## Reasoning about Infinite Computations (1994)

Venue: | Information and Computation |

Citations: | 265 - 58 self |

### BibTeX

@ARTICLE{Vardi94reasoningabout,

author = {Moshe Y. Vardi and Pierre Wolper},

title = {Reasoning about Infinite Computations},

journal = {Information and Computation},

year = {1994},

volume = {115},

pages = {1--37}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate extensions of temporal logic by connectives defined by finite automata on infinite words. We consider three different logics, corresponding to three different types of acceptance conditions (finite, looping and repeating) for the automata. It turns out, however, that these logics all have the same expressive power and that their decision problems are all PSPACE-complete. We also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem. 1 Introduction For many years, logics of programs have been tools for reasoning about the input/output behavior of programs. When dealing with concurrent or nonterminating processes (like operating systems) there is, however, a need to reason about infinite computations. Thus, instead of considering the first and last states of finite computations, we need to consider the infinite sequences of states that the program goes through...

### Citations

1434 |
A Calculus of Communicating Systems
- Milner
- 1980
(Show Context)
Citation Context ... !) are known to be expressively equivalent to star-free !-regular languages [Lad77, Tho79, Tho81]. On the other hand, !-regular sequences are a natural way of describing concurrent processes [Sh79], =-=[Mi80]-=-; and furthermore, the ability to describe !-regular sequences is crucial to the task of program verification [LPZ85]. There are several different ways to extend the expressive power of PTL. We could ... |

1318 |
The temporal logic of programs
- Pnueli
- 1977
(Show Context)
Citation Context ...ring the first and last states of finite computations, we need to consider the infinite sequences of states that the program goes through. Logics to reason about such sequences include temporal logic =-=[Pn77]-=- and temporal-logic-based process logicss[Ni80, HKP80]. In the propositional case, computations can be viewed as infinite sequences of propositional truth assignments. For reasoning about individual p... |

899 | Dynamic Logic [M - Harel, Kozen, et al. - 2000 |

626 |
An automata-theoretic approach to automatic program verification
- Vardi, Wolper
- 1986
(Show Context)
Citation Context ...ormulas easier, we define more specialized classes of automata: subword automata and set-subword automata. They are the specialization to words of subtree automata and set-subtree automata defined in =-=[VW86b]-=-. For completeness sake, we give a detailed treatment of this specialization. Intuitively, a subword automaton checks that, starting at every position in an infinite word, there is a finite word accep... |

517 | Automata on Infinite Objects - Thomas - 1990 |

380 |
Propositional dynamic logic of regular programs
- Fischer, Ladner
- 1979
(Show Context)
Citation Context ... the differences that occur when dealing with ETL l . We first need to define the notion of the closure of an ETL f formula g, denoted cl(g). It is similar in nature to the closure defined for PDL in =-=[FL79]-=-. From now on we identify a formula ::g 0 with g 0 . Given an automaton A = (\Sigma; S; ae; S 0 ; F ), for each s 2 S we define A s to be the automaton (\Sigma; S; ae; fsg; F ). The closure cl(g) of a... |

368 | On the synthesis of a reactive module - Pnueli, Rosner - 1989 |

366 |
On a decision method in restricted second order arithmetic
- Büchi
- 1960
(Show Context)
Citation Context ... to have different expressive powers. One of the main results of this paper is that all these logics are expressively equivalent. They all have the expressive power of !-regular expressions, which by =-=[Bu62]-=- is the same as that of the monadic second-order theory of (N ; !), usually denoted S1S. We also consider the complexity of the decision problem for ETL f and ETL l . It turns out that both logics hav... |

348 |
The complexity of propositional linear temporal logics
- Sistla, Clarke
- 1985
(Show Context)
Citation Context ...d to see that the construction of A g satisfies the conditions of Lemma 2.5. Combining this with the fact that satisfiability for propositional temporal logic (a restriction of ETL f ) is PSPACE-hard =-=[SC85]-=-, we have: Theorem 3.6: The satisfiability problem for ETL f is logspace complete for PSPACE. Let us now consider ETL l . The construction proceeds similarly to the one for ETL f . The first differenc... |

336 | Proving the correctness of multiprocess programs
- Lamport
- 1977
(Show Context)
Citation Context ...hree conditions in [Lan69]. It is known that these three conditions exhaust all the possibilities in the Landweber classification (cf. [Wa79]). 4 The notions of safety and liveness are due to Lamport =-=[Lam77]-=-. Our notion of liveness here corresponds to the notion of guarantee in [MP89]. 5 In [SVW87] it is shown that the decision problem for ETL r is also PSPACE-complete. This result requires significantly... |

