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Outline

Title
Abstract
Figures
Summary of Bounds
Automata Construction and Complexity Results
Conclusions and Future Work
Background
Summary and Open Problems
Related Work
Complexity Analysis and Specialized Construction
Conclusion
Case 2
Case 3
Unranked Case
Complexity of Finding Shortest Synchronizing Words
Introduction
Conclusions
References

Implementation and application of automata

Profile image of Anne Brüggemann-kleinAnne Brüggemann-klein

2012

https://doi.org/10.1007/978-3-319-60134-2
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232 pages

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Abstract

This book constitutes the thoroughly refereed papers of the 17th International Conference on Implementation and Application of Automata, CIAA 2012, held in Porto, Portugal, in July 2012. The 21 revised full papers presented together with 5 invited papers and 7 short papers were carefully selected from 53 submissions. The papers cover various topics such as automata applications in formal verification methods, natural language processing, pattern matching, data storage and retrieval, and bioinformatics, as well as theoretical ...

Figures (53)
22nd International Conference, CIAA 2017 Marne-la-Vallée, France, June 27-30, 2017 Proceedings
22nd International Conference, CIAA 2017 Marne-la-Vallée, France, June 27-30, 2017 Proceedings
Table 1. Monitor and Instrumentation Semantics (a € ACT and yw € AcTU {7} Systems are denoted as processes whose semantics is given in terms of a labeled transition system (LTS). An LTS is a triple (PRoc, (AcTU{rT}), -) where PRoc is a set of process states (p € PROC), ACT is a finite set of observable actions (a € Act), tr ¢ ACT is the distinguished silent action, and —C (PRoc x (ACTU {r}) x Proc) is a transition relation. Monitors are described via the specific syntax given below, but their semantics is also given as an LTS.
Table 1. Monitor and Instrumentation Semantics (a € ACT and yw € AcTU {7} Systems are denoted as processes whose semantics is given in terms of a labeled transition system (LTS). An LTS is a triple (PRoc, (AcTU{rT}), -) where PRoc is a set of process states (p € PROC), ACT is a finite set of observable actions (a € Act), tr ¢ ACT is the distinguished silent action, and —C (PRoc x (ACTU {r}) x Proc) is a transition relation. Monitors are described via the specific syntax given below, but their semantics is also given as an LTS.
Fig. 1. Transforming an automaton into a monitor: DFA to tree unraveling to monitor. to rule VER which ensures that yes S yes, for every t’). Stated otherwise, if L is a regular language on ACT that is recognized by a monitor, then L must be suffiz-closed. Since this property stems from the fact that monitor verdicts are irrevocable, in t he rest of this paper we instead call such languages irrevocable. Now, consider an automaton that recognizes an irrevocable language L. Then, if q is any (reac through a word safely add an a- hable) accepting state of the automaton, and gq can be reached t, then t is clearly in L but so is every word ta. Thus, we can transition from q to an accepting state (for example, itself) if no such transition exists. We call an automaton that can always transition from an accepting state to an accepting state for each a € ACT irrevocable. Note that, in the case of an irrevocable DFA, all transitions from accepting states must go to accepting sta tes.
Fig. 1. Transforming an automaton into a monitor: DFA to tree unraveling to monitor. to rule VER which ensures that yes S yes, for every t’). Stated otherwise, if L is a regular language on ACT that is recognized by a monitor, then L must be suffiz-closed. Since this property stems from the fact that monitor verdicts are irrevocable, in t he rest of this paper we instead call such languages irrevocable. Now, consider an automaton that recognizes an irrevocable language L. Then, if q is any (reac through a word safely add an a- hable) accepting state of the automaton, and gq can be reached t, then t is clearly in L but so is every word ta. Thus, we can transition from q to an accepting state (for example, itself) if no such transition exists. We call an automaton that can always transition from an accepting state to an accepting state for each a € ACT irrevocable. Note that, in the case of an irrevocable DFA, all transitions from accepting states must go to accepting sta tes.
