## MONA: Monadic Second-Order Logic in Practice (1995)

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Venue: | IN PRACTICE, IN TOOLS AND ALGORITHMS FOR THE CONSTRUCTION AND ANALYSIS OF SYSTEMS, FIRST INTERNATIONAL WORKSHOP, TACAS '95, LNCS 1019 |

Citations: | 125 - 19 self |

### BibTeX

@INPROCEEDINGS{Henriksen95mona:monadic,

author = {Jesper G. Henriksen and Ole J.L. Jensen and Michael E. Jørgensen and Nils Klarlund and Robert Paige and Theis Rauhe and Anders B. Sandholm},

title = {MONA: Monadic Second-Order Logic in Practice},

booktitle = {IN PRACTICE, IN TOOLS AND ALGORITHMS FOR THE CONSTRUCTION AND ANALYSIS OF SYSTEMS, FIRST INTERNATIONAL WORKSHOP, TACAS '95, LNCS 1019},

year = {1995},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

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### Abstract

The purpose of this article is to introduce Monadic Second-order Logic as a practical means of specifying regularity. The logic is a highly succinct alternative to the use of regular expressions. We have built a tool MONA, which acts as a decision procedure and as a translator to finite-state automata. The tool is based on new algorithms for minimizing finitestate automata that use binary decision diagrams (BDDs) to represent transition functions in compressed form. A byproduct of this work is a new bottom-up algorithm to reduce BDDs in linear time without hashing. The potential

### Citations

3054 | Graph-Based Algorithms for Boolean Function Manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...aightforward implementation based on explicitly representing the alphabet would only work for very simple examples. Instead, we represent the transition relation using Binary Decision Diagrams (BDDs) =-=[4, 5]-=-. In this way, the alphabet is never explicitly represented. For the external alphabet of ASCII-characters, we choose an encoding based on seven extra tracks holding the binary representation. Thus, c... |

2510 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...duplicate terminals or nonterminals. Such a BDD is just a specialized form of directed acyclic graph that has been compressed by combining structurally isomorphic nodes (see Aho, Hopcroft, and Ullman =-=[1]-=- or Section 3.4 of Cai and Paige [6]). In addition, a reduced BDD has no redundant tests [4]. Such a BDD is obtained by repeatedly pruning every internal vertex v that has both outedges leading to the... |

913 | Symbolic boolean manipulation with ordered binary-decision diagrams
- Bryant
- 1992
(Show Context)
Citation Context ...aightforward implementation based on explicitly representing the alphabet would only work for very simple examples. Instead, we represent the transition relation using Binary Decision Diagrams (BDDs) =-=[4, 5]-=-. In this way, the alphabet is never explicitly represented. For the external alphabet of ASCII-characters, we choose an encoding based on seven extra tracks holding the binary representation. Thus, c... |

486 |
Automata on infinite objects
- Thomas
- 1990
(Show Context)
Citation Context ...or automata. Together with quantification and Boolean connectives, an extraordinary succinct formalism arises. Although it has been known for thirty-five years that M2L defines regular languages (see =-=[16]-=-), the translator from formulas to automata that we describe in this article appears to be one of the first implementations. The reason such projects have not been pursued may be the staggering theore... |

296 |
An nlog n algorithm for minimizing the states in a finite automaton,” The theory of machines and computations
- Hopcroft
- 1971
(Show Context)
Citation Context ... the BDD nodes and the original automaton states, and whose alphabet were zero and one. If this derived automaton had n states, then it could be minimized in O(nlogn) steps using Hopcroft’s algorithm =-=[9]-=-. Unfortunately, such an automaton would be too big. For our purposes, the space savings due to redundant test removal is of crucial importance. But the important ‘skip’ states that arise from redunda... |

107 | A structural induction theorem for processes
- Kurshan, McMillan
- 1989
(Show Context)
Citation Context ...nite-state systems with regular communication topology. We illustrate our method by showing safety and liveness properties for a non-trivial version of the Dining Philosophers’ problem as proposed in =-=[11]-=- by Kurshan and MacMillan. We present MONA, which is our tool that translates formulas in M2L to finitestate machines. We show how BDDs can be used to overcome an otherwise inherent problem of exponen... |

58 |
An iterative approach to language containment
- Balarin, Sangiovanni-Vincentelli
- 1993
(Show Context)
Citation Context ...s and does not provide a single specification language. The work in [14] is closer in spirit to our method in that languages of finite strings are used although not as part of a logical framework. In =-=[2]-=-, another approach is given based on iterating abstractions. The parameterized Dining Philosopher’s problem is solved in [11] by a finite-state induction principle. A tool for M2L on finite, binary tr... |

52 |
Minimisation of acyclic deterministic automata in linear time
- Revuz
- 1992
(Show Context)
Citation Context ...g to the same 14block of partition P. All decision nodes of the BDD are equivalent. The coarsest stable partition Q ′ that respects P ′ for automaton D can be solved in O(m) worst case time by Revuz =-=[13]-=- and Cai and Paige [6], Sec. 3.4. Finding the equivalence classes of states in Q that point to BDD roots belonging to the same block of Q ′ (i.e., finding the coarsest partition Q that respects P) sol... |

