## The Boolean formula value problem is in ALOGTIME (1987)

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Venue: | in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing |

Citations: | 67 - 7 self |

### BibTeX

@INPROCEEDINGS{Buss87theboolean,

author = {Samuel R. Buss},

title = {The Boolean formula value problem is in ALOGTIME},

booktitle = {in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing},

year = {1987},

pages = {123--131}

}

### Years of Citing Articles

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### Abstract

The Boolean formula value problem is in alternating log time and, more generally, parenthesis context-free languages are in alternating log time. The evaluation of reverse Polish notation Boolean formulas is also in alternating log time. These results are optimal since the Boolean formula value problem is complete for alternating log time under deterministic log time reductions. Consequently, it is also complete for alternating log time under AC reductions.

### Citations

241 | Almost optimal lower bounds for small depth circuits
- H˚astad
- 1986
(Show Context)
Citation Context ...ranslation of infix formulas to postfix formulas is in deterministic log time. Indeed, the proofs of Main Theorem 1 and Main Theorem 9 give a fairly complicated translation. A. Yao [18] and J. Hastad =-=[8]-=- prove that there is an oracle which separates the polynomial time hierarchy; this immediately implies that the (unrelativized) log time hierarchy is proper. Hence the computational complexity of the ... |

237 | The Parallel Evaluation of General Arithmetic Expressions
- Brent
- 1974
(Show Context)
Citation Context ...ery formula of size n, there is an equivalent formula of size O(n 2 ) and depth O(log n). An improved construction, which also applied to the evaluation of rational expressions, was obtained by Brent =-=[2]-=-. Spira's result was significant in part because because it implied that there might be a family of polynomial size, log depth circuits for recognizing true Boolean formulas. In other words, that the ... |

152 |
On uniform circuit complexity
- RUZZO
- 1981
(Show Context)
Citation Context ...omial size, O(log n log log n)-depth circuits for the Boolean formula value problem. Our result improves on Cook, Gupta and Ramachandran since alternating log time is a subset of NC 1 . Indeed, Ruzzo =-=[15]-=- showed that alternating log time is the same as UE # -uniform NC 1 , where in this setting UE # -uniformity is a kind of alternating log time uniformity. It is shown below that alternating log time c... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...ue problem might be in (non-uniform) NC 1 . However, it was not known if the transformations of formulas defined by Brent and Spira could be done in NC 1 . Recent progress was made by Miller and Reif =-=[13]-=- who found improved algorithms for computing rational functions on PRAM's in O(log n)-time; however, they did not give improved circuits for the Boolean formula value problem. Very recently, Cook and ... |

94 | On relating time and space to size and depth
- Borodin
- 1977
(Show Context)
Citation Context ...sentence. As mentioned above, N. Lynch [11] first studied the complexity of the Boolean formula problem. It follows from Lynch's work that the Boolean formula value problem is in NC 2 , since Borodin =-=[1]-=- showed that LOGSPACE # NC 2 . Another early significant result on this problem was due to Spira [17] who showed that for every formula of size n, there is an equivalent formula of size O(n 2 ) and de... |

80 |
Models of Computation
- Savage
- 1998
(Show Context)
Citation Context ... a Boolean formula (or: PLOF formula) is in alternating log time. An important area in research in computational complexity is to study the circuit complexity of various decision problems (see Savage =-=[16]-=- or Cook [5] for instance). A related, but less popular, approach to computational complexity is to study the formula complexity of decision problems where a formula is a circuit with fan-out one. It ... |

61 |
The circuit value problem is log space complete for P
- Ladner
- 1975
(Show Context)
Citation Context ...ons. A variant of this problem, the postfix notation Boolean formula value problem is also alternating log time complete under deterministic log time reductions. This complements the result of Ladner =-=[10]-=- that the circuit value problem is complete for polynomial time under log space reductions. It is a longstanding open problem whether for every circuit there is an equivalent formula with the size of ... |

38 |
On time hardware complexity tradeoffs for boolean functions
- Spira
- 1971
(Show Context)
Citation Context ...blem. It follows from Lynch’s work that the Boolean formula value problem is in NC2 , since Borodin [1] showed that LOGSP ACE ⊆ NC2 . Another early significant result on this problem was due to Spira =-=[17]-=- who showed that for every formula of size n, there is an equivalent formula of size O(n2 ) and depth O(log n). An improved construction, which also applied to the evaluation of rational expressions, ... |