Table 3. Bounds on the cost of construction (X signifies that the conversion is trivial The proof of Theorem 4 relies on a result by Chrobak [7] for unary language (languages on only one symbol), who showed that, for every n, there is a unar language Ch,, that is recognized by an NFA with n states, but by no DFA wit eo(vnlogn) states. U,, is t hen the set of words w € {0,1}, such that the 0’s ¢ the 1’s in w are a word from Ch,,. Then, from a deterministic monitor for U, w can extract a unary DFA or Ch, by following the 0*1- or 1*0-transitions of th monitor, until the first time recursion was used (i.e. a back edge was followed Therefore, the first time the deterministic monitor has a back edge is at distanc at least ely nTogn) from t a complete binary tree of he root; so, the deterministic monitor contains at leas height e@(vnlcgn) We provided a method for determinizing monitors. We have focused on monitors for the co-safety fragment of u“HML, as constructed in [11]. We showed that we can add a runtime monitor to a system without having a significant effect or the execution time of the system. Specifically, evaluating a nondeterministic monitor for a runtime trace may amount to keeping track of all possible monitor
Table 3. Bounds on the cost of construction (X signifies that the conversion is trivial The proof of Theorem 4 relies on a result by Chrobak [7] for unary language (languages on only one symbol), who showed that, for every n, there is a unar language Ch,, that is recognized by an NFA with n states, but by no DFA wit eo(vnlogn) states. U,, is t hen the set of words w € {0,1}, such that the 0’s ¢ the 1’s in w are a word from Ch,,. Then, from a deterministic monitor for U, w can extract a unary DFA or Ch, by following the 0*1- or 1*0-transitions of th monitor, until the first time recursion was used (i.e. a back edge was followed Therefore, the first time the deterministic monitor has a back edge is at distanc at least ely nTogn) from t a complete binary tree of he root; so, the deterministic monitor contains at leas height e@(vnlcgn) We provided a method for determinizing monitors. We have focused on monitors for the co-safety fragment of u“HML, as constructed in [11]. We showed that we can add a runtime monitor to a system without having a significant effect or the execution time of the system. Specifically, evaluating a nondeterministic monitor for a runtime trace may amount to keeping track of all possible monitor
Fig. 1. NFA with lookahead constructed as defined in Lemma 5. Wavy and dashed lines represent high and low priority transitions respectively. Negative lookaheads are shown in angle brackets. On the left the expression ab|b*, in the middle the same expression inside an atomic subgroup, getting the prioritized edges augmented by lookaheads on the low-priority case. Right is the full result for an expression discussed in Example 1.
Fig. 1. NFA with lookahead constructed as defined in Lemma 5. Wavy and dashed lines represent high and low priority transitions respectively. Negative lookaheads are shown in angle brackets. On the left the expression ab|b*, in the middle the same expression inside an atomic subgroup, getting the prioritized edges augmented by lookaheads on the low-priority case. Right is the full result for an expression discussed in Example 1.
Fig. 1. A 2-context c, a 2-graph g, and the substitution c[g]. Filled nodes convey the marking of c and g, respectively. Both targets of edges and external nodes of marked graphs are drawn in order from left to right.
Fig. 1. A 2-context c, a 2-graph g, and the substitution c[g]. Filled nodes convey the marking of c and g, respectively. Both targets of edges and external nodes of marked graphs are drawn in order from left to right.
Example 2. Consider the unary regular language
Example 2. Consider the unary regular language
Fig. 2. Schematic drawings of the automata from the proof of Theorem 7. (Left): the NFA B satisfying as(B) = asnra(L) = n and its n-state subautomata depicted by rectangles with dashed lines. (Middle): the multiple entry NFA C build from the n- state subautomata of B. (Right): the NFA A satisfying as(A) = aSnra(L) obtained from the multiple entry NFA C by identifying all initial states.
