# Probabilistic Boolean decision trees and the complexity of evaluating game trees

@article{Saks1986ProbabilisticBD, title={Probabilistic Boolean decision trees and the complexity of evaluating game trees}, author={Michael E. Saks and Avi Wigderson}, journal={27th Annual Symposium on Foundations of Computer Science (sfcs 1986)}, year={1986}, pages={29-38} }

The Boolean Decision tree model is perhaps the simplest model that computes Boolean functions; it charges only for reading an input variable. We study the power of randomness (vs. both determinism and non-determinism) in this model, and prove separation results between the three complexity measures. These results are obtained via general and efficient methods for computing upper and lower bounds on the probabilistic complexity of evaluating Boolean formulae in which every variable appears… Expand

#### Topics from this paper

#### 209 Citations

Randomized Boolean Decision Trees: Several Remarks

- Mathematics, Computer Science
- Theor. Comput. Sci.
- 1998

It is proved that there is a probability distribution over the set of all assignments to variables of a Boolean function with respect to which the average cost of any deterministic decision tree computing that function is at least c. Expand

On the power of randomness in the decision tree model

- Mathematics, Computer Science
- Proceedings Fifth Annual Structure in Complexity Theory Conference
- 1990

Results suggest that there are relations between the decision tree complexity of a Boolean function and its symmetry and consideration is given to the question of what distinguishes graph properties from other highly symmetric Boolean functions, where randomization can help significantly. Expand

Principles of Optimal Probabilistic Decision Tree Construction

- Computer Science
- FCS
- 2006

A method is provided, which can be used to construct the optimal probabilistic decision tree for a given Boolean function, which takes into account the symmetries of given function. Expand

Randomized vs. deterministic decision tree complexity for read-once Boolean functions

- Mathematics, Computer Science
- [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference
- 1991

A lower bound of n/sup 0.51/ on the number of positions that have to be evaluated by any randomized alpha - beta pruning algorithm computing the value of any two-person zero-sum game tree with n final positions is provided. Expand

On Directional vs. General Randomized Decision Tree Complexity for Read-Once Formulas

- Computer Science, Mathematics
- Chic. J. Theor. Comput. Sci.
- 2011

A systematic search for a certain class of functions is conducted and an explicit construction of a read-once Boolean formula f on n variables is provided such that the cost of the optimal directional randomized decision tree for f is Ω(nα) and the cost for the optimal randomized decision trees (without the directional restriction) for f are O(nβ ) with α−β > 0.0101. Expand

On P versus NP
$ \cap $ co-NP for decision trees and read-once branching programs

- Computer Science, Mathematics
- computational complexity
- 1999

It is shown that this simulation cannot be made polynomial: explicit Boolean functions f that require deterministic trees of size exp (\Omega({\rm log^2} N) $ where N is the total number of monomials in minimal DNFs for f and ¬f are exhibited. Expand

On O versus NP \cap co-NP for Decision Trees and Read-Once Branching Programs

- Mathematics, Computer Science
- MFCS
- 1997

It is shown that this simulation cannot be made polynomial: explicit Boolean functions f that require deterministic trees of size exp (Ω(log2N)) where N is the total number of monomials in minimal DNFs for f and -f are exhibited. Expand

Lower Bounds on the Randomized Communication Complexity of Read-Once Functions

- Computer Science, Mathematics
- 2009 24th Annual IEEE Conference on Computational Complexity
- 2009

Information theory methods are used to prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae, which are optimal up to the constant in the base of the denominator. Expand

Randomized vs. deterministic decision tree complexity for read-once Boolean functions

- Mathematics, Computer Science
- computational complexity
- 2005

This work generalizes an existing lower bound technique and combines it with restriction arguments to provide a lower bound ofn0.51 on the number of positions that have to be evaluated by any randomized α-β pruning algorithm computing the value of any two-person zero-sum game tree withn final positions. Expand

On the Monte Carlo Boolean Decision Tree Complexity of Read-Once Formulae

- Mathematics, Computer Science
- Random Struct. Algorithms
- 1995

For a large class of read‐once formulae that this trivial speed‐up is the best that a Monte Carlo algorithm can achieve, a general lower bound is derived on the Monte Carlo complexity of these formULae. Expand

#### References

SHOWING 1-10 OF 17 REFERENCES

The complexity of problems on probabilistic, nondeterministic, and alternating decision trees

- Mathematics, Computer Science
- JACM
- 1985

This work generalizes decision trees in order to study lower bounds on the running times of algorithms that allow probabilistic, nondeterministic, or alternating control. It is shown that decision… Expand

Asymptotic Properties of Minimax Trees and Game-Searching Procedures

- Mathematics, Computer Science
- Artif. Intell.
- 1980

It is shown that a game with WIN-LOSS terminals can be solved by examining, on the average, O [(d) h 2 ] terminal positions if positions if P 0 ≠ P∗ and O [(P∗ (1 − P ∗) ) h ] positionsif P 0 = P∷, the former performance being optimal for all search algorithms. Expand

Nondeterministic versus probabilistic linear search algorithms

- Computer Science
- 26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
- 1985

The proof of the lower bound differs fundamentally from all known lower bounds for LSA's or PLSA's, because it does not reduce the problem to a combinatorial one but argues extensively about e.g. a non-discrete measure for similarity of sets in Rn. Expand

On Recognizing Graph Properties from Adjacency Matrices

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 1976

A non-constructive argument (not based on the construction of an “oracle”) settles this question for d a prime power: at least v216 entries of the adjacency matrix of a v-vertex undirected graph G must be examined in the worst case to determine if G has any given non-trivial monotone graph property. Expand

On the time required to recognize properties of graphs: a problem

- Computer Science
- SIGA
- 1973

In a recent paper [i], Holt and Reingold have proved the following results: any algorithm which, given an n-node graph, detects whether or not the graph enjoys property P must in the worst case probe 0(n 2) entries of the incidence matrix. Expand

Probabilistic computations: Toward a unified measure of complexity

- Computer Science
- 18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
- 1977

Two approaches to the study of expected running time of algoritruns lead naturally to two different definitions of intrinsic complexity of a problem, which are the distributional complexity and the randomized complexity, respectively. Expand

The solution for the branching factor of the alpha-beta pruning algorithm and its optimality

- Computer Science
- CACM
- 1982

When k > top > 2, one can show that the probability for a specific value of top that a[top] = stop(top) is a(top + l) /a( top) , which reduces to (k top + l ) / (n top) so it is very unlikely that the next combination is generated by using the theoretical maxim of operations. Expand

Optimal Search on Some Game Trees

- Computer Science
- JACM
- 1983

It is proved that the dlrecUonal algorithm for solving a game tree is optimal, in the sense of average run trine, for balanced trees (a family containing all uniform trees). This result implies that… Expand

A topological approach to evasiveness

- Mathematics, Computer Science
- Comb.
- 1984

The truth of Karp's conjecture is shown to follow from another conjecture concerning group actions on topological spaces and a special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices. Expand

Lower Bounds on Probabilistic Linear Decision Trees

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 1985

It is shown that the standard arguments used to prove lower bounds on deterministic linear decision trees apply to probabilistic linear decision Trees as well. Expand