Thomas, W.: Automata on Innite Objects. In Leeuwen, J. (ed.): Handbook of Theoretical Computer Science, Volume B, Formal Models and Semantics, Elsevier, Amsterdam, 1990, pp. 133139.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... that the recognition, enumeration, and optimization problems speci ed by an extension of MSOL formula can be solved in linear time [BPT92] This graph variant of MSOL uses Inc (v; e) which means a vertex v is an incident of an edge e, instead of the use of a successor function in ordinary MSOL [Tho90b] Although appealing in theory, these methods are hardly useful in practice due to a huge constant factor for space and time. This arises from the manipulation of huge tables: 101 The construction of tables re ects the decomposition of the property description into primitive ones; previous ....

.... graphs, a restricted class of graphs, from logical description of properties by a graph variant of monadic second order formula [BPT92] This graph variant of MSOL uses Inc (v; e) which means a vertex v is an incident of an edge e, instead of the use of a successor function in ordinary MSOL [Tho90b] Although appealing in theory, these methods are hardly useful in practice due to a huge constant factor for space and time. Program Transformation System There are several program transformation system. One of them is stratego [Vis01] which is a language for describing a program ....

Wolfgang Thomas. Automata on innite objects. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 4, pages 133-192. Elsevier Science Publishers, 1990.

....realized by a transducer, the domain of f is rational. However, it is not true that any rational set of in nite words is recognized by a deterministic B uchi automaton. Landweber s theorem states that a set of in nite words is recognized by a deterministic B uchi automaton i it is rational and G [29]. Recall that a set is said to be G is it is equal to a countable intersection of open sets for the usual topology of A . It is worth pointing out that the domain of a function realized by a transducer may be any rational set al..though it is supposed that all states of the transducer are nal. ....

Thomas, W. Automata on innite objects. In Handbook of Theoretical Computer Science, J. van Leeuwen, Ed., vol. B. Elsevier, 1990, ch. 4, pp. 133-191.

.... each sequential component) to a notion of step admitting sequences of transitions, we distinguish between input states (starting and ending a transition sequence) and non input states (intermediate states of the transition sequence) The acceptance condition is a Muller acceptance condition (see [7]) De nition 1. A Timed Cooperating Automaton (TCA) is a tuple M = hM 1 ; Mn ; I; Fi; where: each sequential automaton M i (with 1 i n) is a triple hQ i ; q 0 i ; i i such that 3 1. Q i is a nite set of states (Q 1 ; Qn are required to be pairwise disjoint and Q is ....

Thomas, W., Automata on Innite Objects, in J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, Elsevier Science Publishers, Amsterdam, 1990, pp. 134-191. 12

....and is computable. Proof. Let (t) be a MLO past formula with only one free variable t. A structure S here is de ned as an in nite word on the alphabet = 2 L where L is the set of monadic symbols of (t) The property de ned by (t) depends only on the pre x of size t 1 of a model. Thus [Tho90] there exists a nite complete deterministic automaton A on the alphabet accepting a language of nite words L(A) such that S; n j= t) i the pre x of S of size n 1 belongs to L(A) Now, given the automaton A and the Finite Probabilistic Process M with initial state s 0 we build a new ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, pages 131-191. North-Holland,, 1990. 13

....We say that a formula is a future (past) formula if it contains only state formulas, boolean connectives and future (past) temporal operators. We say that a formula is a general safety formula if it is of the form 0 , for a past formula . 2. 2 Alternating Automata Automata on in nite words [Tho90] are a convenient way to represent temporal formulas. For every linear temporal formula there exists an automaton on in nite words such that a sequence of states satis es the temporal formula if and only if it is accepted by the corresponding automaton. Thus to check whether a trace satis es a ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, vol. B, pages 133{ 191. Elsevier Science Publishers (North-Holland), 1990. 59 Finkbeiner and Sipma

....are of special interest as they in many cases require fairness assumptions. The liveness property required is that as many items should be delivered to the data target as are read from the data source. If the system is connected to a data source which generates the regular language (see e.g. [34]) 0 [ 0 Delta 1, using data independence [38, 27] and properties of regular languages Kaivola argues that it is then enough to verify the property: LIVE Either innitely many 0:s are delivered to the target or 1 is delivered. The property can be expressed in LTL in the following way: ....

W. Thomas. Automata on innite objects. In Handbook of Theoretical Computer Science, volume B, pages 133191. Elsevier, 1990.

....and distributed transition systems of various kinds [30] The theory of traces is well developed [3] This theory may be viewed as a smooth generalization of the classical theory of sequences. It turns out that most of the algebraic, automata theoretic and logical results concerning sequences [27] have a natural extension to the setting of traces. There has been however one prominent gap to date in the logical theory of traces. Our main result concerning LTrL lls this gap. To bring this out, we recall the famous theorem of Kamp [13] extended by Gabbay et.al. 8] It says that LTL, the ....

