Optimal Proof Systems and Sparse Sets Harry Buhrman, CWI Steve Fenner, South Carolina Lance Fortnow, NEC/Chicago Dieter van Melkebeek, DIMACS/Chicago

Convergence of Theory This talk talks about relating some very different looking concepts in complexity theory: – Optimal proof systems. – Complete sets for the class of sparse NP languages. – Reductions of sparse sets to tally sets.

The Great Book Does Erdös’ great book really exist?

Tautologies A tautology is a formula that is true no matter what assignment is used. Formulas that are not tautologies have easy proofs of this fact.

Tautologies If we set x 1 to TRUE, x 2 to TRUE and x 3 to FALSE then formula is false. Focus on tautologies in Disjunctive Normal Form—OR of ANDs. How about proofs that a formula is a tautology?

Proof Systems Cook and Reckhow (1979) defined proof systems for tautologies. A proof system is a way of describing easily verifiable proofs that a formula is a tautology. For example, a truth-table of all the possible inputs will prove that a formula is a tautology. – These proofs are quite large though.

Resolution Proofs Consider the following two formula: The first formula is a tautology if and only if the second one is a tautology. This process is called resolution.

Resolution Proofs Every tautology can be resolved to a DNF with an empty clause. The list of resolutions forms a proof system. Haken (1985) showed that resolution requires large proofs.

Proof Systems A proof system is an efficiently computable function mapping onto the tautologies. For a given proof system f and tautology , the size of a proof for  is the length of the shortest x such that f(x)= .

Proof Systems and Complexity Cook and Reckhow: Tautologies have polynomial-size proof systems if and only if NP = co-NP. – Idea: Guess polynomial-size proof. Can separate NP and co-NP and thus P from NP by showing that tautologies do not have small proof systems.

Comparing Proof Systems We say a proof system f is as good as a proof system g if for every proof of a tautology in g there is a proof in f that is not much longer. – Formally: There is a polynomial p such that for all strings x there is a y, |y| < p(|x|), and f(y) = g(x). Resolution is as good as truth-table.

f is as good as g Formulaf-proofs g-proofs

Optimal Proof Systems A proof system is optimal if it is as good as any other proof system. Similar to the notion of NP-completeness, because it measures the largest member of a class. If you have an optimal proof system f, then NP = co-NP if and only if f has polynomial- size proofs for all tautologies.

Do optimal proof systems exist? If NP = co-NP then tautology has polynomial-size proof which are trivially optimal. Even if tautology has no short proof systems, there still might be an optimal one. Let us first look at a variation of optimal proof systems.

P-optimal Proof Systems A proof system f is P-optimal if for any proof system g, tautology  and proof p for , (g(p) =  ), we can efficiently compute from p a f-proof q of . Every P-optimal proof system is optimal though the other direction is not clear. Do there exist P-optimal proof systems?

f is an optimal proof system Formulaf-proofs g-proofs

f is a p-optimal proof system Formulaf-proofs g-proofs

UP-Complete Sets UP consists of the languages accepted by nondeterministic Turing machines having at most one accepting path. – Examples include primality, factoring. – One-way functions exist if and only if P  UP. L is UP-complete if L is in UP and for every A in UP there is a function f such that

Do UP-complete sets exist? The typical complete set: L = { | M i (x) accepts in j steps} If M 1, M 2, …enumerate the NP machines then L may not be in UP. We need to enumerate UP machines, i.e., machines that have at most one accepting path for all inputs.

Do UP-complete sets exist? We need to enumerate UP machines, i.e., machines that have at most one accepting path for all inputs.

Do UP-complete sets exist? We need to enumerate UP machines, i.e., machines that have at most one accepting path for all inputs. Determining whether a given nondeterministic machine M is a UP machine is undecidable.

Do UP-complete sets exist? We need to enumerate UP machines, i.e., machines that have at most one accepting path for all inputs. Determining whether a given nondeterministic machine M is a UP machine is undecidable. For a better understanding we turn to oracles and relativization.

Turing Machine M INPUT TAPE WORK TAPE

Oracle Turing Machines M INPUT TAPE WORK TAPE ORACLE TAPE q?qyqnq?qyqn

Oracle Turing Machines M INPUT TAPE WORK TAPE QUERY q?qyqnq?qyqn

Oracle Turing Machines M INPUT TAPE WORK TAPE QUERY q?qyqnq?qyqn

Oracle Turing Machines M INPUT TAPE WORK TAPE q?qyqnq?qyqn

Oracle Turing Machines M INPUT TAPE WORK TAPE q?qyqnq?qyqn

Oracle Turing Machines M INPUT TAPE WORK TAPE ORACLE TAPE q?qyqnq?qyqn The Oracle is the set of “Yes” answers.

Relativization We appear quite far from separating any real complexity classes such as P and NP. Baker, Gill and Solovay (1975) noticed that proofs in complexity theory relativize, that is the proofs go through if all the machines involved have access to same oracles.

Relativization and P vs NP Baker, Gill and Solovay (1975) show there are oracles A and B such that P A = NP A P B  NP B Techniques currently used would not settle the P versus NP question.

