## Time-Space Tradeoffs for Satisfiability (1997)

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Venue: | Journal of Computer and System Sciences |

Citations: | 30 - 1 self |

### BibTeX

@ARTICLE{Fortnow97time-spacetradeoffs,

author = {Lance Fortnow},

title = {Time-Space Tradeoffs for Satisfiability},

journal = {Journal of Computer and System Sciences},

year = {1997},

volume = {60},

pages = {2000}

}

### Years of Citing Articles

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### Abstract

We give the first nontrivial model-independent time-space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for log-space uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...

### Citations

11418 |
Computers and Intractability: A Guide to the Theory of NP -completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ... P, NP and the polynomial-time hierarchy have been well studied. Definitions and basic results of these classes can be found in basic textbooks such as Hopcroft and Ullman [HU79] or Garey and Johnson =-=[GJ79]-=-. We use the multitape model of Turing machines though as we discuss in Section 3, our lower bounds for SAT hold in more general models. The results given only hold if the time and space bounds have t... |

4100 |
Introduction to Automata Theory, Languages and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...ussed in this paper like NL, P, NP and the polynomial-time hierarchy have been well studied. Definitions and basic results of these classes can be found in basic textbooks such as Hopcroft and Ullman =-=[HU79]-=- or Garey and Johnson [GJ79]. We use the multitape model of Turing machines though as we discuss in Section 3, our lower bounds for SAT hold in more general models. The results given only hold if the ... |

824 | The complexity of theorem-proving procedures
- Cook
- 1971
(Show Context)
Citation Context ...cepted by Turing machines using simultaneously t(n) time and s(n) space. NTISP[t(n); s(n)] is the nondeterministic version of this class. SAT consists of the set of satisfiable boolean formulae. Cook =-=[Coo71]-=- and Levin [Lev73] independently show that SAT is NP-complete. Later Cook [Coo88], building on work of Pippenger and Fischer [PF79] and Hennie and Stearns [HS66], shows how to reduce nondeterministic ... |

318 |
Relationships between nondeterministic and deterministic tape complexities
- Savitch
- 1970
(Show Context)
Citation Context ...elativization models that make NL = NP, these same models also can collapse NL and AP (alternating polynomial time) even though we know these classes differ (PSPACE = AP [CKS81], NL ` DSPACE[log 2 n] =-=[Sav70]-=- and PSPACE strictly contains DSPACE[log 2 n] [HS65]). Research done while on leave at the Centrum voor Wiskunde en Informatica in Amsterdam. URL: http://www.cs.uchicago.edu/~fortnow. Email: fortnow@c... |

239 |
Some connections between non-uniform and uniform complexity classes
- Karp, Lipton
- 1980
(Show Context)
Citation Context ...ur algorithm will use only linear time. 2 5 Circuits and Branching Programs Theorem 3.1 gives a lower bound for satisfiability on log-time uniform circuits. Harry Buhrman shows how to use Karp-Lipton =-=[KL80]-=- combined with the techniques of Section 4 to improve the bound to log-space uniformity. Theorem 5.1 (Buhrman) SAT cannot be solved by logarithmic-space uniform NC 1 circuits of size n 1+o(1) . Proof ... |

237 | Almost Optimal Lower Bounds for Small Depth Circuits, volume 5
- H˚astad
- 1989
(Show Context)
Citation Context ...AT A ` DTIME A [n 1+ffl ] but P A 6= \Sigma p;A s(n) . This proof builds on techniques used by Ko [Ko89] who presented a relativized world where P = NP 6= PSPACE which in turn built on work by Hastad =-=[Has89]-=-. In particular we make use of the following bounds for parity. Lemma 6.4 (Hastad) For sufficiently large m and d, no depth d circuit of size 2 m 1=2d can compute parity of m input variables. However,... |

199 |
On the computational complexity of algorithms
- Hartmanis, Stearns
- 1965
(Show Context)
Citation Context ...dels also can collapse NL and AP (alternating polynomial time) even though we know these classes differ (PSPACE = AP [CKS81], NL ` DSPACE[log 2 n] [Sav70] and PSPACE strictly contains DSPACE[log 2 n] =-=[HS65]-=-). Research done while on leave at the Centrum voor Wiskunde en Informatica in Amsterdam. URL: http://www.cs.uchicago.edu/~fortnow. Email: fortnow@cs.uchicago.edu. Supported in part by NSF grant CCR 9... |