....be denoted FO( and it will have the same syntax as FO( I) The only di erence is that the binary relation is to be viewed as the linear order relation over the positions of a word. Viewed di erently, FO( is just FO( I) with I = The basic details concerning FO( can be found in [27]. We will not reprove here the famous equivalence in expressive power between FO( and LTL( 13] Rather, we will use this result and work with trace consistent fragment of LTL( instead of FO( I) This is possible by the following lemma which was observed in a slightly di erent setting in ....

W. Thomas. Automata on innite objects. In J. van Leeuven, editor, Handbook of Theoretical Computer Science Vol.B, pages 133-192. Elsevier, 1990.

.... [ XnT mi n : 3) Obviously, the mirror images of solutions of (3) are exactly the solutions of (2) To test (3) for solvability, we build a looping tree automaton B, i.e. a B uchi tree automaton where all states are nal. Let us brie y introduce in nite trees and looping tree automata (see [14] for details) Let be a nite alphabet and, w.l.o.g. NR = f1; kg. A labeled k ary in nite tree t is a mapping from N R into . In particular, the nodes of t can be viewed as words over NR . In case is a singleton, t is called unlabeled. A looping tree automaton A is a tuple ....

....(3) where the size of the regular sets S mi i and T mi i are measured by the size of nondeterministic nite automata accepting these sets. Since the emptiness problem for B uchi tree automata (and thus looping tree automata) can be solved in polynomial time in the size of the automaton [14] (and actually in linear time for looping automata) this yields the desired exponential time algorithm for deciding the solvability of equation (3) However, the existence of a solution does not a priori imply that there is also a regular one. Thus, we must still show that regular solvability can ....

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W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 133-191. Elsevier Science Publishers, Amsterdam, 1990.

....illustrative purposes, show how we encoded the sender process of the i protocol in the tool s input language. We also discuss how the property of potential livelock was encoded and veri ed. 4. 1 Cospan Cospan [HHK96] is a model checker for synchronous systems based on the theory of automata [Tho90] In this framework, the system to be veri ed is speci ed as an automaton P , the task the system is intended to perform is speci ed as an automaton T , and veri cation is accomplished by checking for containment of the language L(P ) in the language L(T ) Cospan performs this check by ....

W. Thomas. Automata on innite objects. In Handbook of Theoretical Computer Science, Volume B. Elsevier Science Publishers, 1990.

....these sets. Moreover, all states of the B uchi automaton are nal. Thus, we are using a restricted form of B uchi automata (sometimes called looping tree automata in the literature) Since the emptiness problem for B uchi tree automata can be solved in polynomial time in the size of the automaton [9] (and actually in linear time for looping automata) this yields an exponential time algorithm deciding whether an equation of the form (3) has a solution. However, the existence of a solution does not a priori imply that there is also a regular one. It is well known [9] that the set of trees ....

....the size of the automaton [9] and actually in linear time for looping automata) this yields an exponential time algorithm deciding whether an equation of the form (3) has a solution. However, the existence of a solution does not a priori imply that there is also a regular one. It is well known [9] that the set of trees accepted by a B uchi automaton contains a regular (or rational) tree t. As also stated in [9] a tree t is regular i for every label the set fv j t(v) g is regular. In our setting this means that the language fv j t(v) b 0 b n g for some xed label b 0 b ....

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W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 133-191. Elsevier Science Publishers, Amsterdam, 1990. 10

....into k subsets, 0 [ k . A ranked tree over is a nite, ordered tree where every node with i children is labeled with a symbol from i . We denote T the set of ranked 45372 A tree automaton is a device that accepts or rejects ranked trees. The de nition below is adapted from [Tho90] and slightly modi ed. De nition 2.2 A non deterministic top down tree automaton is A = Q; q 0 ; P ) where Q is a nite set of states, q 0 2 Q is the initial state and P S i=1;k i Q Q i . We write (a; q) q 1 q 2 : q i whenever (a; q; q 1 ; q i ) 2 P . Given t 2 T , ....

....provided that we restrict the output type as follows: is a DTD. every regular expression occurring in is star free . Theorem 4.1 [MSV00] Typechecking is decidable for TreeQL over databases with functional dependencies, and for star free output DTDs. Star free regular expressions [Tho90] are much more expressive than their name suggests. We are not allowed to use the Kleene closure, however we can use the complement, denoted compl, and the empty set ; For example, if = fa; b; cg, then compl( denotes , compl( b: j :c: denotes a . Thus, ....