Interpreting Relativization Be careful in interpreting these results: – A very few number of results do not relativize, most notably in the area of interactive proofs. – Space and large time classes do not have clean enough oracle models for these results. – Relativization results are not impossibility results, nor do they give an indication whether a particular statement is true or false.

UP and Relativization Hartmanis and Hemachandra (1984) show that UP does not have complete sets relative to an oracle. – Note that if P = NP then P = UP = NP and UP does have complete sets. How does this relate to P-optimal proof systems?

P-Optimal and UP Messner and Torán (1998) show that if P-optimal proof systems exist then UP has complete sets. Combining with Hartmanis-Hemachandra gives relativized world where there do not exist P-optimal proof systems.

Sparse Sets and NP A set of strings over {0,1}* can have 2 n strings of length n. A sparse set is a small set with at most n c strings at length n for some fixed c. Are there complete sets for the sparse NP sets?

NP  SPARSE-complete Sets Mahaney (1978) shows that if there is a sparse set that is NP-complete then P = NP. Is there a set that is NP, sparse and hard for only the other sparse sets in NP? Similar to the UP case, it is impossible to decide whether a given NP machine accepts a sparse set.

Optimal Proof Systems Messner and Torán also show that if optimal proof systems exist then NP  SPARSE has complete sets. They could not conclude that there exists relativized worlds where no optimal proof systems exist because the oracle question for NP  SPARSE remained open.

Our Result There exists a relativized world where NP  SPARSE does not have complete sets. Corollary: There exists a relativized world where there are no optimal proof systems.

Other Types of Reductions Results described so far are for many-one reductions, where we say that A reduces to B if there exists a polynomial-time function f such that We can also consider other reductions.

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

Truth-Table Reducibility B... A set A Truth-Table reduces to B if we can answer questions to A by asking arbitrary nonadaptive questions to B. A...

Truth-Table Reducibility B... A set A Truth-Table reduces to B if we can answer questions to A by asking arbitrary nonadaptive questions to B. A...

Turing Reductions Hartmanis and Yesha (1984) show that there exists a tally set in NP that is Turing- hard for every sparse NP set. – A tally set is a subset of 1 *. – Every tally set is sparse. What is the relationship between sparse and tally sets?

SPARSE to TALLY A sparse set has at most a polynomial number of strings at any length. A tally set can only have 1 n at length n. In some sense both sets can encode same amount of information. However the strings in a sparse set could be “hidden” making more complex sets.

SPARSE to TALLY Book and Ko (1988) show – Every sparse sets truth-table reduces to some tally set. – There is some sparse sets that does not truth- table reduce to a tally set if the number of queries is fixed.

SPARSE to TALLY Ko (1989) – There is some sparse sets that does not truth- table reduce to a tally set if the reduction is disjunctive—accepts if any of the queries are in the tally set. Buhrman-Longpré-Spaan (1995) – Every sparse set can be conjunctively reduced to a tally set—accepts if all queries are in the tally set.

SPARSE to TALLY Schöning (1993) gives a probabilistic reduction from sparse to tally. If A is sparse and p a polynomial, there is a tally set B and a randomized efficiently computable function f such that – If x is in A then f(x) is always in B. – If x is not in A then the probability that f(x) is in B is at most 1/p(|x|).

NP  SPARSE-complete Sets These proofs preserve NP-ness. If A is sparse and in NP and p a polynomial, there is a tally set B in NP and a randomized efficiently computable function f such that – If x is in A then f(x) is always in B. – If x is not in A then the probability that f(x) is in B is at most 1/p(|x|).

NP  TALLY-complete Sets NP  TALLY has complete sets. T = { 1 | M i (1 n ) accepts in k steps} T is complete for NP  SPARSE via – Turing-reductions – Truth-table reductions – Conjunctive truth-table reductions – Randomized reductions

NP  SPARSE-complete Sets Open: Can NP  SPARSE have complete sets but no complete tally sets? Our relativization techniques force any NP  SPARSE-complete set to look like a tally set. We can then apply negative results for SPARSE to TALLY to the NP  SPARSE- complete set problem.

Relativization Results There exists relativized worlds where – There do not exist any NP  SPARSE-complete sets under disjunctive reductions. – There do not exist any NP  SPARSE-complete sets under truth-table reductions asking only o(n/log n) queries. – There exists a sparse set that does not reduce to any tally set by any truth-table reduction using o(n/log n) queries.

Tight Result For any constant c > 0, there exists a relativized world where NP  SPARSE has no complete sets under truth-table reductions using o(n/log n) queries and O(log n) bits of advice.

Tight Result Under a reasonable assumption, for all values of k, NP  SPARSE has a complete set under conjunctive truth-table reductions using n/(k log n) queries and O(log n) bits of advice. Uses derandomization techniques of Klivans and van Melkebeek. Similar results for SPARSE to TALLY.

Further Directions Tight bounds for Turing reductions? Eliminate “reasonable assumption” needed for derandomization. How does NP  SPARSE compare with other “promise classes” like UP, BPP and NP  co-NP. – Differences in enumerations and time- hierarchy.

Conclusions Often very different looking questions on complexity theory tie together. We also use many different techniques from Kolmogorov complexity to state-of-the-art derandomization results. Still no strong evidence for or against the existence of optimal proof systems.