Fig. 2. Schematic drawings of the automata from the proof of Theorem 7. (Left): the NFA B satisfying as(B) = asnra(L) = n and its n-state subautomata depicted by rectangles with dashed lines. (Middle): the multiple entry NFA C build from the n- state subautomata of B. (Right): the NFA A satisfying as(A) = aSnra(L) obtained from the multiple entry NFA C by identifying all initial states.
Fig. 2. DFA D,,,n(a, 6, c1, c2, di, dz, e) of Definition 2 when n — 1 — k is odd, k is even, and both are at least 2; missing transitions are self-loops.
Fig. 2. DFA D,,,n(a, 6, c1, c2, di, dz, e) of Definition 2 when n — 1 — k is odd, k is even, and both are at least 2; missing transitions are self-loops.
The semantics of the vectorial formula y , for a Markov chain M and a valuation p is the semantics of its first component over the initial location s’” of M and it is equivalent to the value of the configuration (s’",t (y’")) in the semantics game Guy, That is, [p4 lilh,(s'”) = valay,, (s'”,t(y”)). It is worth noticing that only a maximum of n out of mx (n+1) variables are bound within the formula y,. Therefore, y, can be seen as a system of n+ 1 equations in n variables that can be reduced by substitution and Gauss elimina- tion techniques to a single scalar j?-calculus sentence (see [22]). In particular, it is sufficient to derive a solution, or expression, for each of the n variables and if 2; is even:
The semantics of the vectorial formula y , for a Markov chain M and a valuation p is the semantics of its first component over the initial location s’” of M and it is equivalent to the value of the configuration (s’",t (y’")) in the semantics game Guy, That is, [p4 lilh,(s'”) = valay,, (s'”,t(y”)). It is worth noticing that only a maximum of n out of mx (n+1) variables are bound within the formula y,. Therefore, y, can be seen as a system of n+ 1 equations in n variables that can be reduced by substitution and Gauss elimina- tion techniques to a single scalar j?-calculus sentence (see [22]). In particular, it is sufficient to derive a solution, or expression, for each of the n variables and if 2; is even:
Table 1. A summary of complexity of bifix-free languages for n > 6 with the minimal sizes of the alphabet required to meet the bounds Table 2. The minimal sizes of the alphabet in a universal most complex stream(s) for some of the studied subclasses of regular languages
Table 1. A summary of complexity of bifix-free languages for n > 6 with the minimal sizes of the alphabet required to meet the bounds Table 2. The minimal sizes of the alphabet in a universal most complex stream(s) for some of the studied subclasses of regular languages
Table 1. Path-structured GCID systems of size (3;1,1,0;1,1,1) simulating type-0 grammars G in SGNF. In the table, c’ € {A1, Ao, «’} and c € {Bi, Bo, Kk} UT, f,r are rule markers, while A is a dummy symbol that was not part of the alphabet of G.
Table 1. Path-structured GCID systems of size (3;1,1,0;1,1,1) simulating type-0 grammars G in SGNF. In the table, c’ € {A1, Ao, «’} and c € {Bi, Bo, Kk} UT, f,r are rule markers, while A is a dummy symbol that was not part of the alphabet of G.
Table 2. GCID rules of size (4;1,1,0;1,1,0) with axiom «KS and cE N” UT U {x}. rase CF: The intended simulation works as follows: Proof. Consider a type-0 grammar G = (N,T, P, S) in SGNF as in Definition 1 We construct a GCIDp system IT = (4,V,T, {«S}, H,1,1,R) such that L(IZ) = L(G). The four columns of the table correspond to the four components of JIT. The rows correspond to the simulation of r: X > Y,Y2, f : AB — X and of the context-free deletion rule h : S” — \. The last row deletes the left-end marker introduced in the axiom. The alphabet of IT is V C NUT U {p,p’,p",p'”: p € [(1...|P|]} U{«}. R is defined as shown in Table 2, depending on G. Clearly, I has size (4; 1,1,0;1, 1,0). We now prove that L(G) C L(IZ). To this end, we show that if w > w’ in G, with w,w’ € (N UT)*, then (Kw); =’ (Kw’), according to I. From this fact, the claim follows by a simple induction argument. As the claim is evident for rule h, we only need to discuss w = w’ due to using ¢ context-free rule (Case CF) or due to using a non-context-free rule (Case CF).
