S. Fortune, J. Hopcroft, and E.M. Schmidt, "The Complexity of Equivalence and Containment for Free Single Variable Program Schemes", in Automata, Languages, and Programming, Goos, Hartmannis, Ausiello, and Boehm, eds., Springer-Verlag, Lecture Notes in Computer Science, Vol. 62, 1978, pp. 227-240.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is defined by imposing restrictions on the the Binary Decision Diagram (BDD) representation introduced by Lee [Lee 1959] and Akers [Akers 1978] such that the resulting form is canonical. These restrictions and the resulting canonicity were first recognized by Fortune, Hopcroft, and Schmidt [Fortune et al. 1978]. Functions are represented as directed acyclic graphs, with internal vertices corresponding to the variables over which the function is defined and terminal vertices labeled by the function values 0 and 1. Although the OBDD representation of a function may have size exponential in the number of ....

Fortune, S., Hopcroft, J., and Schmidt, E. M. 1978. The complexity of equivalence and containment for free single variable program schemes. Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 62, G. Goos, J. Hartmanis, Ausiello, and Boehm, eds. Springer-Verlag, Berlin, pp. 227-240.

....that although the synthesis operations of constructing OBDD s for fg and fg are intractable if the two given OBDD s do not obey the same ordering (as shown below) this condition is not necessary for testing equivalence. There is a polynomial time algorithm due to Fortune, Hopcroft, and Schmidt [FHS78] for testing whether an OBDD is equivalent to a read once program, which can be used in this case. 2.2.1 Read once programs Read once programs do not enjoy quite the same degree of manipulability as their oblivious version, OBDD s. The read once property implies that the program is satisfiable ....

....in the literature. Further work There are many open questions surrounding the topics of this thesis, some of which have already been mentioned. We will describe some of these problems that we consider to be the most important, interesting, or tractable. The oldest of these problems, open since [FHS78], is 65 Open Question 1 Is there a deterministic polynomial time algorithm for determining the equivalence of two read once programs The answer to this question loses some practical significance in light of the lower bound for multiplication and the intractability of the synthesis operations, ....

S. Fortune, J. Hopcroft, and E. M. Schmidt. The complexity of equivalence and containment for free single variable program schemes. Springer-Verlag LNCS No. 62, 1978, pp. 227--240.

....closed class if we can efficiently determine whether two different representations of a concept class are functionally equivalent then an EV oracle is not needed. Such equivalence tests exist for monotone DNF, read once branching programs [36] Horn sentences, free branching programs [14, 21], read once boolean formulas [28] and readtwice DNF [34] Also, if the number of unspecified attributes is O(log n) then evaluation can be done in polynomial time by simply considering all completions of x. Likewise, when h is a unate formula, then it is easily seen that the evaluation (for any ....

....closed class for whichwe can efficiently determine whether two different representations of a concept class are functionally equivalent then an EV oracle is not needed. Such equivalence tests exist for monotone DNF, read once branching programs [36] Horn sentences, free branching programs [14, 21], read once boolean formulas [28] and read twice DNF [34] Corollary 7 Let C be exactly learnable in polynomial time using the EQ and MQ oracles with a hypothesis class H where H is projection closed and for which there is a polynomial time algorithm to determine if h 1 2 H and h 2 2 H are ....

S. Fortune, J. Hopcroft, and E. Schmidt. The complexity of equivalence and containment for free single variable program schemes. In Lecture Notes in Computer Science 62: Proceedings of Automata, Languages and Programming, pages 227--240. Springer-Verlag, 1978.

....element set. By Corollary 2.3 (applied to p = p B Gamma p B 0 ) it follows that p B and p B 0 agree on Q. Now, choosing T in Corollary 2.4 as T = f1; 2ng, we will detect an inequivalence with probability more than 1 2 . It follows that 1 BPE 2 coRP. Fortune, Hopcroft, and Schmidt [FHS78] have shown that if one of two given one time only branching programs is even ordered , then the equivalence can be deicided in polynomial time. In particular, the equivalence problem for ordered branching programs is solvable in polynomial time. 7 Two branching programs B and B 0 are ....

....NP. Hence we get the main result of this section. Corollary 3.4 1 BPI is not NP hard, unless PH = Sigma p 2 . Since ordered branching programs , OBDD s, are a restricted form of onetime only branching programs, Corollary 3.4 can be applied. Since the equivalence problem for OBDD s is in P [FHS78] the isomorphism problem for OBDD s is in NP. Corollary 3.5 The isomorphism problem for OBDD s is not NP complete, unless PH = Sigma p 2 . 4 Extension to Arithmetic Circuits In this section we extend the above protocol for branching programs to an interactive proof to decide the ....