W. Thomas. Automata on innite objects. In Formal Models and Semantics, volume B of Handbook of Theoretical Computer Science, chapter 4, pages 133{ 192. Elsevier, Amsterdam, 1990. 17

....sets. Moreover, all states of the B uchi automaton are nal. Thus, we are using a restricted form of B uchi automata (sometimes called looping tree automata in the literature [11] Since the emptiness problem for B uchi tree automata can be solved in polynomial time in the size of the automaton [10] (and actually in linear time for looping automata) this yields an exponential time algorithm deciding whether an equation of the form (1) has a solution. However, this does not a priori mean that there is also a regular one. It is well known [10] that the set of trees accepted by a ....

....in polynomial time in the size of the automaton [10] and actually in linear time for looping automata) this yields an exponential time algorithm deciding whether an equation of the form (1) has a solution. However, this does not a priori mean that there is also a regular one. It is well known [10] that the set of trees accepted by a B uchi automaton contains a regular (or rational) tree t if it is nonempty. In our setting this implies that the language fv j t(v) b 0 b n g for some xed label b 0 b n 2 f0; 1g n 1 is regular. In particular, the nite union [ b0 b1 b ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 133-191. Elsevier Science Publishers, Amsterdam, 1990.

....and an implementation of HOL (higher order logic) in the Isabelle system. There are compelling reasons for investigating this combination. WS1S belongs to a class of monadic logics that are among the more expressive decidable logics known and many decision problems can be embedded in WS1S [ Thomas, 1990 ] The logic is also well suited for reasoning about many kinds of systems that can be modeled using automata [ Basin and Klarlund, 1998 ] However, as with all decidable logics, its expressiveness is limited. In contrast, HOL is a very expressive foundation for reasoning about programs and ....

.... 1 e) d 2 e) d: e = bool ) bool) d e) d9x : e = 9 : nat ) bool) bool) x : nat:d e) d9X : e = 9 : nnat ) bool) bool) X : nnat:d e) 4 The problem of determining if WS1S sentences are true under the above interpretation is decidable (see, e.g. Thatcher and Wright, 1967; Thomas, 1990 ] The decision procedure is semantically based; it translates a WS1S formula to an automaton A that, essentially, recognizes valuations for the free variables under which is true. It is easy to determine from the resulting automaton A whether a sentence is true in the above structure. ....

Wolfgang Thomas. Automata on innite objects. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 4. MIT Press/Elsevier, 1990.

.... graphs, a restricted class of graphs, from logical description of properties by a graph variant of monadic second order formula [5] This graph variant of MSOL uses Inc (v; e) which means a vertex v is an incident of an edge e, instead of the use of a successor function in ordinary MSOL [17]. Although appealing in theory, these methods are hardly useful in practice due to a huge constant factor for space and time. 7 Conclusions In this paper, we propose an important theorem (generation rule) for generating ecient algorithms for solving maximum multi marking problems, which can e ....

Wolfgang Thomas. Automata on innite objects. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 4, pages 133-192. Elsevier Science Publishers, 1990.

....3 Automata on In nite Words Automata on in nite words di er from automata on nite words in their acceptance mechanism: there are no nal states; instead, acceptance is determined w.r.t. the set of states that are visited in nitely often. Di erent types of acceptance conditions are studied (see [Tho94] for an overview) In the following we will work with B uchi conditions. De nition 1. A (nondeterministic) B uchi automaton A = h ; Q; i consists of a nite input alphabet , a nite set of states Q, a set of initial states , a transition function : Q 2 Q and a set of accepting ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science. Elsevier Science Publishers (North-Holland), 1994.

....want to show Mw j= t anbn . As a witnessing element for the leading existential quanti er we use n. The rest is straight forward. The question which classes of words can be de ned in rst order logic and in monadic second order logic has been intensively investigated, surveys may be found e.g. in [16,17,13]. By a well known result of B uchi, the sets of nite words de nable by monadic second order logic are precisely the regular sets of words. Now, aa n bb n denotes a context free but not regular language, thus Lemma 3 tells us that L it 6 LMonSO . We have already observed that LFO L it is ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science. Vol. B : Formal Models and Semantics, pages 135-192. Elsevier, Amsterdam, 1990.

....so far, it is more ecient to develop a general notion of diagram that can be applied to arbitrary properties. This is achieved by generalized veri cation diagrams (GVD s) 16] which are applicable to any state quanti ed temporal logic formula. Like temporal formulas and automata (see [61]) gvd s have an associated set of computations, constrained by an acceptance condition. Veri cation conditions similar to those for invariance diagrams establish that all runs of the system S are represented in the diagram. Fairness properties and ranking functions justify the gvd acceptance ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 133-191. Elsevier Science Publishers (North-Holland), 1990.