Table 2. GCID rules of size (4;1,1,0;1,1,0) with axiom «KS and cE N” UT U {x}. rase CF: The intended simulation works as follows: Proof. Consider a type-0 grammar G = (N,T, P, S) in SGNF as in Definition 1 We construct a GCIDp system IT = (4,V,T, {«S}, H,1,1,R) such that L(IZ) = L(G). The four columns of the table correspond to the four components of JIT. The rows correspond to the simulation of r: X > Y,Y2, f : AB — X and of the context-free deletion rule h : S” — \. The last row deletes the left-end marker introduced in the axiom. The alphabet of IT is V C NUT U {p,p’,p",p'”: p € [(1...|P|]} U{«}. R is defined as shown in Table 2, depending on G. Clearly, I has size (4; 1,1,0;1, 1,0). We now prove that L(G) C L(IZ). To this end, we show that if w > w’ in G, with w,w’ € (N UT)*, then (Kw); =’ (Kw’), according to I. From this fact, the claim follows by a simple induction argument. As the claim is evident for rule h, we only need to discuss w = w’ due to using ¢ context-free rule (Case CF) or due to using a non-context-free rule (Case CF).
Table 3. GCID rules of size (4; 1,1,0;1,0,1) simulating a type-O0 grammar in SGNF This shows that L(G) C LUI). The main complication for the correctness proof is the fact that we may return to Cl with strings containing rule markers. This brings along a detailed discussion of four different situations for w when considering (5); =/, (w), =’ (w’); according to IT. A detailed explanation of these different situations follows a similar argument as in Theorem 1 and is omitted here in view of page constraint. oO
Table 3. GCID rules of size (4; 1,1,0;1,0,1) simulating a type-O0 grammar in SGNF This shows that L(G) C LUI). The main complication for the correctness proof is the fact that we may return to Cl with strings containing rule markers. This brings along a detailed discussion of four different situations for w when considering (5); =/, (w), =’ (w’); according to IT. A detailed explanation of these different situations follows a similar argument as in Theorem 1 and is omitted here in view of page constraint. oO
Table 4. GCID rules of size (3; 2,1,0;1,0,1) simulating a type-O grammar in SGNF.
Table 4. GCID rules of size (3; 2,1,0;1,0,1) simulating a type-O grammar in SGNF.
Table 5. GCID rules of size (3;2,1,0;1,1,0) simulating a type-O grammar in SGNF.
Table 5. GCID rules of size (3;2,1,0;1,1,0) simulating a type-O grammar in SGNF.
Fig. 1. Random automata of size 10.
Fig. 1. Random automata of size 10.
Jne may wonder whether the two optimisations, namely the loop complexity 1euristic and minimising the subset automaton, indeed provide a speed-up. As in evidence that it does, we report on the following experiment. We enumerated 00 automata with three states and computed their star height. The computation vas considered an overflow when the number of pending matrix products was >5 yillions. After more than 50 years of research, the star height problem still had the repu- tation to be intractable, even for very small automata. By implementing state- of-the-art algorithms together with new heuristics and optimisations, we reached a new step in the understanding of this problem. In particular, we discovered
Jne may wonder whether the two optimisations, namely the loop complexity 1euristic and minimising the subset automaton, indeed provide a speed-up. As in evidence that it does, we report on the following experiment. We enumerated 00 automata with three states and computed their star height. The computation vas considered an overflow when the number of pending matrix products was >5 yillions. After more than 50 years of research, the star height problem still had the repu- tation to be intractable, even for very small automata. By implementing state- of-the-art algorithms together with new heuristics and optimisations, we reached a new step in the understanding of this problem. In particular, we discovered
Fig. 1. Two trees t and ¢’ and its optimal alignment A
Fig. 1. Two trees t and ¢’ and its optimal alignment A
Table 2. The nondeterministic state complexity of operations on free classes; com- plementation is from [13]. The dot means that the complexity is the same as in the previous column. Fig. 1. Binary factor-free witnesses for intersection.
Table 2. The nondeterministic state complexity of operations on free classes; com- plementation is from [13]. The dot means that the complexity is the same as in the previous column. Fig. 1. Binary factor-free witnesses for intersection.
Fig. 2. Binary prefix-free witnesses for union meeting the upper bound m+ n.
Fig. 2. Binary prefix-free witnesses for union meeting the upper bound m+ n.
Table 1. Closure properties of the transduction families discussed.
Table 1. Closure properties of the transduction families discussed.