274 |
Results on the propositional -calculus
- Kozen
- 1983
(Show Context)
Citation Context ...l dynamic logic in general. It is known that PDL is less expressive than PDL+ loop (Pratt, see [HS82]), which is less expressive than PDL + repeat [HS82], which is less expressive than PDL + fixpoint =-=[Ko83]-=-, which can be shown to be less expressive than PDL+ quantification. Also, the last language can even be shown to be highly undecidable (\Pi 1 1 \Gamma complete). It should be noted that the automata-... |

252 |
Finite automata and their decision problems
- Rabin, Scott
- 1959
(Show Context)
Citation Context ...sted within another automata connective, and complementing automata, when we have a negated automata connective. Thus, the construction can be viewed as combining the classical subset construction of =-=[RS59] and Choue-=-ka's "flag construction" in [Ch74]. This should be contrasted with the treatment of ETL r in [SVW87], where a special construction is needed to complement Buchi automata. As a result, the si... |

243 |
On the Temporal Analysis of Fairness
- Gabbay, Pnueli, et al.
- 1980
(Show Context)
Citation Context ...53-802, 650 Harry Rd., San Jose, CA 95120-6099,USA, vardi@almaden.ibm.com z Address: Institut Mont'efiore, B28, Universit'e de Li`ege, B-4000 Li`ege Sart-Tilman, Belgium, pw@montefiore.ulg.ac.be 1 In =-=[GPSS80]-=- propositional temporal logic (PTL) was shown to be expressively equivalent to the monadic first-order theory of (N ; !), the natural numbers with the less-than relation. This was taken as an indicati... |

238 | Automatic Verification of Probabilistic Concurrent Finite-State Programs - Vardi - 1985 |

228 | Semantical considerations on Floyd-Hoare Logic - Pratt - 1976 |

220 | Real-Time Logics: Complexity and Expressiveness - Alur, Henzinger - 1992 |

212 |
Temporal logic can be more expressive
- Wolper
- 1983
(Show Context)
Citation Context ...damental notion of regularity of sets of event sequences is not first-order; even the simple assertion that "the proposition p holds at least in every other state on a path" is not expressib=-=le in PTL [Wo83]-=-. In fact, PTL and the first-order theory of (N ; !) are known to be expressively equivalent to star-free !-regular languages [Lad77, Tho79, Tho81]. On the other hand, !-regular sequences are a natura... |

210 |
Automata-theoretic techniques for modal logics of programs
- Vardi, Wolper
- 1986
(Show Context)
Citation Context ...plexity of ETL r see Section 3. 6 The automata-theoretic approach described here can be viewed as a specialization of the automatatheoretic approach to decision problems of dynamic logic described in =-=[VW86a]-=- (see also [ES84, St82]). Note, however, that while the tree automata constructed in [ES84, St82, VW86a] accept only some models of the given formulas, the automata constructed here accept all models ... |

174 | Synthesis of Communicating Processes from Temporal Logic Specifications - Manna, Wolper - 1984 |

144 |
The glory of the past
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- 1985
(Show Context)
Citation Context ...d, !-regular sequences are a natural way of describing concurrent processes [Sh79], [Mi80]; and furthermore, the ability to describe !-regular sequences is crucial to the task of program verification =-=[LPZ85]-=-. There are several different ways to extend the expressive power of PTL. We could add some non-first-order construct, such as least-fixpoint or second-order quantification, but we prefer here to add ... |

137 |
P.: The complementation problem for Büchi automata with applications to temporal logic
- Sistla, Vardi, et al.
- 1987
(Show Context)
Citation Context ...ities in the Landweber classification (cf. [Wa79]). 4 The notions of safety and liveness are due to Lamport [Lam77]. Our notion of liveness here corresponds to the notion of guarantee in [MP89]. 5 In =-=[SVW87]-=- it is shown that the decision problem for ETL r is also PSPACE-complete. This result requires significantly more complicated automata-theoretic techniques that are beyond the scope of this paper. For... |

130 |
Characterizing correctness properties of parallel programs using fixpoints
- Emerson, Clarke
- 1980
(Show Context)
Citation Context ...again nonelementary. The extension of PTL closest to the one discussed in this paper is probably that of the temporal -calculus, in which PTL is extended with fixpoint operators. It was introduced in =-=[EC80]-=-; see also [BKP85, BKP86]. It has the same expressive power as ETL and, as is shown in [Va88], also has a PSPACE-complete decision problem (see also [BB89]). It is interesting to note however that thi... |

97 |
Testing and generating infinite sequences by a finite automaton
- McNaughton
- 1966
(Show Context)
Citation Context ...ata. These last two classes are incomparable. The languages accepted by Buchi (repeating acceptance) automata are often called the !-regular languages. By results of Buchi [Bu62] (see also McNaughton =-=[McN66]-=-), this class of languages is closed under union, intersection and complementation and is equivalent to the class of languages describable in the monadic second-order theory of one successor (S1S) and... |

