# Logspace vs Polynomial time

One of the primary goals of complexity theory is separating complexity classes, a.k.a proving lower bounds. Embarrassingly we have only a handful of unconditional separation results. Separating P from NP is of course the mother of all such goals. Anybody who understands the philosophical underpinnings of the P vs NP problem would love to LIVE to see its resolution. Towards resolving this, we made some (“anti”)-progress (Eg : Relativization, Natural proofs, Algebrization) and have a new geometric complexity theory approach which relies on an Extended-Extended-Extended-Extended-Riemann-Hypothesis !! For more information about the history and status of P vs NP problem read Sipser’s paper [Sipser’92], Allender’s status report [Allender’09] or Fortnow’s article [Fortnow’09].

Today’s post is about the Logspace (L) vs Polynomial time (P) problem, which (in my opinion) is right next to the P vs NP problem in its theoretical importance. I guess many researchers believe that $L \neq P$. Did we make any progress/anti-progress towards resolving the $L \neq P$ conjecture ?  Here are two attempts both based on branching programs and appeared in MFCS with a gap of 20 years !!

1) A conjecture by Barrington and McKenzie (BM’89): The problem $GEN$ is defined as follows : $GEN$ : Given an $n \times n$ table filled with entries from $\{1,2,\dots,n\}$, which we interpret as the multiplication table of an $n$-element groupoid, and a subset $S$ of $\{1,2,\dots,n\}$ which includes element 1, determine whether the subgroupoid $$, defined as the closure of $S$ under the groupoid product, includes element $n$.

Barrington-McKenzie Conjecture : For each $n > 1$, a branching program in which each node can only evaluate a binary product within an $n$-element groupoid, branching $n$ ways according to the $n$ possible outcomes, must have at least $2^{n-2}$ nodes to solve all $n \times n$ $GEN$ instances with singleton starting set $S$.

The problem $GEN$ is known to be P-complete [JL’76]. Barrington-McKenzie Conjecture would imply that $GEN \notin DSPACE({{\log}^k}n)$ for any $k$. In particular, it would imply that $L \neq P$. I don’t know if there is any partial progress towards resolving this conjecture.

2) Thrifty Hypothesis : This is a recent approach by Braverman et. al [BCMSW’09] towards proving a stronger theorem $L \neq LogDCFL$. Stephen Cook presented this approach at Valiant’s 60th birthday celebration and Barriers Workshop. He also announced a $100 prize for solving an intermediate open problem mentioned in his slides. Tree Evaluation Problem (TEP): The input to the problem is a rooted, balanced $d$-ary tree of height $h$, whose internal nodes are labeled with $d$-ary functions on $[k] = \{1, . . . , k\}$, and whose leaves are labeled with elements of $[k]$. Each node obtains a value in $[k]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root. In their paper they show that $TEP \in LogDCFL$ and conjecture that $TEP \notin L$. They introduce Thrifty Branching Programs and prove that TEP can be solved by a thrifty branching program. A proof of the following conjecture implies that $L \neq LogDCFL$. For more details, read this paper. Thrifty Hypothesis : Thrifty Branching Programs are optimal among deterministic branching programs solving TEP. Open Problems : • My knowledge about the history of L vs P problem is limited. Are there other approaches/attempts in the last four decades to separate L from P ? • An intermediate open problem is mentioned in the last slide of these slides. The authors announced$100 prize for the first correct proof. Read their paper for more open problems.

References :

• [BM’89] David A. Mix Barrington, Pierre McKenzie: Oracle Branching Programs and Logspace versus P. MFCS 1989: 370-379
• [BCMSW’09] Mark Braverman, Stephen A. Cook, Pierre McKenzie, Rahul Santhanam, Dustin Wehr: Branching Programs for Tree Evaluation. MFCS 2009: 175-186
• [Sipser’92] Michael Sipser: The History and Status of the P versus NP Question STOC 1992: 603-618
• [Allender’09] Eric Allender: A Status Report on the P Versus NP Question. Advances in Computers 77: 117-147 (2009) [pdf]
• [Fortnow’09] Lance Fortnow: The status of the P versus NP problem. Commun. ACM 52(9): 78-86 (2009) [pdf]
• [JL’76] Neil D. Jones, William T. Laaser: Complete Problems for Deterministic Polynomial Time. Theor. Comput. Sci. 3(1): 105-117 (1976)

## 7 thoughts on “Logspace vs Polynomial time”

1. Neil Jones |

Hello Kintali

I haven’t solved that open problem despite many tries over the years. However a characterisation might interest you: L = NP iff read-only recursive programs are equivalent to read-only tail recursive programs, i.e., iff there is no need for the recursion stack in read-only computation.

An earlier first-order version in TCS:

A later higher-order version in JFP:

Regards and good luck!

Neil

• kintali |

Hi Neil. Thanks for the interesting links.

2. vzn |

Im working on a variation of the P vs L problem & came up with the following formulation. I believe it has deep connections to the theory at best and at least is a nice general framework for studying time/space tradeoffs, complexity class separations, etc.; if anyone agrees its a valid open problem please upvote my question, Id appreciate it.

http://cstheory.stackexchange.com/questions/9680/repetition-in-compressibility-of-tm-run-sequences

my original idea that led to this was a proof sketch along the lines that if P==L then P!=NP. I am still debating whether to write up this proof sketch somewhere in cyberspace (cstheory moderators/audience didnt react favorably so far)

3. vegafrank |

Hello Kintali,

I have solved L versus P just proving that L = P. Look to my proof and do not hesitate to comment about it, please. Thanks in advance…