Hopcroft, J.E., "An n log n algorithm for minimizing the states of a finite automaton," The Theory of Machines and Computations, pp. 189-196 (1971).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is mainly due to complex edge labels, as in our application, rather than to intricate sequential structure. Automata Minimization Algorithm. One of the main nontrivial operations on automata is minimization. In Mosel this is realized via a modification of the well known Hopcroft algorithm [4], working on sets of labels instead of single labels. 4 Efficiency Already 30 years ago Alonzo Church proposed monadic second order logic on strings as an appropriate specification formalism for reasoning about sequences of bitvectors [2] This logic is among the most succinct decidable logics ....

J. Hopcroft: "An n log n algorithm for minimizing states in a finite automaton," Proc. Int. Symp. on Theory of Machines and Computations, Technion, Haifa (IL), Aug. 1971, pp.189-196.

.... (2 # , 1,3,4,5,6,7,8,9,10) 1,2,4 # , 3,5,6,7,8,9,10) 1,2,3,4,7 # , 6,5 # , 8,9,10) 3.2 Circuit Graph Partition Algorithm Figure 4 gives the pseudo code for computing the coarsest partition B. Our approach is based on the classical partition refinement algorithm given by Hopcroft [22] which is broadly used for many similar purposes. The algorithm uses a greatest fixpoint computation for successively refining a given partition. The initial set of equivalence classes is based on a classification according to inputs and non inputs and the number of fan ins and fan outs. Next, the ....

J. Hopcroft, "An n log n algorithm for minimizing states in a finite automaton," in Theory of Machines and Computations (Z. Kohavi and A. Paz, eds.), (New York), pp. 186--196, Academic Press, 1971.

....algorithm is theoretically exponential in the number of states of the automaton, but works well in practice for the sizes of automata typically needed for verification. Efficient (O(n log(n) where n is the number of states in the automaton) minimization algorithms also exist for finite automata [ 14, 15]. Before we apply these algorithms to the automata we generate, we need to make their labels typical of such automata. We perform this by applying the following transformation to the labels of the automata we generate. Assume that A = a. an is the set of propositions in the alphabet of an ....

Hopcroft, J. "An nlogn algorithm for minimizing states in a finite automaton", in Proc. of the Theory of Machines and Computations. 1971, New York. Academic Press, pp. 189196. Z. Kohavi, Ed.

....to make the problem of estimating complexity tractable, two approaches are used. The first approach is based upon Finite Automata. A Finite Automata whose smallest accepted string is the bit string whose complexity is to be determined, shown in Definition 6. 3, is minimized using techniques such as [20,21] where L(FA) is the set of languages accepted by the automaton FA and l(FA) is its size. Finite Automata Estimate of FA l x K FA L x where L(FA) is the set of languages accepted by the automaton FA, and l(FA) is its size. Figure 6.1 illustrates the uncompressed representation of an ....

Hopcroft J.E., "An n Log n algorithm for minimizing the states in a finite automaton", The Theory of Machines and Computations (Z. Kohavi, ed.), pp. 189-196, Academic Press, New York, 1971.

....circuit are the minimization of the number of machine s internal states and their assignment. Optimal solution to each of those problems is very complex and, classically, have received an independent treatment. State reduction of completely specified FSM can be achieved in O(n log n) steps [1] whereas state minimization of incompletely specified FSM is already NP complete [2] This means that most probably there will never be an algorithm of less than exponential complexity and therefore heuristic techniques must be used. This problem has received new attention in last years and very ....

J. Hopcroft: "An nlogn Algorithm for Minimizing States in a Finite Automaton ", in "Theory of Machines and Computation", Kohavi ed., pp. 189-196, Academic Press, 1971.

....resulting from the goaldirected strategy is so small that minimization of cloning is not an important issue. Partitioning Algorithm The algorithm for merging equivalent CloningVectors is related to the algorithm for minimizing the number of states in a Deterministic Finite Automaton (DFA) Hop71] It is very similar to an algorithm used in ParaScope to minimize the number of implementations of a procedure required when multiple definitions of the procedure occur in the program composition [CKT 86c] The partitioning algorithm is presented in Figure 5.3. Initially, all clones of a ....