86 |
Models of program logics
- Pratt
- 1979
(Show Context)
Citation Context ...VW87] for ETL r are exponential, the exponent for ETL f and ETL l is linear, while it is nonlinear for ETL r . It is interesting the compare our technique here to Pratt's model construction technique =-=[Pr79]-=-. There one starts by building a maximal model, and then one eliminates states whose eventualities are not satisfied. Our local automata correspond to those maximal models. However, instead of elimina... |

84 |
A temporal fixpoint calculus, in
- Vardi
- 1988
(Show Context)
Citation Context ...bly that of the temporal -calculus, in which PTL is extended with fixpoint operators. It was introduced in [EC80]; see also [BKP85, BKP86]. It has the same expressive power as ETL and, as is shown in =-=[Va88]-=-, also has a PSPACE-complete decision problem (see also [BB89]). It is interesting to note however that this last result was obtained using an extension of the automata-theoretic techniques presented ... |

83 |
Finite automata: Behavior and synthesis
- Trakhenbrot, Barzdin
- 1973
(Show Context)
Citation Context ...if there are a finite word w = w 1 : : : w k in \Sigma ? and a finite sequence s 0 ; : : : ; s k of states in S such that s 0 = s, s k = t, and s i+1 2 ae(s i ; w i+1 ) for 0sisk \Gamma 1. Lemma 2.3: =-=[TB73]-=- A Buchi automaton accepts some word iff there is an accepting state of the automaton that is reachable from some initial state and is reachable from itself. We can now prove the following: Theorem 2.... |

80 |
Space-bounded reducibility among combinatorial problems
- Jones
- 1975
(Show Context)
Citation Context ...m requires only logarithmic space. To show NLOGSPACE hardness, given Lemma 2.3, it is straightforward to construct a reduction from the graph accessibility problem, proved to be NLOGSPACE complete in =-=[Jo75]-=-. To solve the satisfiability problem for extended temporal logic, we will proceed as follows: build a Buchi automaton accepting the models of the formula and determine if that automaton is nonempty. ... |

77 | A.: A really abstract concurrent model and its temporal logic - Barringer, Kuiper, et al. - 1986 |

67 |
On the complexity of !-automata
- Safra
- 1988
(Show Context)
Citation Context ..., where a special construction is needed to complement Buchi automata. As a result, the size of the automaton A g constructed in [SVW87] for an ETL r formula g is 2 O(jgj 2 ) . Using results by Safra =-=[Sa88]-=-, this can be improved to 2 O(jgj log jgj) , which is provably optimal. Thus, while the construction given here for ETL f and ETL l as well as the construction given in [SVW87] for ETL r are exponenti... |

65 |
Deciding full branching time logic
- Emerson, Sistla
- 1984
(Show Context)
Citation Context ...l logic formulas, we will often have to take the intersection of several Buchi automata. The following lemma is a special case of Theorem A.1 in [VW86a] and extends a construction of [Ch74] (see also =-=[ES84]). Theorem-=- 2.6: Let A 0 ; : : : ; A k\Gamma1 be Buchi automata. There is a Buchi automaton A with k \Theta \Pi k\Gamma1 i=0 jA i j states such that L(A) = L(A 0 ) " : : : " L(A k\Gamma1 ). Proof: Let ... |

61 | First Order Dynamic Logic - HAREL - 1979 |

53 | The anchored version of the temporal framework
- Manna, Pnueli
- 1989
(Show Context)
Citation Context ... the possibilities in the Landweber classification (cf. [Wa79]). 4 The notions of safety and liveness are due to Lamport [Lam77]. Our notion of liveness here corresponds to the notion of guarantee in =-=[MP89]-=-. 5 In [SVW87] it is shown that the decision problem for ETL r is also PSPACE-complete. This result requires significantly more complicated automata-theoretic techniques that are beyond the scope of t... |

53 | On the relation of programs and computations to models of temporal logic - Wolper - 1987 |

49 |
A near-optimal method for reasoning about action
- Pratt
- 1980
(Show Context)
Citation Context ...g the satisfaction of eventualities. This construction always yields automata whose size is exponential in the size of the formula. We could also construct our automata using the tableau technique of =-=[Pr80]-=-. This technique can sometimes be more efficient than the maximal model technique. A major feature of our framework here is the use of set-subword automata to check for eventualities. A direct constru... |

48 | Propositional Dynamic Logic of Looping and Converse is Elementary Decidable - Streett - 1982 |

41 |
Weak Alternating Automata Give a Simple Explanation of Why Most Temporal and Dynamic Logics are Decidable in Exponential Time
- MULLER, SAOUDI, et al.
- 1988
(Show Context)
Citation Context ...ormula-checking automata. His approach eliminates the need for distinction between the local automaton and the eventuality automaton. Another approach, using weak alternating automata is described in =-=[MSS88]-=-. In that approach not only there 20 is no distinction between the local automaton and the eventuality automaton, but the automaton is constructed by a simple induction on the structure of the formula... |