Fig. 1. The transformation of Hadamard expressions into rotating automata. This paper presents a generic framework to extend any algorithm that con- verts a rational expression to a (weighted) one-way automaton, into an algorithm hat converts a Hadamard expression to a rotating automaton. More precisely, ve show that every Hadamard expression E can be rewritten into a rational xpression p(F) such that the conversion of p(E) into a (one-way) automaton A xy some algorithm o can be interpreted as a rotating automaton A” that realises he series denoted by the original Hadamard expression E. These transformations ire presented in Fig. 1. The proof of our main theorem amounts to prove that this liagram — where [.] is the interpretation of expressions and | -| is the behaviour yf automata — is actually commutative. We then apply this result to two meth- yds of automata synthesis: the derivation automaton and the follow automaton, ind prove rules that allow in each of these cases to directly handle Hadamard “xpressions. —s i = ra - ma gmk ZA
Fig. 1. The transformation of Hadamard expressions into rotating automata. This paper presents a generic framework to extend any algorithm that con- verts a rational expression to a (weighted) one-way automaton, into an algorithm hat converts a Hadamard expression to a rotating automaton. More precisely, ve show that every Hadamard expression E can be rewritten into a rational xpression p(F) such that the conversion of p(E) into a (one-way) automaton A xy some algorithm o can be interpreted as a rotating automaton A” that realises he series denoted by the original Hadamard expression E. These transformations ire presented in Fig. 1. The proof of our main theorem amounts to prove that this liagram — where [.] is the interpretation of expressions and | -| is the behaviour yf automata — is actually commutative. We then apply this result to two meth- yds of automata synthesis: the derivation automaton and the follow automaton, ind prove rules that allow in each of these cases to directly handle Hadamard “xpressions. —s i = ra - ma gmk ZA
A derivation with respect to the Hadamard operators is also defined:
A derivation with respect to the Hadamard operators is also defined:
Like for derivatives, the functions First, Last and Follow can be extended to {adamard expressions in order to get a direct construction. Notice that, for very Hadamard expression E, an occurrence of letter r appears in p(E) for each {tadamard operator which appears in E. This leads to extend the positions to he occurrences of Hadamard operators. The extension of function Null is done n Definition 6.
Like for derivatives, the functions First, Last and Follow can be extended to {adamard expressions in order to get a direct construction. Notice that, for very Hadamard expression E, an occurrence of letter r appears in p(E) for each {tadamard operator which appears in E. This leads to extend the positions to he occurrences of Hadamard operators. The extension of function Null is done n Definition 6.
Table 1. Table 1-S (left) and Table 1-M (right) Notice that Table 1-S defines a determinist ic automaton since A is deter. ministic. For instance, the transitions 2 and 3 are not in conflict since they ar associated with sequences {d;} with different properties. Indeed, the former cor. responds to a sequence {d;} with d; > 0 for al which changes the state of the counter from 0 that |{d; = 1}| > |{d; = —1}| > 0, and then from 0 to 2. Transitions 8 and 11 correspond to 7 and d. = 1 for at least one e to 1, whereas the latter implies the state of the counter change: negative cycles. Here the decre ment d corresponds to the current value c of the counter. Since a computatior traverses a negative cycle at most once, a symbol o” occurs at most once in < word of A’.
Table 1. Table 1-S (left) and Table 1-M (right) Notice that Table 1-S defines a determinist ic automaton since A is deter. ministic. For instance, the transitions 2 and 3 are not in conflict since they ar associated with sequences {d;} with different properties. Indeed, the former cor. responds to a sequence {d;} with d; > 0 for al which changes the state of the counter from 0 that |{d; = 1}| > |{d; = —1}| > 0, and then from 0 to 2. Transitions 8 and 11 correspond to 7 and d. = 1 for at least one e to 1, whereas the latter implies the state of the counter change: negative cycles. Here the decre ment d corresponds to the current value c of the counter. Since a computatior traverses a negative cycle at most once, a symbol o” occurs at most once in < word of A’.
