On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs

On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs

Error-free affine, unitary, and probabilistic OBDDS

© IFIP International Federation for Information Processing 2018. We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata versions of these models

Reordering method and hierarchies for quantum and classical ordered binary decision diagrams

© Springer International Publishing AG 2017.We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between determin-istic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is 2Ω(n/logn) and present quantum OBDD of width O(n2). It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs. Additionally, we prove the width hierarchy for complexity classes of Boolean functions for bounded error probabilistic OBDDs. And using “reordering” method we extend a hierarchy for k-OBDD of polynomial size, for k = o(n/log3n). Moreover, we proved a similar hierarchy for bounded error probabilistic k-OBDD. And for deterministic and proba-bilistic k-OBDDs of superpolynomial and subexponential size

On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs

On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs

On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

© 2016, Pleiades Publishing, Ltd.The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read ktimes Branching programs for k=o(logn/loglogn) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003. We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a “functional description” of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs. Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs