Perspective on Lower Bounds: Diagonalization Lance Fortnow NEC Research Institute

A Theorem Permanent is not in uniform TC0. Papers: Allender ’96. Caussinus-McKenzie-Thérien-Vollmer ’96. Allender-Gore ’94.

Counting Hierarchy PP – Class of languages accepted by probabilistic machines with unbounded error. Counting Hierarchy

Counting Hierarchy in TC0 If Permanent is in uniform TC0 then Permanent is in P and PP in P. Counting Hierarchy collapses to P. Permanent is AC0-hard for P. All of P and thus the entire counting hierarchy collapses to uniform TC0.

Threshold Machines Alternating machines that ask “Do a majority of my paths accept?” Polynomial-time unbounded thresholds is equivalent to PSPACE. Polynomial-time constant thresholds is the counting hierarchy. Log-time constant thresholds is uniform-TC0.

Almost done For any k, there exists a language L accepted by a polynomial-time k-threshold machine that is not accepted by any log-time k-threshold machine. Not yet done: Could be that L is accepted by a log time r-threshold machine for some r > k.

Finishing Up SAT is accepted by log-time k-threshold machine. All of NP is accepted by some log-time k-threshold machine. All of the counting hierarchy is accepted by some log-time k-threshold machine. Contradiction!

Diagonalization Want to prove separation. Assume collapse. Get other collapses. Keep collapsing until we have collapsed two classes that can be separated by diagonalization.

Diagonalization - Positives Diagonalization works! Diagonalization is not “natural” or at least it avoids the Razborov-Rudich natural proof issues. Proofs are simple—sometimes require clever ideas but rarely hard combinatorics.

Diagonalization - Negatives Only weak separations so far. Relativization Probably will not settle P = NP. Can only get nonrelativizing separations by using nonrelativizing collapses. Hard to diagonalize against nonuniform models of computation.

Diagonalization Cantor (1874) – There is no one-to-one function from the power set of the integers to the integers. Proof: Suppose there was. Then we could enumerate the power set of the integers: S1, S2, S3, …

Proof of Cantor’s Theorem 1 2 3 4 5 6  S1 In Out S2 S3 S4 S5 S6  

Proof of Cantor’s Theorem 1 2 3 4 5 6  S1 In Out S2 S3 S4 S5 S6  

Proof of Cantor’s Theorem 1 2 3 4 5 6  S1 Out In S2 S3 S4 S5 S6  

A Brief History 600 BC - Epimenides Paradox. All cretans are liars…One of their own poets has said so. 400 BC - Eubulides Paradox. This statement is false. 1200 AD – Medieval Study of Insolubia. I am a liar.

A Brief History 1874 – Cantor. 1901 - Russell’s Paradox. The set of reals is not countable. 1901 - Russell’s Paradox. The set of all sets that does not contain itself as a member. 1931 - Gödel’s Incompleteness. This statement does not have a proof.

A Brief History 1936 – Turing. 1956 – Friedberg-Muchnik. The halting problem is undecidable. 1956 – Friedberg-Muchnik. There exist incomplete Turing degrees. 1965 – Hartmanis-Stearns. More time gives more languages.

Time and Space Hierarchies Nondeterministic Space Hierarchy. Ibarra (1972), IS (1988). First to use multiple collapses. Nondeterministic Time Hierarchy. Cook (1973), SFM (1978), Žàk (1983). Unbounded collapses necessary. Almost-everywhere Hierarchies. Open: Probabilistic, Quantum.

Delayed Diagonalization Ladner ’75 If P  NP then there exists a set in NP that is not in P and not complete. To keep the language in NP we wait until we have fulfilled the previous diagonalization step.

Diagonalization is it! Kozen (1980) Any proof that P is different from NP is a diagonalization proof. Says more about the difficulty of formalizing the notion of diagonalization than of the possibility of other types of proofs.

Nonrelativizing Separations Buhrman-Fortnow-Thierauf (1998). Exponential version of MA does not have polynomial size circuits. Relativized world where it does have polynomial size circuits. Proof uses EXP in P/poly implies EXP in MA (Babai-Fortnow-Lund).

The Next Great Result Logspace is strictly contained in NP. No good reason to think this is hard. Several possible approaches. Four ways to separate NP from L. 1. Autoreducibility. 2. Intersections of Finite Automata. 3. Anti-Impagliazzo-Wigderson. 4. Trading Alternation, Time and Space.

