J. E. ]topcroft and R. M. Karp. An Algorithm for Testing the Equivalence of Finite Automata. Technical Report TR-71-114, Dept. of Cronpurer Science, Cornell University, ltfiaca, New York, USA, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Figure 4. A node structure consists of live fields: tsymbol for a type symtol, arcs for a set of feature value pairs, ifeatures for a set of inhibited features, duodes for a set of disagreement liodcs i.e. disagreement classcs, aud foard. The field forwa is used for the Union Find algoritlnn[9] to calculate unions of clses iu the salHe manner [luet s algorithm[10] By traversiug two DGs nodes with the same feature mldrs simultaneously, catcu lating the uniou of their ches, aud copying arcs, their unificaiou can be calculated iu Figure 5. The function Unify takes two input nodes aud ....

J. E. ]topcroft and R. M. Karp. An Algorithm for Testing the Equivalence of Finite Automata. Technical Report TR-71-114, Dept. of Cronpurer Science, Cornell University, ltfiaca, New York, USA, 1971.

....of words accepted by M , where, as usual, we say that M accepts a word w if the path, which starts at the initial state and follows the edges labeled with the letters of w, ends at a final accepting state. Two DFA s are called equivalent if they accept the same language. It is well known (see [12, 1]) that the minimization of M , i.e. the problem of finding an equivalent DFA M whose decision diagram has a minimal number of states, can be solved in time O(n log n) The following slight modification of this problem is however NP hard: MinRep(DFA) Given two DFA s M;M d , find the smallest DFA ....

....and time O(m) for the retrieval of the witness when B red is given. In our application, n 1 and n 2 are bounded above by nn d . The design of a procedure WITNESS for DFA s is more subtle. Of course, the mere equivalence test can be done with a classical algorithm due to Hopcroft and Karp (see [12, 1]) which uses the UNION FIND data structure and runs in almost linear time. We were however not able to retrieve a witness when running this procedure. Our method for retrieving a witness makes use of another classical algorithm due to Hopcroft (see [11] which minimizes a given DFA in time O(n log ....

J. E. Hopcroft and R. M. Karp. An Algorithm for Testing the Equivalence of Finite Automata. Research Report TR--71--114, Cornell University, Ithaca, N.Y., 1971.

....we have discussed how coinductively defined sets can be turned into inductively defined ones using the fixpoint rule. 5.2. Related work 5.2.1. Recursive type equality and subtyping The present work was inspired by an observation that meant that the standard unification closure algorithm [HK71, Hue76] (see [ASU86, Section 6.7] for a presentation) can be turned into an axiomatization of recursive type equality simply by adding the type equation to be proved to the assumptions in the premises; see Figure 4. Unification closure works by building a bisimulation between two recursive types. This ....

J. Hopcroft and R. Karp. An algorithm for testing the equivalence of finite automata. Technical Report TR-71-114, Dept. of Computer Science, Cornell U., 1971.

....Finally, we have discussed how coinductively defined sets can be turned into inductively defined ones using the fixpoint rule. 5.2. Related work 5.2.1. Recursive type equality and subtyping The present work was inspired by an observation that meant that the standard unification closure algorithm [HK71, Hue76] (see [ASU86, Section 6.7] for a presentation) can be turned into an axiomatization of recursive type equality simply by adding the type equation to be proved to the assumptions in the premises; see Figure 4. Unification closure works by building a bisimulation between two recursive types. This ....

J. Hopcroft and R. Karp. An algorithm for testing the equivalence of finite automata. Technical Report TR-71-114, Dept. of Computer Science, Cornell U., 1971.

....the case where we are dealing only with conjunctions of equality constraints, efficient graph unification algorithms exist. The graph unification algorithm presented by Ait Kaci is a node merging process using the UNION FIND method (originally used for testing the equivalence of finite automata [Hopcroft Karp 71] It has its analogue in the unification algorithm for rational terms based on a fast procedure for congruence closure [Huet 76] Node merging is a destructive operation Since actual merging of nodes to build new node equivalence classes modifies the argument DGs, they must be copied before ....

J. E. Hopcroft and R. M. Karp. An Algorithm for testing the Equivalence of Finite Automata. Technical report TR-71114, Dept. of Computer Science, Cornell University, Ithaca, NY, 1971.

....for an expression e of size n we can generate in linear or almost linear time on a RAM (depending on the encoding of variables) a set E of monotype equations of size O(n) such that e is typable if and only if E is unifiable. E can be checked for unifiability in linear [89,72] or almost linear time [43]. This leads to a linear or almost linear upper bound for the time complexity of deciding typability in the Hindley Calculus. Since the additional inequational constraints in the Milner Calculus seem rather innocuous at first sight, this may have led researchers to incorrectly claim linear or ....

....can be made precise by defining arrow graph morphisms and proving uniqueness and minimality by induction on the depth with respect to dag edges of the arrow graph. Proposition 43 G is polynomial time computable. Proof: A simple adaptation of the union find based unification algorithm [43,1] yields an algorithm that executes in time O(knff(n; n) where ff is an extremely slow growing function (see [115] We can now define a reduction relation on normalized, downward closed arrow graphs simply by executing rule 4b (Figure 6.6) with subsequent exhaustive application of rules 1, 2, 3, ....

J. Hopcroft and R. Karp. An Algorithm for Testing the Equivalence of Finite Automata. Technical Report TR-71-114, Dept. of Computer Science, Cornell U., 1971.

....efficient constraint normalization algorithm algorithm can be refined to accommodate constraints of the form ff b ff 0 . Term graphs with equivalence classes have been used for fast implementations of unification. We shall not go into details, but refer the reader to the literature; e.g. HK71,AHU74,Hue76, PW78,MM82,ASU86] The equivalence classes are represented by a system of equivalence class representatives (ecr s) and there are two operations available on ecr s: find(n) for node n is a function that returns the ecr of the equivalence class to which n belongs; union(n; n 0 ) ....

J. Hopcroft and R. Karp. An algorithm for testing the equivalence of finite automata. Technical Report TR-71-114, Dept. of Computer Science, Cornell U., 1971.

.... (b) The disjoint sets of (a) after performing union(1; 3) and union(5; 2) c) The disjoint sets of (b) after performing union(1; 7) followed by union(4; 1) d) The disjoint sets of (c) after performing union(4; 5) ometry problems [31, 45, 46] testing equivalence of finite state machines [3, 26], string algorithms [4, 29] logic programming and theorem proving [6, 7, 28, 63] and several combinatorial problems such as solving dynamic edge and vertex connectivity problems [66] computing least common ancestors in trees [2] solving off line minimum problems [20, 27] finding ....

J. E. Hopcroft, R. M. Karp, An algorithm for testing the equivalence of finite automata, TR-71-114, Dept. of Computer Science, Cornell University, Ithaca, N.Y., 1971.

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