Fig. 1. The reversal bounded automaton A and the associated s-automaton A’. automaton A € DFCM(1,0,1) at the top of Fig. 1, where each transition con- sumes an input symbol (i.e. the input head always moves to the right) and an arrow from p to q with label o,c,d stands for 6(p,o,c) = (q, R,d). By Defin- ition5, one has Qo = {q,p,t}, Qi = {¢,p}, Qo = {r,s} and Q3 = {r,s, th}. Since A does not have a negative cycle, by Definition 6 the alphabet and the set of states of the s-automaton A’ of A are I = {ao,a1,b-1,60,b_,} and Q’ = {qo, Po, to, 1; P1; 72; 82,73, $3, t3}, respectively. Note that I is a proper subset of the alphabet specified in Definition 6, as we consider only symbols associated with existing transitions (e.g. a”, b” or a!_, are not added to I’). The automaton A’ is shown at the bottom of Fig. 1.
Fig. 1. The reversal bounded automaton A and the associated s-automaton A’. automaton A € DFCM(1,0,1) at the top of Fig. 1, where each transition con- sumes an input symbol (i.e. the input head always moves to the right) and an arrow from p to q with label o,c,d stands for 6(p,o,c) = (q, R,d). By Defin- ition5, one has Qo = {q,p,t}, Qi = {¢,p}, Qo = {r,s} and Q3 = {r,s, th}. Since A does not have a negative cycle, by Definition 6 the alphabet and the set of states of the s-automaton A’ of A are I = {ao,a1,b-1,60,b_,} and Q’ = {qo, Po, to, 1; P1; 72; 82,73, $3, t3}, respectively. Note that I is a proper subset of the alphabet specified in Definition 6, as we consider only symbols associated with existing transitions (e.g. a”, b” or a!_, are not added to I’). The automaton A’ is shown at the bottom of Fig. 1.
Fig. 1. The three types of transformations in wr (n) from Remark 2. Semigroup Wee (n). For n > 3 we define the semigroup:
Fig. 1. The three types of transformations in wr (n) from Remark 2. Semigroup Wee (n). For n > 3 we define the semigroup:
Let s be the transformation illustrated in Fig. 2 and defined by:
Let s be the transformation illustrated in Fig. 2 and defined by:
Fig. 3. Map of the (sub)cases of Supercase 3 in the proof of Theorem 10 (part 1). Internal injectivity: Let t be any transformation that fits in this case and rest in the same s; we will show that t = t. From (a), there is the unique collid pair {p,z} focused to a state in a cycle, hence {p, 2} = {p, z}. Moreover, p ¢ p are not in this cycle, so p = p and 2 = z, which means that Ot = Ot = p. Si there is no state gq 4 0 such that qt = p, the only state mapped to p by s is hence pt = pt. From (d) for i=1,...,k —1, state pt’t is uniquely determin hence pt'+! = pt'+!, Finally, for i = k there is no state colliding with pt’! « mapped to pt*, hence pé*+! = pt*t! = n —1. Since the other transitions i are defined exactly as in t and ft, we have f = t. Then we have four other cases, which together cover all possibilities for ¢
Fig. 3. Map of the (sub)cases of Supercase 3 in the proof of Theorem 10 (part 1). Internal injectivity: Let t be any transformation that fits in this case and rest in the same s; we will show that t = t. From (a), there is the unique collid pair {p,z} focused to a state in a cycle, hence {p, 2} = {p, z}. Moreover, p ¢ p are not in this cycle, so p = p and 2 = z, which means that Ot = Ot = p. Si there is no state gq 4 0 such that qt = p, the only state mapped to p by s is hence pt = pt. From (d) for i=1,...,k —1, state pt’t is uniquely determin hence pt'+! = pt'+!, Finally, for i = k there is no state colliding with pt’! « mapped to pt*, hence pé*+! = pt*t! = n —1. Since the other transitions i are defined exactly as in t and ft, we have f = t. Then we have four other cases, which together cover all possibilities for ¢

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Susan Rodger

1997

Abstract We present a collection of new and enhanced tools for experimenting with concepts in formal languages and automata theory. New tools, written in Java, include JFLAP for creating and simulating finite automata, pushdown automata and Turing machines; P&acirc; t&eacute; for parsing restricted and unrestricted grammars and transforming context-free grammars to Chomsky Normal Form; and PumpLemma for proving specific languages are not regular.

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