Fortune, Hopcroft, and Schmidt. The complexity of equivalence and containment for free single variable program schemes. In ICALP: Annual International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 62, pages 227--240. Springer-Verlag, 1978.

....order dynamically during the construction of the OBDD, based on the exchange of adjacent variables [Rud93] In this paper we address the problem how to work with OBDDs of di erent variable orders, especially w.r.t. the application of a binary Boolean operator to two OBDDs. Even though in [FHS78] an ecient algorithm for the equivalence test of Free BDDs, i.e. BDDs without the ordering constraint is given, this algorithm cannot be used for computing arbitrary Boolean operations. We address the similar problem by working with OBDDs of di erent variable orders, but we extend it by providing ....

S. Fortune, J. Hopcroft, and E. M. Schmidt. The Complexity of Equivalence and Containment for Free Single Variable Program Schemes, in ICALP'78 - Automata Languages and Programming, LNCS 62, pp.227-240, Springer, 1978

....2 conclusions are presented in Section 5. 2 Design Verification Ordered Binary Decision Diagrams (BDDs) are directed acyclic graphs which form a canonical representation of Boolean functions [1] They can be used to prove the equivalence of Boolean functions by ordering the variables of the BDD [11]. There are two phases for the construction of BDDs in the verification process: one where the BDDs for the individual circuits are built, and the second during which the equivalence between them is verified. As the sizes of circuits become large, the state space of the circuits increases in which ....

L. Fortune, J. Hopcroft and E. Schmidt, The Complexity of Equivalence and Containment for Free Single Variable Program Schemes, Vol. 62 of LNCS, Springer-Verlag, 1978.

....lower bounds are known, however, even for read twice only branching programs computing an explicit Boolean function, and no such bound will appear in this paper. In connection with the history of read once only branching programs we should mention a paper by Fortune, Hopcroft and Schmidt [FHS]. In the context of program schemes, they gave an exp(c p n) lower bound for computing an explicit function by read once only branching programs satisfying the additional restriction that the variables have to be examined in precisely the same order along each computation path. Without this ....

....an Omega Gamma 47 S(n) lower bound on the space complexity of the family f f n : n = 1; 2; g on any reasonable model of computation. The Fortune Hopcroft Schmidt result mentioned above corresponds to on line space complexity: the input bits are read once and in a given order only. The [FHS] result provides an Omega Gamma p n) space lower bound for such computation (independently of the given order of input bits) General read once only branching programs suggest the following machine model which we call eraser RAM. This is a RAM with a special read only input tape. The machine ....

S. Fortune, J. Hopcroft, E. M. Schmidt, The complexity of equivalence and containment free single variable program schemes, Cornell Univ. TR 77-310

....(nonoblivious) read once programs are not as easily manipulated as OBDDs. It is not known how to test equivalence in polynomial time, though there is a randomized (co RP) algorithm [BCW80] There is also a deterministic algorithm to test the equivalence of an OBDD and a readonce program [FHS78]. Moreover, the synthesis operations are provably intractable there exist functions f and g that each have polynomial size read once programs but whose conjunction f # g requires exponential size read once programs. For example, determining whether a 0, 1 matrix is a permutation matrix ....

S. Fortune, J. Hopcroft, and E. M. Schmidt, The complexity of equivalence and containment for free single variable program schemes, Lecture Notes in Comput. Sci. 62, Springer-Verlag, New York, 1978, pp. 227--240.

....parents. Note that the two subfunctions rooted at the ffl vertex do not have any common inputs. In the figure, attributed edges are indicated by dots on them. 3. 2 Probabilistic Equivalence of FBDs It has been shown that the noncontainment problem for FBDs without function vertices, is NPcomplete [8]. However Blum, Chandra and Wegman [1] have given a polynomial time algorithm to probabilistically decide the equivalence of two FBDs without and and xor vertices. A brief description of the bottom up version of the algorithm follows. Let F be an algebraic field with at least 2n elements. Let S ....

S. Fortune, J. Hopcroft, and E. M. Schmidt. The Complexity of Equivalence and Containment for Free Single Variable Program Schemes. In Goos, Hartmanis, Ausiello and B'ohm Eds., Lecture Notes in Computer Science, volume 62, pages 227--240, 1978.

....of discrete functions have be presented and studied extensively. A survey of the literature on the subject by Moret [7] cites over 100 references, but none of these describe a sufficient set of 3 algorithms to implement a Boolean function manipulation program. Fortune, Hopcroft, and Schmidt [8] studied the properties of graphs obeying similar restrictions to ours, showing that two graphs could be tested for functional equivalence in polynomial time and that some functions require much larger graphs under these restrictions than under milder restrictions. Payne [9] describes techniques ....