1. Autoreducibility Autoredubile sets are sets with a certain amount of redundacy in them. Whether certain complete problems are autoreducible can separate complexity classes. Burhman, Fortnow, van Melkebeek and Torenvliet ’95

Reducibility A set A (Turing) reduces to B if we can answer questions to A by asking arbitrary adaptive questions to B. A ... B ...

Autoreducibility A set A is autoreducible if we can answer questions to A by asking arbitrary adaptive questions to A. A ... A ...

Autoreducibility A set A is autoreducible if we can answer questions to A by asking arbitrary adaptive questions to A except for the original question. A ... A ...

Autoreducibility and NL  NP If EXPSPACE-complete sets are all autoreducible then NL  NP. If PSPACE-complete sets are all nonadaptively autoreducible then NL also differs from NP.

Diagonalization Helps! Assume NP = NL. We then create a set in A such that A is in EXPSPACE. A is hard for EXPSPACE. A diagonalizes against all autoreductions. NP = NL implies a EXPSPACE-complete sets that is not autoreducible.

2. Intersecting Finite Automata Finite automata can capture pieces of a computation. Intersecting them can capture the whole computation. Karakostas-Lipton-Viglas 2000.

Intersecting Finite Automata Does a given automata ever accept? Check in time linear in size. Do a given collection of k automata of size s have a non-empty intersection? Can do in time sk. If one can do substantially better, complexity separation occurs.

Simulating Computation Input Tape Finite Control Work Tape

Simulating Computation Input Tape Finite Control F1 F2 F3 Work Tape

Simulating Computation Input Tape G Finite Control F1 F2 F3 Work Tape

Results Given k finite automata with s states and one finite automata with t states. If we can determine if there is a common intersection in time so(k)t then P is different from L.

Results Given k finite automata with s states and one finite automata with t states. If we can determine if there is a common intersection by a circuit of size so(k)t then NP is different from L.

Diagonalization Helps Quick simulations of the intersections of finite automata allow us to solve logarithmic space in time n1+ which is strictly contained in P.

3. Anti-Impagliazzo-Wigderson If deterministic 2O(n) time (E) does not have 2o(n) size circuits then P = BPP. Assumption very strong: We are allowed to use huge amounts to nonuniformity to save a little time. To prove assumption false would separate P from NP.

P = NP and Small Circuits for E P = NP implies P = PH. P = PH implies E = EH. Kannan ’81: EH contains languages that do not have 2o(n) size circuits. E does not have 2o(n) size circuits.

L = NP and Linear Space If every language in linear space has 2o(n) size circuits then L is different than NP. We don’t even know if SAT has 2o(n) size circuits. If SAT does not have 2o(n) size circuits than L is different from NP.

How to show L  NP Assuming SAT has very small, low-depth circuits show that Linear Space has slightly small circuits.

4. Alternation, Time and Space Use relationships between alternation, time and space to get the collapses needed for a contradiction. Kannan ’84. Fortnow ’97. Lipton-Viglas ’99. Fortnow-van Melkebeek ‘00. Tourlakis ‘00.

Lower Bounds on 2 2-Linear time cannot be simulated by a machine using n1.99 time and polylogarithmic space.

Suppose it could… logc n n1.99

Suppose it could… logc n n0.995 n0.995 n1.99 n0.995 n0.995

Suppose it could… logc n n0.995 n0.995 n1.99 n0.995 n0.995

Separations Generalize: j-linear time requires nearly nj time on small space machines. If one could show j-linear time requires nk time with small space for all k then NP is different from L.

Lower Bounds on SAT Satisfiability cannot be solved by any machine using no(1) space and na time for any a less than the golden ratio, about 1.618. Various time-space tradeoffs.

Razborov – “It’s not dead yet” Circuit Complexity – 5 years Diagonalization Complexity Theory – 35 years Computability – 65 years Proof Technique – 125 years Concept – 2600 years … and “It’s not dead yet”

Steve Mahaney “Diagonalization is a tool for showing separation results, but not a power tool.”

Steve Mahaney “Diagonalization is a tool for showing separation results, but not a power tool.”

Conclusions Diagonalization still produces new lower bounds and possibilities for the future. The actual diagonalization step is easy. The trick is doing the collapses to get the diagonalization. Hard combinatorics not required. Is NP  L just around the corner?