40 | On Equations for Regular Languages, Finite Automata - Brzozowski, Leiss - 1980 |

39 |
Weak monadic second order theory of successor is not elementary recursive
- Meyer
- 1975
(Show Context)
Citation Context ...e decision problem for ETL f and ETL l . It turns out that both logics have the same complexity as PTL: polynomial space (cf. [HR83, SC85]). In contrast, the decision problem for S1S is nonelementary =-=[Me75]-=-, as is the emptiness problem for regular expressions with complement [MS73]. 5 An important contribution of this paper is to show that temporal logic formulas can be directly compiled into equivalent... |

38 |
Theories of automata on ~-tapes: A simplified approach
- CHOUEKA
- 1974
(Show Context)
Citation Context ... complementing automata, when we have a negated automata connective. Thus, the construction can be viewed as combining the classical subset construction of [RS59] and Choueka's "flag construction=-=" in [Ch74]-=-. This should be contrasted with the treatment of ETL r in [SVW87], where a special construction is needed to complement Buchi automata. As a result, the size of the automaton A g constructed in [SVW8... |

35 |
Temporal logic with fixed points
- Banieqbal, Barringer
- 1987
(Show Context)
Citation Context ...ith fixpoint operators. It was introduced in [EC80]; see also [BKP85, BKP86]. It has the same expressive power as ETL and, as is shown in [Va88], also has a PSPACE-complete decision problem (see also =-=[BB89]-=-). It is interesting to note however that this last result was obtained using an extension of the automata-theoretic techniques presented in this paper. The relation between the various types of accep... |

32 | Infinite sequences and finite machines - Muller - 1963 |

29 | Application of model theoretic games to discrete linear orders and finite automata - Ladner - 1977 |

27 |
Decision Problems for !-Automata
- Landweber
- 1969
(Show Context)
Citation Context ... nondeterministic 3 For an extensive study of acceptance conditions for !-automata see [Ch74, Ka85, Lan69, MY88, Sta87, Tho90, Wa79]. Our three conditions corresponds to the first three conditions in =-=[Lan69]-=-. It is known that these three conditions exhaust all the possibilities in the Landweber classification (cf. [Wa79]). 4 The notions of safety and liveness are due to Lamport [Lam77]. Our notion of liv... |

26 |
On Omega Regular Sets
- Wagner
- 1979
(Show Context)
Citation Context ...87, Tho90, Wa79]. Our three conditions corresponds to the first three conditions in [Lan69]. It is known that these three conditions exhaust all the possibilities in the Landweber classification (cf. =-=[Wa79]-=-). 4 The notions of safety and liveness are due to Lamport [Lam77]. Our notion of liveness here corresponds to the notion of guarantee in [MP89]. 5 In [SVW87] it is shown that the decision problem for... |

23 |
Yet another process logic
- Vardi, Wolper
- 1983
(Show Context)
Citation Context ...dynamic logic; see [Va85a, VW86a]. Also, the temporal logics discussed here can be combined with dynamic logic to yield expressive process logics which are also amenable to automata-theoretic methods =-=[VW83]-=-. In conclusion, we note that our results have an interesting interpretation from a purely automata-theoretic point of view. The ability to have an automaton operator nested within another automaton o... |

20 | The Propositional Dynamic Logic of Deterministic, Wellstructured Programs - Halpern, Reif - 1983 |

19 |
Succinct Representation of Regular Languages by Boolean Automata II
- Leiss
- 1985
(Show Context)
Citation Context ...1. The run forest is accepting if x is accepting whenever x is a leaf. A accepts w if it has an accepting run forest on w. Afa's define regular languages. Nevertheless, it follows from the results in =-=[Le81]-=-, [CKS81] that they can be exponentially more succinct than NFA's. That is, given any n-state AFA, one can construct an 2 n -state NFA that accepts the same language. Furthermore, for each n there is ... |

16 | A combinatorial approach to the theory of #-automata - Thomas - 1981 |

12 | A compositional temporal approach to a CSP-like language - Barringer, Kuiper, et al. - 1985 |

12 |
Process logic with regular formulas
- Harel, Peleg
- 1985
(Show Context)
Citation Context ...f [SVW87] to prove the analogue of Theorems 5.1 and 5.2 (though with a nonlinear exponent in the exponential bound) for ATL r . 30 6 Concluding Remarks There are other approaches to extending PTL. In =-=[HP85]-=- the language is extended by regular operators corresponding to concatenation and the Kleene star. This, however, pushes the decision problem for their language to nonelementary complexity. Furthermor... |