S. Fortune, J. Hopcroft, and E.M. Schmidt, "The Complexity of Equivalence and Containment for Free Single Variable Program Schemes", in Automata, Languages, and Programming, Goos, Hartmannis, Ausiello, and Boehm, eds., Springer-Verlag, Lecture Notes in Computer Science, Vol. 62, 1978, pp. 227-240.

.... by Sieling and Wegener [106] It is easy to see that the consistency test for the syntactic variant of read k times branching programs is in P, whereas it is coNP complete for non syntactic read k times branching programs if k 2 (this follows by a simple application of a well known reduction from [39]) There are some investigations on variants of branchings programs at the borderline between syntactic and non syntactic models. A (1; k) branching program is a read once branching program where for each computation path there may be a set of at most k variables which do not fulfill the ....

....arbitrary read k times branching programs as the class of representations, the answer to the above question is negative. First, it has been known for quite a long time that the satisfiability problem is NP complete already for read 2 times branching programs (Fortune, Hopcroft, and M. Schmidt [39]) For read once branching programs, satisfiability can be tested in polynomial time. But another problem with read k times branching programs even occurs in the case k = 1. This is the fact that the Boolean synthesis of two input graphs of polynomial size may lead to an exponentially large ....

S. Fortune, J. Hopcroft, and E. Meineche Schmidt. The complexity of equivalence and containment for free single variable program schemes. In Proc. of the 5th Int. Coll. on Automata, Languages, and Programming (ICALP), LNCS 62, 227--240. Springer-Verlag, 1978.

....ordering, which represents the same function as B (see [4] Thus the following two problems are motivated. 1. General equivalence test problem. Check whether a OBDD B 1 and a oeOBDD B 2 represent the same function. A polynomial time algorithm which solves this problem has been presented in [6]. 2. Variable ordering transformation problem. Compute starting from a OBDD B 1 and a permutation oe a oeOBDD B 2 representing the same function as B 1 . An efficient solution of this problem is important for the improvement of given variable orderings by simulated annealing or genetic ....

....ae (left) is a set of nodes of B 1 , and ae (right) is a set of equations fx i 1 = e i 1 ; x i = e i g ; for i 1 ; i pairwise distinct indices. The elements of the set L, which can be canonically regarded as pairs of PhiOBDDs, play an analogous role as in the OBDD case (see [6]) They must be equivalent in order that B 1 and B 2 be equivalent. But here, moreover, it is necessary to remove linear dependent pairs after having checked whether the left and the right components fulfill the same linear equations. Algorithm Inititialization: Let var be an integer ....

St. Fortune, J. Hopcroft, E. M. Schmidt, The complexity of equivalence and containment for free single variable program schemes, in: Proc. ICALP 1978, Lecture Notes in Computer Sci. 62, Springer--Verlag 1978, pp. 227--240.

....a different environment and with a different variable order. Memory and time requirements usually are lower but different variable order have to be coped with at the same moment. Another related problem is testing the equivalence of two given OBDDs which do not share the same variable order. In [FHS78] a polynomial time algorithm for deciding this problem is given, but this algorithm does not provide the difference equivalence of the OBDDs, i.e. the set of the input assignments on which the functions differ agree that is desired in some applications. In this paper we concentrate on the ....

S. Fortune, J. Hopcroft and E. Schmidt, The Complexity of Equivalence and Containment for Free Single Variable Program Schemes, Proc. of ICALP'78 Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, 227-240, 1978.

....a BDD generated in a different environment with a different variable order. Memory and time requirements usually are lower but different variable orders have to be coped with simultaneously. A related problem is the test for equivalence of two given OBDDs not sharing the same variable order. In [FHS78] a polynomial time algorithm for deciding this problem is given, but the algorithm does not compute the difference equivalence of the OBDDs, i.e. the set of the input assignments on which the functions differ agree that is desired in some applications. In this paper we concentrate on the ....

S. Fortune, J. Hopcroft and E. Schmidt, The Complexity of Equivalence and Containment for Free Single Variable Program Schemes, Proc. of ICALP'78 Automata, Languages and Programming, volume 62 of Lecture Notes in Computer Science, 227-240, 1978.

....the variable order dynamically during the construction of the OBDD, based on the exchange of adjacent variables [Rud93] In this paper we address the problem how to work with OBDDs of different variable orders, especially the combination of two OBDDs by a binary Boolean operator. Even though in [FHS78] an efficient algorithm for the equivalence operation applied to Free BDDs, i.e. BDDs without the ordering constraint is given, this algorithm cannot be used for computing an arbitrary Boolean operation. We address a similar problem by working with OBDDs that have different variable orders, but ....