....appropriate representation for StateVector and for CloningVectors resulting from call sites, the expected time required for partitioning can be done in time linear in the number of elements being partitioned. A different representation would yield O(nlogn) time, even for worst case performance [Hop71] As a possibility, the set representations can be treated as strings, with some canonical order imposed on the set elements. Then, partitioning can be performed by hashing to a location matching the string representation of the set. If two sets hash to the same location and have the same set ....

J. Hopcroft. An nlogn algorithm for minimizing states in a finite automaton. In Z. Kohavi and A. Paz, editors, Theory of Machines and Computations, pages 189--196. Academic Press, New York, NY, 1971.

....However, in order to make the problem of estimating complexity tractable, two approaches are used. The first approach is based upon Finite Automata. A Finite Automata whose smallest accepted string is the bit string whose complexity is to be determined is minimized using techniques such as [4, 5] as shown in Definition 1 where L(FA) is the set of languages accepted by the automaton FA and l(FA) is its size. min ) min FA l x K FA L x = Definition 1: Finite Automata Estimate of Complexity. Figure 1 illustrates the uncompressed representation of an arbitrary ....

Hopcroft J.E., "An n Log n algorithm for minimizing the states in a finite automaton", The Theory of Machines and Computations (Z. Kohavi, ed.), pp. 189-196, Academic Press, New York, 1971.

....exponentially smaller. Consider regular expression R = ajb) a(ajb) n (ajb) The DFA MR constructed by subset construction has an exponential number of states; but the min state DFA equivalent to MR has only a linear number of states. DFA minimization algorithms currently in use are off line [13,24]. We have to construct CHAPTER 7. CONCLUSION 114 the whole DFA before minimization. It seems that an on line version of min state DFA construction algorithm uses less auxiliary space; but unfortunately, it uses exponential auxiliary space at the worst case. Consider another regular expression R ....

Hopcroft, J., " An nlogn Algorithm for Minimizing states in a Finite Automata", in Theory of Machines and Computation, ed. Kohavi and Paz, pp. 189-196, Academic Press, New York, 1971.

....an algorithm for DFA minimization in 1950 s. Their algorithm runs in time O(kn 2 ) on DFA s with n states and k input symbols and was adequate for most of the classical applications. Many variations of this algorithm have appeared over the years; see [54] for a comprehensive summary. Hopcroft [18] developed a significantly faster algorithm of time complexity O(kn log n) in early 1970 s. Recently, Blum [3] presented a simpler algorithm of the same time complexity. In two special cases, namely when input alphabet contains one symbol and in the case of acyclic DFA s (i.e. when the language ....

J. Hopcroft, "An n log n Algorithm for Minimizing States in a Finite Automaton ", Theory of Machines and Computations, Academic Press, New York, pp.189-196, 1971.

....functions for q s important expressions, we obtain only two distinct state vectors. 4.2.3 Partitioning Algorithm The algorithm for merging equivalent cloning vectors appears in Figure 6. It is related to the algorithm for minimizing the number of states in a Deterministic Finite Automaton (DFA) [14]. It is also similar to an algorithm used to minimize the number of implementations of a procedure required when multiple definitions of the same procedure occur in a program [11] The algorithm partitions the cloning vectors for a procedure according to the values for their state vectors. It ....

....canonical order imposed on its elements. If we test for equality by hashing the strings, the partitioning step for each procedure has an expected time linear in the number of its cloning vectors. An approach based on state minimization would yield O(n log n) time, even for worst case performance [14]. Phase 3. The final phase of the cloning algorithm is accomplished by a single top down pass over the call graph. The number of clones created is less than the total number of cloning vectors. Thus, Phase 3 is also bounded by the number of cloning vectors. Given that the time required by each ....

Hopcroft, J. An nlogn algorithm for minimizing states in a finite automaton. In Z. Kohavi and A. Paz, editors, Theory of Machines and Computations, pages 189--196. New York: Academic Press, 1971.