37 | Using multiset discrimination to solve language processing problems without hashing, Theoret
- CAI, PAIGE
- 1995
(Show Context)
Citation Context ... Such a BDD is just a specialized form of directed acyclic graph that has been compressed by combining structurally isomorphic nodes (see Aho, Hopcroft, and Ullman [1] or Section 3.4 of Cai and Paige =-=[6]-=-). In addition, a reduced BDD has no redundant tests [4]. Such a BDD is obtained by repeatedly pruning every internal vertex v that has both outedges leading to the same vertex w, and redirecting all ... |

25 | Hardware Verification using Monadic SecondOrder Logic
- Basin, Klarlund
(Show Context)
Citation Context ...as three states and accepts the language (1 ∪ 10) ∗ ,which is the regular expression that one would obtain by reasoning about the circuit. For more advanced applications to hardware verification, see =-=[3]-=-. 4.3. Equivalence testing. A closed formula φ is a tautology if L(φ) =L(Σ ∗ ), i.e. if all strings over Σ satisfy φ. The equivalence of formulas φ and ψ then amounts to whether φ ⇔ ψ is a tautology. ... |

22 | Reduction of OBDDs in linear time - Sieling, Wegener - 1993 |

16 | Parametric circuit representation using inductive boolean functions
- Gupta, Fisher
- 1993
(Show Context)
Citation Context ...inimization algorithm works very fast in practice thanks to a simple generalization of the unary apply operation of BDDs. Comparisons to other work. Parameterized circuits are described using BDDs in =-=[8]-=-. This method relies on formulating inductive steps as finite-state devices and does not provide a single specification language. The work in [14] is closer in spirit to our method in that languages o... |

16 | An n log n algorithm for minimizing states in a nite automaton - Hopcroft |

14 |
Automatic generation of network invariants for the verification of iterative sequential systems
- Rho, Somenzi
- 1987
(Show Context)
Citation Context ...rk. Parameterized circuits are described using BDDs in [8]. This method relies on formulating inductive steps as finite-state devices and does not provide a single specification language. The work in =-=[14]-=- is closer in spirit to our method in that languages of finite strings are used although not as part of a logical framework. In [2], another approach is given based on iterating abstractions. The para... |

12 |
Three Efficient Algorithms Based on Partition Refinement
- Paige, Tarjan
- 1987
(Show Context)
Citation Context ...block is first created. The index of the first node placed in a newly created block is the index for that block. Work is in progress for exploring the “processing the smaller half” idea found in e.g. =-=[12]-=-. We should mention, however, that the current implementation of the minimization algorithm in practice seems to run faster than the procedures for constructing product and subset automata. MONA featu... |

8 | Automata on innite objects - Thomas - 1990 |

5 |
Symbolic model checking and constraint logic programming: a cross-fertilisation
- Corsini, Rauzy
(Show Context)
Citation Context ... by a finite-state induction principle. A tool for M2L on finite, binary trees has been developed at the University of Kiel [15]. Apparently, this tool has only been used for very simple examples. In =-=[7]-=-, a programming language for finite domains based on a fixed point logic is described and used for verification of non-parameterized finite systems. Contents. In Section 2, we explain the syntax and s... |

4 |
Übersetzung von logischen Ausdrücken in Baumautomaten: Entwicklung eines Verfahrens und seine Implementierung. Unpublished
- Steinmann
- 1993
(Show Context)
Citation Context ...stractions. The parameterized Dining Philosopher’s problem is solved in [11] by a finite-state induction principle. A tool for M2L on finite, binary trees has been developed at the University of Kiel =-=[15]-=-. Apparently, this tool has only been used for very simple examples. In [7], a programming language for finite domains based on a fixed point logic is described and used for verification of non-parame... |

2 | Monadic second-order logic for parameterized verification
- Jensen, Jrgensen, et al.
- 1994
(Show Context)
Citation Context ...y either • may be overlooked, as in the case of parameterized verification of distributed finite-state systems with a regular communication topology; or 1 This article is a heavily revised version of =-=[10]-=-. 2 BRICS, Centre of the Danish National Research Foundation for Basic Research in Computer Science, Department of Computer Science, University of Aarhus. 3 The corresponding author is Nils Klarlund, ... |

1 | Hardware veri��cation using monadic second-order logic - Basin, Klarlund - 1995 |

1 | Three eOEcient algorithms based on partition re��nement - Paige, Tarjan - 1987 |

1 | Automatic generation of network invariants for the veri��cation of iterative sequential systems - Rho, Somenzi - 1993 |

1 | bersetzung von logischen Ausdr#cken in Baumautomaten: Entwicklung eines Verfahrens und seine Implementierung. Unpublished - Steinmann - 1993 |