....: j n 2 f 0; 1 g ) For an example, if t = f(g(a; h(g(b; c) c) a) and v is the (unique) leaf with label b, then its access word is (f; 1) g; 2) h; 1) g; 1)b. Let A be the nite tree automaton constructed from (by the classical result of Doner, Thatcher, Wright; see for instance Thomas [Tho90]) Let Q be its set of states. Let us run A over t 0 = t[ Each node u of t (ie of t 0 ) gets one and only one state q(u) 2 Q. Let v be a leaf of t, with access word: f 1 ; i 1 ) f 2 ; i 2 ) f n ; i n )c. The corresponding nodes of the path are say u 1 ; u 2 ; un ; v ....

Wolfgang Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, chapter 4, pages 133-191. Elsevier Science Publishers B. V., 1990. 19

....by a MSO[ formula is the set of words satisfying it. The correspondence between languages and MSO[ formulas is given by B uchi Theorem which states that the class of regular languages is precisely the class of languages de ned by MSO[ formulas. The theory MSO[ is decidable (cf. [24]) The theory of k successors. If we consider the case where D is the set of words over the alphabet f0; k 1g, U is the empty set, and B = f k ; succ 0 ; succ k 1 g, 82 Theories of Layered Metric Temporal Structures with k interpreted as the standard strict pre x relation ....

.... P 1 ; Pn ( xed) subsets of f0; k 1g , a model theoretic structure for MSO[ k ; succ 0 ; succ k 1 ] can be viewed as a lea ess, perfectly balanced k ary tree labelled over the alphabet = f0; 1g n The theory MSO[ k ; succ 0 ; succ k 1 ] is decidable (cf. [24]) The theory MSO[ pow k ; hp k ] In [18, 19, 23] we de ned a (decidable) extension of MSO[ called MSO[ pow k ; hp k ] which we proved to be the logical counterpart of the operational de nition of systolic languages. The theory MSO[ pow k ; hp k ] extends MSO[ with a binary ....

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W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Vol. B, pages 133-191. Elsevier Science Publishers, 1990.

....of these events. For example, both a variable assignment and a method call could be examples of events. A number of formalisms for specifying properties have been proposed, including temporal logics [9, 19] process algebras [4, 15] and various forms of regular languages and nite state automata [18, 21]. FLAVERS uses deterministic nite state automata for specifying properties to be checked on terminating executions of a system. A deterministic FSA can be represented as a tuple (S; s 0 ; A) S is the set of all states of the FSA, including the unique start state s 0 . is called the ....

....used in our LIE FLAVERS algorithm, present an overview of the approach, and then give the details of the LIE algorithm itself. 5.1. Representing Liveness Properties Since FSAs can encode only nite event sequences, we need a di erent formalism to describe in nite behaviors. automata [21] provide such a formalism. Usually, for an in nite trace sequence to be accepted by an automaton, some in nite pattern of accept states of this automaton must be observed on the traces of this automaton on . In particular, we use a well known subclass of automata, B uchi automata. A ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, chapter 4, pages 133-191. Elsevier Science Publishers B. V., 1990.

....boolean combinations of existential and universal regular trace properties [11] which include Ltl. We can characterize the expressive power of 1 dg similarly. Let 1 g be the state logic that consists of all formulas of the form hh1iiK, where K is an regular expression with constants from [19]. We identify K with the set of in nite words over the alphabet 2 that satisfy K. Let G be a game structure whose observables contain . A state s of G is in [ hh1iiK] G if player 1 has a strategy f 1 such that for all strategies f 2 of player 2, we have L f1 ;f 2 (s) K. Given a formula ....

.... set of states, Q 0 Q is the set of initial states, Q 2 Q is the transition relation, is the input alphabet, Q is a state labeling, and : Q f0; n 1g is 5 The solution of the regular control problem on game structures requires deterministic automata (see, e.g. [19]) whereas nondeterministic (and hence B uchi) automata suce for the regular veri cation problem on the underlying transition structures, as in [11] 12 the acceptance condition. An execution of C on the in nite word w 0 w 1 w 2 : 2 is an in nite sequence e = q 0 q 1 q 2 : of ....

W. Thomas. Automata on innite objects. In J. van Leeuwen, ed., Handbook of Theoretical Computer Science, volume B, pp. 133-191. Elsevier, 1990.

No context found.

Thomas, W.: Automata on Innite Objects. In Leeuwen, J. (ed.): Handbook of Theoretical Computer Science, Volume B, Formal Models and Semantics, Elsevier, Amsterdam, 1990, pp. 133139.

No context found.

W. Thomas, Automata on Innite Objects, in: Handbook of Theoretical Computer Science, Vol. B (ed. J. van Leeuwen), North-Holland, pp. 133191, 1990.

No context found.

W. Thomas. Automata on innite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Vol. B, pages 133-191. Elsevier Sci. Pub., 1990.

No context found.

W. Thomas. Automata on innite objects. in Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics, J. van Leeuwen, Ed. Cambridge, MA: MIT Press, 1990, pp. 133-191.

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