## Reasoning about The Past with Two-Way Automata (1998)

Citations: | 134 - 12 self |

### BibTeX

@INPROCEEDINGS{Vardi98reasoningabout,

author = {Moshe Y. Vardi},

title = {Reasoning about The Past with Two-Way Automata},

booktitle = {},

year = {1998},

pages = {628--641},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

The µ-calculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the µ-calculus is EXPTIME-complete. This upper bound, however, is known for a version of the logic that has only forward modalities, which express weakest preconditions, but not backward modalities, which express strongest postconditions. Our main result in this paper is an exponential time upper bound for the satisfiability problem of the µ-calculus with both forward and backward modalities. To get this result we develop a theory of two-way alternating automata on infinite trees.

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Citation Context ...spond to reasoning about the past. There is now a significant body of evidence of the usefulness of reasoning about the past in the context of program correctness [19,27]. For example, it is shown in =-=[32,33]-=- that past temporal connective can conveniently replace history variables in compositional verification. Backward modalities also have a counterpart in description logics, where they correspond to inv... |

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Citation Context ..., leading to industrial acceptance of computer-aided verification [1]. More recently, the ��-calculus has found a new application domain in the theory of description logics in Artificial Intellige=-=nce [14]. As-=- a result of this prominence, the ��-calculus has been the subject of extensive research; in particular, researchers focused on the truth problem and the satisfiability problem. In the truth probl... |

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Citation Context ...ce. For a point w E W and a sentence T, we say that 9 holds at w in K, denoted K, w ~ 9 iff w E 9 K. 2.2 Alternating Tree Automata For an introduction to the theory of automata oil infinite trees see =-=[33]-=-. An infinite tree is a set T C IN+, such that if x 9 c E T where x E IN* and c E IN, then also x E T, and, if the tree is full, then also x 9 d E T for all 0 < c I < c, (Here we use IN to denote the ... |

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Citation Context ...ted to equivalent one-way nondeterministic tree automata with an exponential blowup. The emptiness problem can then be solved by using known algorithms for emptiness of nondeterministic tree automata =-=[9,26,34]. Th-=-is yields an exponential time upper bound for the emptiness problem for alternating tree automata, resulting in a bound of the same complexity for satisfiability of the full ��-calculus. 2 Prelimi... |

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Citation Context ... The ��-calculus has gained further prominence with the discovery that its formulas can be evaluated symbolically in a natural way [4], leading to industrial acceptance of computer-aided verificat=-=ion [1]. More r-=-ecently, the ��-calculus has found a new application domain in the theory of description logics in Artificial Intelligence [14]. As a result of this prominence, the ��-calculus has been the su... |

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Citation Context ...conditions. Backward modalities correspond to reasoning about the past. There is now a significant body of evidence of the usefulness of reasoning about the past in the context of program correctness =-=[19,27]-=-. For example, it is shown in [32,33] that past temporal connective can conveniently replace history variables in compositional verification. Backward modalities also have a counterpart in description... |

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(Show Context)
Citation Context ... has been established as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics such as PDL [13], process logics suc=-=h as YAPL [50], an-=-d temporal logics such as CTL* [8]. The ��-calculus has gained further prominence with the discovery that its formulas can be evaluated symbolically in a natural way [4], leading to industrial acc... |

22 |
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(Show Context)
Citation Context ...ies, since they are one-way automata. To deal with backward modalities we introduce two-way alternating automata on infinite trees, based on an analogous notion of two-way automata on finite trees in =-=[38]-=-. It remains then to solve the emptiness problem for two-way alternating tree automata. Alternating tree automata can be viewed as infinite games [29]; this holds for both one-way and two-ways automat... |

22 |
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Citation Context ...e-model property for the full ��-calculus. Note that it follows from [39] that the finite-model property does not hold for the full ��-calculus; in contrast, it does hold for the standard ��=-=��-calculus [23]-=-. The tree-model property asserts that if a sentence is satisfiable then it is satisfiable by a bounded-degree infinite tree structure. A tree structure is a Kripe structure hW; R; Li where W is a tre... |

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(Show Context)
Citation Context ...propositional variables, a set Prog of atomic programs, and a subset DProg of Prog of deterministic atomic programs. Deterministic atomic programs correspond to functional roles in description logics =-=[15]-=-. Determinism plays a role in the semantics, but not in the syntax. As we will see, having to deal with deterministic programs introduces an additional difficulty that we will have to overcome in orde... |

14 |
Alternating Automata: Unifying Truth and Validity Checking for Temporal Logics
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(Show Context)
Citation Context ... formulas to nondeterministic automata is nontrivial; for example, the translation in [51] is exponential and consists of a sequence of successive translations. As argued in [28] and then utilized in =-=[2,47]-=-, it is easier to translate formulas to alternating automata. Alternating tree automata generalize nondeterministic tree automata by allowing multiple successor states to go down along the same branch... |

13 |
Exponential determinization for !-automata with strong-fairness acceptance condition
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(Show Context)
Citation Context ...f the acceptance condition of B 0 is still linear in the size of the acceptance condition of A. We now co-determinize B 0 , i.e., determinize it and complement it in a singly-exponential construction =-=[36,37,43]-=- to obtain a deterministic parity word automaton B 00 that rejects violating downward paths. The deterministic tree automaton B 000 is obtained by simply running B 00 in parallel over all branches of ... |

11 |
A decision procedure for the propositional -calculus
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(Show Context)
Citation Context ...tified. An exponential time lower time bound follows from the lower bound for PDL in [13], and an exponential time upper time bound was shown in [9], following a sequence of improving upper bounds in =-=[24,40,49]. Th-=-e exponential time upper bound for the ��-calculus was shown, however, only for a version of the logic that has only forward modalities. The formula hai' holds in a state s of a Kripke structure M... |

8 |
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Citation Context ...y problem for the full ��-calculus is decidable in exponential time. 5 Concluding Remarks Over the last few years there has been a significant interest in logics with bounded number of variables (=-=see [6,21]-=-). One surprising result in this line of research is that while FO 3 , 3-variable first-order logic, is already undecidable, the satisfiability problem for FO 2 , 2-variable first-order logic, is NEXP... |

8 |
What makes modal logic so robustly decidable
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(Show Context)
Citation Context ...he automata-theoretic approach advocated in [9,39,51]. We first show that even though the full ��-calculus does not have the finite-model property, it does have the tree-model property. (As argued=-= in [48]-=-, the tree-model property, which asserts that if a formula is satisfiable then it is satisfiable by a bounded-degree infinite tree structure, offers an explanation for the robust decidability of many ... |

6 |
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Citation Context ...ulus is a propositional modal logic augmented with least and greatest fixpoint operators. It was introduced in [22], following earlier studies of fixpoint calculi in the theory of program correctness =-=[7,30,35]. Over the pas-=-t 15 years, the ��-calculus has been established as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics such ... |

5 |
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Citation Context ...to solve the emptiness problem for two-way alternating tree automata. Alternating tree automata can be viewed as infinite games [29]; this holds for both one-way and two-ways automata. It is shown in =-=[20]-=- that under certains conditions, which hold here, the winning player has a memoryless strategy in these games. We use this to show that two-way alternating tree automata can be translated to equivalen... |