19], termination tool [21], coherence checker [20], and sufficient completeness checker =-=[25]-=-. Our present experience with case studies, involving different abstractions discussed in the literature, suggests a fairly wide applicability for this method. After summarizing prerequisites on Kripk... |

3 | Maude: Speci and programming in rewriting logic. Theoretical Computer Science - Clavel, Duran, et al. - 2001 |

2 |
Adrián Riesco. Introducing the ITP tool: a tutorial
- Clavel, Palomino
(Show Context)
Citation Context ...that can be discharged by theorem proving methods. Our approach can be mechanized using the rewriting logic language Maude [11, 12] and its associated LTL model checker [22], inductive theorem prover =-=[13]-=-, Church-Rosser checker [19], termination tool [21], coherence checker [20], and sufficient completeness checker [25]. Our present experience with case studies, involving different abstractions discus... |

2 | Mechanical Veri of Reactive Systems - Manolios - 2000 |

2 | T.E.: Abstraction-Based Deductive-Algorithmic Verification of Reactive Systems - Restrepo - 1998 |