190 |
PP is as hard as the polynomial-time hierarchy
- Toda
- 1991
(Show Context)
Citation Context ...E[2 f(n) ]. By assumption we also have P = NL so DTIME[n] ` NSPACE[logn]. By padding we have DTIME[2 f(n) ] ` NSPACE[f(n)]. The lemma follows. 2 Theorem 6.1 also gives limitations to improving Toda's =-=[Tod91]-=- theorem. Toda showed that any constant level of the polynomial-time hierarchy can be reduced to the complexity class PP. We show that extending his result would yield a nice separation. Corollary 6.6... |

176 | Natural Proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ...he Dutch Foundation for Scientific Research (NWO) and a Fulbright Scholar award. 1 Diagonalization over uniform classes also avoids the limits of combinatorial proofs described by Razborov and Rudich =-=[RR97]-=-. We make partial progress by giving some new time-space tradeoffs for satisfiability. We prove a general result: For r(n) any unbounded function such that r(n) = O( log n log log n ) and ffl ? 0, SAT... |

175 |
The complexity of finite functions
- Boppana, Sipser
- 1990
(Show Context)
Citation Context ... classes remains the most important and difficult of problems in theoretical computer science. Circuit complexity and other techniques on finite functions have seen some exciting early successes (see =-=[BS90]-=-) but have yet to achieve their promise of separating complexity classes above logarithmic space. Other techniques based on logic and geometry also have given us separations only on very restricted mo... |

102 |
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
- Babai, Nisan, et al.
- 1992
(Show Context)
Citation Context ...g n s(n) \Gamma \Sigma LIN O(1) . 2 Note that no nonlinear lower bound on SAT is known for nonuniform NC 1 circuits. Using a similar proof, we can get lower bounds for uniform branching programs (see =-=[BNS92]-=-). Corollary 5.4 SAT cannot be computed on log-space uniform branching programs of size n 1+o(1) . 6 NP versus NL Lemma 4.2 gives an immediate separation of nondeterministic logarithmic space and unbo... |

102 | Relations among complexity measures - Fischer, Pippenger - 1979 |

57 | Two-tape simulation of multitape Turing machines
- Hennie, Stearns
- 1966
(Show Context)
Citation Context ...of satisfiable boolean formulae. Cook [Coo71] and Levin [Lev73] independently show that SAT is NP-complete. Later Cook [Coo88], building on work of Pippenger and Fischer [PF79] and Hennie and Stearns =-=[HS66]-=-, shows how to reduce nondeterministic time to a satisfiability question of a small formula. Lemma 2.2 (Cook) Let M be a nondeterministic Turing machine running in time t(n). There is a O(t(n) log t(n... |

41 | The role of relativization in complexity theory
- Fortnow
- 1994
(Show Context)
Citation Context ...al-time (NP). We have no inherent reason to believe that diagonalization will not succeed in separating NP from NL. Relativization results for space-bounded classes are hard to interpret (see Fortnow =-=[For94]-=-). While there are relativization models that make NL = NP, these same models also can collapse NL and AP (alternating polynomial time) even though we know these classes differ (PSPACE = AP [CKS81], N... |

41 |
On determinism versus nondeterminism and related problems
- Paul, Pippenger, et al.
- 1983
(Show Context)
Citation Context ...s allowed. Kannan shows that NTIME[n] 6` DTIME[n] " NL in the sequential input model. Like our paper, Kannan makes use of Nepomnjascii's Theorem [Nep70]. Later Paul, Pippenger, Szemer'edi and Tro=-=tter [PPST83]-=- separate NTIME[n] and DTIME[n] using different techniques. Kannan also proves the following curious related result. Theorem 1.1 (Kannan) There is a k such that for all t(n) such that n k = o(t(n)) an... |

40 |
The Recognition Problem for the Set of Perfect Squares
- Cobham
- 1966
(Show Context)
Citation Context ...ses might be not nearly as difficult as previously believed, perhaps considerably easier than separating P from NP. 1.1 Related Work The question of time-space tradeoffs goes back to 1966 when Cobham =-=[Cob66]-=- showed that palindromes required quadratic time-space tradeoffs on a Turing machine with a single-head on a readonly input tape. Much of the work on time-space tradeoffs deals with restricted machine... |

38 | A survey of Russian approaches to Perebor (brute-force search) algorithms - Trakhtenbrot - 1984 |

37 |
Universal’nyĭe perebornyĭe zadachi (Universal search problems
- Levin
- 1973
(Show Context)
Citation Context ...achines using simultaneously t(n) time and s(n) space. NTISP[t(n); s(n)] is the nondeterministic version of this class. SAT consists of the set of satisfiable boolean formulae. Cook [Coo71] and Levin =-=[Lev73]-=- independently show that SAT is NP-complete. Later Cook [Coo88], building on work of Pippenger and Fischer [PF79] and Hennie and Stearns [HS66], shows how to reduce nondeterministic time to a satisfia... |