S. Fortune, J. Hopcroft, and E. M. Schmidt. The Complexity of Equivalence and Containment for Free Single Variable Program Schemes, in ICALP'78 - Automata Languages and Programming, LNCS 62, pp.227--240, Springer, 1978

.... the theorem follows as this problem is NP complete [4] The process is as follows: given an obdd O, run the learning algorithm; membership queries are solved by evaluating them over O; equivalence queries are be solved in polynomial time using the algorithm of Fortune, Hopcroft, and Schmidt [6], which works even if the two obdds use different variable orderings. 2 By the same argument, we can show the following: suppose we have a polynomialtime learning algorithm that not necessarily outputs the optimum obdd, but is guaranteed to output a polynomial approximation to it (this is the ....

S. Fortune, J. Hopcroft, and E. Schmidt: "The complexity of equivalence and containment for free single variable program schemes". Proc. 5th Intl. Colloquium on Automata, Languages, and Programming. Springer-Verlag Lecture Notes in Computer Science 62 (1978), 227--240.

....Our conclusions are presented in Section 5. 2 Design Verification Ordered Binary Decision Diagrams (BDDs) are directed acyclic graphs which form a canonical representation of Boolean functions [1] They can be used to prove the equivalence of Boolean functions by ordering the variables of the BDD [12]. There are two phases for the construction of BDDs in the verification process: one where the BDDs for the individual circuits are built, and the second during which the equivalence between them is verified. As the sizes of circuits become large, the state space of the circuits increases in ....

J. H. L. Fortune and E. Schmidt. The complexity of equivalence and containment for free single variable program schemes. In LNCS, volume 62, 1978.

....closed class 1 if we can efficiently determine whether two different representations of a concept class are functionally equivalent then an EV oracle is not needed. Such equivalence tests exist for monotone DNF, read once branching programs [36] Horn sentences, free branching programs [14, 21], read once boolean formulas [28] and readtwice DNF [34] Also, if the number of unspecified attributes is O(log n) then evaluation can be done in polynomial time by simply considering all completions of x. Likewise, when h is a unate formula, then it is easily seen that the evaluation (for any ....

....closed class for whichwe can efficiently determine whether two different representations of a concept class are functionally equivalent then an EV oracle is not needed. Such equivalence tests exist for monotone DNF, read once branching programs [36] Horn sentences, free branching programs [14, 21], read once boolean formulas [28] and read twice DNF [34] Corollary 7 Let C be exactly learnable in polynomial time using the EQ and MQ oracles with a hypothesis class H where H is projection closed and for which there is a polynomial time algorithm to determine if h 1 2 H and h 2 2 H are ....

S. Fortune, J. Hopcroft, and E. Schmidt. The complexity of equivalence and containment for free single variable program schemes. In Lecture Notes in Computer Science 62: Proceedings of Automata, Languages and Programming, pages 227--240. Springer-Verlag, 1978.

....total ordering requirement of ROBDDs. One such relaxation is to allow variables to occur in any order but at most once along any path from the root to the terminal. Such BDDs are called Free BDDs (Figure 1(b) In general free BDDs are not canonical and their manipulation is an intractable problem [22]. However, in [25] it was shown that restricted forms of free BDDs known as typed Free BDDs are canonical and can be easily manipulated. In typed Free BDDs, for any given variable assignment, the resulting paths in all graphs contain variables in the same order. The variable ordering for different ....

L. Fortune et. al. The complexity of equivalence and containment for free single variable program schemes. Lecture Notes in Computer Science 62, Springer-Verlag, pages 227--240, 1978.

....branching programs. There are many papers presenting lower bound methods for FBDDs. The first ones are due to Wegener [20] and Zak [21] and in the paper of Simon and Szegedy [18] most previous approaches are handled in a unified way. Already in the early paper of Fortune, Hopcroft and Schmidt [7] it was shown that FBDDs are exponentially more powerful than OBDDs by presenting an example of a function with polynomial FBDD size but exponential OBDD size. The algorithmic aspects of FBDDs are investigated by Sieling and Wegener [17] and Gergov and Meinel [9] It turned out that many but not ....

Fortune, S., Hopcroft, J. and Schmidt, E.M. (1978). The complexity of equivalence and containment for free single variable program schemes. In Proc. of 5th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 62, 227--240.

No context found.

S. Fortune, J. Hopcroft, and E.M. Schmidt, "The Complexity of Equivalence and Containment for Free Single Variable Program Schemes", in Automata, Languages, and Programming, Goos, Hartmannis, Ausiello, and Boehm, eds., Springer-Verlag, Lecture Notes in Computer Science, Vol. 62, 1978, pp. 227-240.

No context found.

S. Fortune, J. Hopcroft, and E. Schmidt. The complexity of equivalence and containment for free single program schemes. In Proceedings Automata, pages 227-240, 1978.

No context found.

Fortune, S., Hopcroft, J., and Schmidt, E. M. (1978). The complexity of equivalence and containment for free single variable program schemes. In Proc. of 5th ICALP, LNCS 62, 227--240.

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