....However, our technique can reduce the size of the representation because we can use one represenative memory locations to represent all the memory locations pointed to by that formal parameter pointer. Our algorithm for computing equivalence classes is one of the partitioning algorithms (e.g. [5, 9]) that compute a partition of some items according to some criteria. A partitioning algorithm first initializes the partition with an overestimate. The algorithm then iteratively refines the partition until the partition becomes safe, that is, if two items t1 and t2 are in the same set, then t1 ....

....until the partition becomes safe, that is, if two items t1 and t2 are in the same set, then t1 and t2 should be indistinguishable according to the criteria. The details of the efficient implementation and complexity analysis of the refinement step (Update in our algorithm) can be found in [9]. 5 Conclusions We presented equivalence analysis, a technique that can be used to improve many data flow analyses in the presence of pointers. We also presented an efficient algorithm to compute the equivalence classes. We conducted several studies to evaluate the performance and effectiveness ....

J. E. Hopcroft. An nlogn algorithm for minimizing states in finite automata. In Theory of Machines and Computations. Academic Press, 1971.

....q, we discover that two of the three state vectors are equivalent and can be merged. 4.2.2 Partitioning Algorithm The algorithm for merging equivalent cloning vectors appears in Figure 6. It is related to the algorithm for minimizing the number of states in a Deterministic Finite Automaton (DFA) [14]. It is also similar to an algorithm used to minimize the number of implementations of a procedure required when multiple definitions of the same procedure occur in a program [10] 1. Initially, all CloningVectors for a particular procedure are placed in the same partition. 2. In reverse ....

....a string with some canonical order imposed on its elements. If we test for equality by hashing the strings, the partitioning step for each procedure has an expected time linear in the number of its cloning vectors. A different approach would yield O(n log n) time, even for worst case performance [14]. Phase 3. The final phase of the cloning algorithm is accomplished by a single top down pass over the call graph. The number of clones created is less than the total number of cloning vectors. Thus, Phase 3 is also bounded by the number of cloning vectors. Given that the time required by each ....

J. Hopcroft. An nlogn algorithm for minimizing states in a finite automaton. In Z. Kohavi and A. Paz, editors, Theory of Machines and Computations, pages 189--196. Academic Press, New York, NY, 1971.

....process described in this section is the classical approach [80] It is by no means the only algorithm. Regarding its efficiency, for example, the worst case complexity (when no reduction is achievable) is O(N 2 ) for an N state FSM. In comparison, the Hopcroft algorithm proposed in 1970 [81, 92] has complexity O(N log 2 N ) Surprisingly little work has been done in this area since 1970. No further simplified algorithm has been found by the author, according to literature search including a private communication with Prof. Hopcroft in February 1993 [82] Some new extensions to the ....

....2 ) where S is the number of states of the FSM to be minimised. This happens when there is no state reduction. For the distance generating FSM, the complexity is O(N 4 ) So far only one simplified algorithm has appeared in the literature, that is, the Hopcroft O(N 2 log N 2 ) algorithm [92]. Further simplification is still possible. 52 4.3.2 State Minimisation by Known Properties Because of the complexity of the state minimisation algorithm itself, it is often not desirable or possible to use, especially for some large codes. Therefore, when possible we should use the known ....

J. Hopcroft, "An n log n algorithm for minimizing states in a finite automaton," in Theory of machines and computations (Z. Kohavi and A. Paz, eds.), pp. 189--196, Academic Press, 1971.

....theory is the state minimisation process credited to Huffman (1954) 49] Mealy (1955) 50] and Moore (1956) 51] Various aspects of the theory were refined through the early 1960 s [80] Some improved algorithms were developed as late as 1970. One such is the Hopcroft state minimisation algorithm [81], although it should be noted that this is still not optimal [82] FSM minimisation algorithm by state partitioning is, however, not commonly seen in text books on computer algorithms, except [83] An FSM can also be considered as a directed graph in graph theory [84, 85] Some authors therefore ....