35 | Parenthesis grammars - McNaughton - 1967 |

32 |
Deterministic CFL’s are accepted simultaneously in polynomial time and log squared space
- Cook
- 1979
(Show Context)
Citation Context ...formity conditions on circuits. We shall use a condition called ALOGT IME-uniformity for formulas. This is similar in spirit to the UBC -uniformity of Ruzzo, originally introduced by Borodin and Cook =-=[4]-=-; except that alternating log time computablility is substituted for log space computability. Definition: A family of formulas is ALOGT IME-uniform if and only if the predicate F is in ALOGT IME where... |

23 | Log space recognition and translation of parenthesis languages
- Lynch
- 1977
(Show Context)
Citation Context ... 0 reductions. 1. Introduction The Boolean formula value problem is to determine the truth value of a variable-free Boolean formula, or equivalently, to recognize the true Boolean sentences. N. Lynch =-=[11]-=- gave log space algorithms for the Boolean formula value problem and for the more general problem of recognizing a parenthesis context-free grammar. This paper shows that these problems have alternati... |

11 |
Separating the polynomial time hierarchy by oracles
- Yao
- 1985
(Show Context)
Citation Context ... that the natural translation of infix formulas to postfix formulas is in deterministic log time. Indeed, the proofs of Main Theorem 1 and Main Theorem 9 give a fairly complicated translation. A. Yao =-=[18]-=- and J. Hastad [8] prove that there is an oracle which separates the polynomial time hierarchy; this immediately implies that the (unrelativized) log time hierarchy is proper. Hence the computational ... |

8 |
On time hardware complexity tradeo#s for Boolean functions
- Spira
- 1971
(Show Context)
Citation Context ...lem. It follows from Lynch's work that the Boolean formula value problem is in NC 2 , since Borodin [1] showed that LOGSPACE # NC 2 . Another early significant result on this problem was due to Spira =-=[17]-=- who showed that for every formula of size n, there is an equivalent formula of size O(n 2 ) and depth O(log n). An improved construction, which also applied to the evaluation of rational expressions,... |

3 |
A fast parallel algorithm for recognition of parenthesis languages
- Gupta
- 1985
(Show Context)
Citation Context ...ound improved algorithms for computing rational functions on PRAM's in O(log n)-time; however, they did not give improved circuits for the Boolean formula value problem. Very recently, Cook and Gupta =-=[7]-=- and, independently, Ramachandran [14] were successful in giving polynomial size, O(log n log log n)-depth circuits for the Boolean formula value problem. Our result improves on Cook, Gupta and Ramach... |

2 |
Notes on log space representation. typewritten manuscript
- Dowd
- 1986
(Show Context)
Citation Context ...s from a node to its parent and children. In this formulation the formula value problem is equivalent to the reachability problem and hence is logspace complete --- this was shown by Dowd and Statman =-=[6]-=- and independently by Tompa. 6. Parenthesis Languages are in ALOGT IME Parenthesis languages, first studied by McNaughton [12], are context-free grammars of the form (V, T , P, S) which have two disti... |

2 |
Restructuring formula trees. Unpublished manuscript
- Ramachandran
- 1986
(Show Context)
Citation Context ...g rational functions on PRAM's in O(log n)-time; however, they did not give improved circuits for the Boolean formula value problem. Very recently, Cook and Gupta [7] and, independently, Ramachandran =-=[14]-=- were successful in giving polynomial size, O(log n log log n)-depth circuits for the Boolean formula value problem. Our result improves on Cook, Gupta and Ramachandran since alternating log time is a... |

1 |
On some languages in nc (summary
- Ibarra, Jiang, et al.
- 1987
(Show Context)
Citation Context ... a string of symbols from #, determines if the input is a Boolean formula. The answer is yes; the ALOGT IME algorithm uses the fact that counting is ALOGT IME computable. (Ibarra, Jiang and Ravikumar =-=[9]-=- use a di#erent approach and obtain the stronger result that one-sided Dyck languages on k letters are in ALOGT IME.) For our purposes, it is convenient to describe the alternating log time Turing mac... |

1 |
On some languages in nc 1 (summary
- Ibarra, Jiang, et al.
- 1987
(Show Context)
Citation Context ... a string of symbols from Σ, determines if the input is a Boolean formula. The answer is yes; the ALOGT IME algorithm uses the fact that counting is ALOGT IME computable. (Ibarra, Jiang and Ravikumar =-=[9]-=- use a different approach and obtain the stronger result that one-sided Dyck languages on k letters are in ALOGT IME.) For our purposes, it is convenient to describe the alternating log time Turing ma... |