36 | Satisfiability is quasilinear complete in NQL
- Schnorr
- 1978
(Show Context)
Citation Context ...(n) question of length O(t(n) log t(n)). Gurevich and Shelah [GS89] show how to simulate a nondeterministic random-access machines by multitape Turing machines. Their proof builds on Schnorr's result =-=[Sch78]-=- that one can sort m elements on a multitape Turing machine in O(m log O(1) m) time. Theorem 2.3 (Gurevich-Shelah-Schnorr) Every language that is accepted by a nondeterministic random-access machine u... |

26 |
On spectra
- BENNETT
- 1962
(Show Context)
Citation Context ...ammaffl ] ` [ k \Sigma LIN k While credit is usually given to Nepomnjascii for Theorem 4.4, the result has quite an interesting history. Smullyan [Smu61, p. 147] defined the rudimentary sets. Bennett =-=[Ben62]-=- proves a result on rudimentary sets that implies that every language computed in polynomial time and square root space is rudimentary. Nepomnjascii [Nep70] extended Bennett's work to Turing machines ... |

25 |
Near-optimal Time-Space Tradeoff for Element Distinctness
- Yao
- 1994
(Show Context)
Citation Context ...atic time-space tradeoffs on a Turing machine with a single-head on a readonly input tape. Much of the work on time-space tradeoffs deals with restricted machine models such as comparison models (see =-=[Yao94]-=-) and JAG models (see [EP95]). For Turing machines, lower bounds typically require restricted access to the input, usually with heads that read the input tape sequentially (see [GS90, DG84, Kar86]) bu... |

24 | Using autoreducibility to separate complexity classes
- Buhrman, Fortnow, et al.
(Show Context)
Citation Context ...f our current understanding of complexity classes. Acknowledgments Much of the research of this paper was motivated by the author's research with Harry Buhrman and Leen Torenvliet on autoreducibility =-=[BFT95]-=-. The author thanks Harry Buhrman for many initial discussions on this paper and for allowing the inclusion of Theorem 5.1 and its proof in this paper. The author thanks Eric Allender for pointing out... |

22 |
Towards separating nondeterminism from determinism
- Kannan
- 1984
(Show Context)
Citation Context ... Bennett's work to Turing machines and showed that every set in NTISP[n O(1) ; n 1\Gammaffl ] is rudimentary. Wrathall [Wra78] later showed that the rudimentary sets are exactly \Sigma LIN k . Kannan =-=[Kan84]-=- rediscovers Theorem 4.4 and shows that it holds for random-access machines. Kannan [Kan84] and independently Woods [Woo86] generalize this theorem to show constant alternating polynomial time and n 1... |

21 |
Rudimentary predicates and relative computation
- WRATHALL
- 1978
(Show Context)
Citation Context ...lynomial time and square root space is rudimentary. Nepomnjascii [Nep70] extended Bennett's work to Turing machines and showed that every set in NTISP[n O(1) ; n 1\Gammaffl ] is rudimentary. Wrathall =-=[Wra78]-=- later showed that the rudimentary sets are exactly \Sigma LIN k . Kannan [Kan84] rediscovers Theorem 4.4 and shows that it holds for random-access machines. Kannan [Kan84] and independently Woods [Wo... |

20 |
Relativized polynomial time hierarchies having exactly k levels
- Ko
- 1989
(Show Context)
Citation Context ...n) computable in time polynomial in n and every constant ffl ? 0, there exists an oracle A such that SAT A ` DTIME A [n 1+ffl ] but P A 6= \Sigma p;A s(n) . This proof builds on techniques used by Ko =-=[Ko89]-=- who presented a relativized world where P = NP 6= PSPACE which in turn built on work by Hastad [Has89]. In particular we make use of the following bounds for parity. Lemma 6.4 (Hastad) For sufficient... |

15 |
An O(T log T) reduction from RAM computations to satisfiability
- Robson
- 1991
(Show Context)
Citation Context ...machine running in time t(n). There is a O(t(n) log t(n)) time and O(log t(n)) space algorithm that maps inputs x of length n to formulae OE of size O(t(n) log t(n)) such that x 2 L , OE 2 SAT Robson =-=[Rob91]-=- shows that Lemma 2.2 holds even for random-access Turing machines. Let the language QBF s(n) consist of quantified boolean formulae restricted to s(n) \Gamma 1 alternations where the first quantifier... |