....process described in this section is the classical approach [80] It is by no means the only algorithm. Regarding its efficiency, for example, the worst case complexity (when no reduction is achievable) is O(N 2 ) for an N state FSM. In comparison, the Hopcroft algorithm proposed in 1970 [81, 92] has complexity O(N log 2 N ) Surprisingly little work has been done in this area since 1970. No further simplified algorithm has been found by the author, according to literature search including a private communication with Prof. Hopcroft in February 1993 [82] Some new extensions to the ....

J. E. Hopcroft, "An n log n algorithm for minimizing states in a finite automaton," Technical Report CS-190, Stanford University, 1970.

....lexicographic sorting algorithm [1] We show how multiset dag discrimination can be used to obtain an improved solution to acyclic instances of the many function coarsest partition problem. The many function coarsest partition problem, used by Hopcroft to model the problem of DFA minimization [12], has applications in program optimization and program integration. It can be formulated 14 as follows. Given a directed multi graph (V, E 1 , E k ) where V is the set of vertices, and E 1 , E k are sets of edges) and an initial partition P = V 1 , V s of V, find a ....

....such that for each block C in P and each i = 1, k, there exists a block C 0 in P such that the image set E i [C ] C 0 , where E i [C ] y : x, y ]E i and xC . Here we assume that for each i = 1, k, the outdegree of each vertex v V in (V, E i ) is at most 1. An algorithm was given in[12] that solves this problem in time Q(k V log V ) and space Q(k V ) in the worst case, which is true even when the graph (V, E 1 . E k ) is acyclic. However, when the graph (V, E 1 . E k ) is acyclic, we can solve the problem in time and space O (k V ) using a solution to ....

Hopcroft, J., "An n log n Algorithm for Minimizing States in a Finite Automaton," in Theory of Machines and Computations, ed. Kohavi and Paz, pp. 189-196, Academic Press, New York, 1971.

....all the p inputs for each of the n states, and there are no more than n 1 rounds of splitting, since there are n states. Therefore, the total time complexity of a straightforward implementation of this state partitioning is O(pn 2 ) A modification of an algorithm for automata minimization [Hop] gives a fast algorithm with complexity O(pnlog n) After partitioning the states of the machine M into blocks of equivalent states, say r blocks B 1 , B r , we can construct the equivalent minimized machine M as follows. We project each block into one state: B i t i , and let the set ....

J. E. Hopcroft, "An n log n algorithm for minimizing states in a finite automaton," Theory of Machines and Computations, Z. Kohavi and A. Paz, Ed. Academic Press, pp. 189-196, 1971.

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Hopcroft, J.E., "An n log n algorithm for minimizing the states of a finite automaton," The Theory of Machines and Computations, pp. 189-196 (1971).

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J.E. Hopcroft, "An n log n algorithm for minimizing the states in a finite automaton, " in The Theory of Machines and Computations, eds. Z. Kohavi and A. Paz (Academic Press, New York, 1971) 189--196.

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J. Hopcroft, "An nlogn algorithm for minimizing states in a finite automaton," in Theory of Machines and Computation, J. Kohavi, Ed. New York: Academic, 1971, pp. 189--196.

No context found.

John E. Hopcroft. An nlogn algorithm for minimizing the states of a finite automaton. The Theory of Machines and Computations, pages 189--196, 1971.

No context found.

John E. Hopcroft. An nlogn algorithm for minimizing the states of a finite automaton. The Theory of Machines and Computations, pages 189--196, 1971.

No context found.

J. E. Hopcroft, "n log n algorithm for minimizing states in finite automata," Stanford Univ., Satnford, CA, Tech. Rep. CS 71/190, 1971.

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J. E. Hopcroft, "n log n Algorithm for Minimizing States in Finite Automata, " Stanford Univ., Stanford, CA, Tech. Rep. CS 71/190, 1971.

No context found.

Hopcroft, J.E., "An n log n algorithm for minimizing the states of a finite automaton," The Theory of Machines and Computations, pp. 189-196 (1971).

No context found.

J.E. Hopcroft. An nlogn algorithm for minimizing states in a finite automaton. In Theory of Machines and Computations, Ed. by Zvi Kohavi and Azaria Paz, pages 189--196. Academic Press, 1971.

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