15 |
Rudimentary predicates and turing calculations
- Nepomnjaˇsčiĭ
- 1970
(Show Context)
Citation Context ...time then we can collapse more than a constant number of levels of the polynomial-time hierarchy to a small amount of nondeterministic time. We then consider an extension of the work of Nepomnjaˇsčiĭ =-=[Nep70]-=- that shows that any language computable in nondeterministic time n α(n) and space n 1−ɛ can be solved in α(n) alternations and linear time for any α(n) = n o(1) . Our main results follow by combining... |

14 |
Nearly-linear time
- Gurevich, Shelah
- 1989
(Show Context)
Citation Context ...articular, using the ideas of the proof of Cook's Lemma (Lemma 2.2), we get that any language in \Sigma t(n) s(n) can be reduced to a QBF s(n) question of length O(t(n) log t(n)). Gurevich and Shelah =-=[GS89]-=- show how to simulate a nondeterministic random-access machines by multitape Turing machines. Their proof builds on Schnorr's result [Sch78] that one can sort m elements on a multitape Turing machine ... |

12 | Two Time-Space Tradeoffs for Element Distinctness,” Theoretical - Karchmer - 1986 |

11 |
propositional formulas represent nondeterministic computations
- Short
- 1988
(Show Context)
Citation Context ...n); s(n)] is the nondeterministic version of this class. SAT consists of the set of satisfiable boolean formulae. Cook [Coo71] and Levin [Lev73] independently show that SAT is NP-complete. Later Cook =-=[Coo88]-=-, building on work of Pippenger and Fischer [PF79] and Hennie and Stearns [HS66], shows how to reduce nondeterministic time to a satisfiability question of a small formula. Lemma 2.2 (Cook) Let M be a... |

11 |
Rudimentary predicates and turing calculations
- Nepomnjascii
- 1970
(Show Context)
Citation Context ... time then we can collapse more than a constant number of levels of the polynomial-time hierarchy to a small amount of nondeterministic time. We then consider an extension of the work of Nepomnjascii =-=[Nep70]-=- that shows that any language computable in nondeterministic time n ff(n) and space n 1\Gammaffl can be solved in ff(n) alternations and linear time for any ff(n) = n o(1) . Our main results follow by... |

11 | Einfuhrung in die Komplexitatstheorie - Reischuk - 1990 |

9 |
A nearly optimal time-space lower bound for directed stconnectivity on the NNJAG model
- Edmonds, Poon
- 1995
(Show Context)
Citation Context ...a Turing machine with a single-head on a readonly input tape. Much of the work on time-space tradeoffs deals with restricted machine models such as comparison models (see [Yao94]) and JAG models (see =-=[EP95]-=-). For Turing machines, lower bounds typically require restricted access to the input, usually with heads that read the input tape sequentially (see [GS90, DG84, Kar86]) but these results break down i... |

7 | Nondeterministic linear-time tasks may require substantially nonlinear deterministic time in the case of sublinear work space - Gurevich, Shelah - 1990 |

7 | Theory of Formal Systems, volume 47 of Annals of Mathematical Studies - Smullyan - 1961 |

6 | A time-space tradeoff for language recognition - Duris, Galil - 1984 |

3 |
Relativized isomorphisms of NP-complete sets
- Goldsmith, Joseph
- 1993
(Show Context)
Citation Context ...Turing machine or a random-access machine. By Lemma 2.2 we have that SAT is complete for NQL under quasi-linear time reductions. We use SAT A to represent a relativized version of satisfiability (see =-=[GJ93]-=-). Relativized SAT A has several extra predicate A 0 ; A 1 ; : : : such that Am (x 1 ; : : : ; xm ) has the property that x 1 : : : xm 2 A , Am (x 1 ; : : : ; xm ) 4 For every oracle A, SAT A has the ... |

3 |
Emde Boas. The problem of space invariance for sequential machines
- Slot, van
- 1988
(Show Context)
Citation Context ...hine using the same number of alternations. For space one can simulate such a RAM on a multitape machine by simply rereading the input and the entire memory used every time a memory call is made (see =-=[Sv88]-=-). Theorem 2.4 Every language accepted by a random-access Turing machine using space s(n) and time t(n) can be simulated by a multitape Turing machine using s(n) space and t 2 (n) time. We say a funct... |

3 |
Bounded arithmetic formulas and Turing machines of constant alternation
- Woods
- 1986
(Show Context)
Citation Context ...78] later showed that the rudimentary sets are exactly \Sigma LIN k . Kannan [Kan84] rediscovers Theorem 4.4 and shows that it holds for random-access machines. Kannan [Kan84] and independently Woods =-=[Woo86]-=- generalize this theorem to show constant alternating polynomial time and n 1\Gammaffl space is contained in S k \Sigma LIN k . Reischuk [Rei90, p. 282] gives an even broader generalization (Theorem 4... |