Alternation-Trading Proofs, Linear Programming, and Lower Bounds

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proof-by-contradiction strategy that we call alternation-trading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a small-scale theorem prover based on this result, we extract new human-readable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques.Comment: To appear in STACS 2010, 12 page

Hierarchies of turing machines with restricted tape alphabet size

It is shown that for any real constants b>a≥0, multitape Turing machines operating in space L1(n)=[bnr] can accept more sets than those operating in space L2(n)=[anr] provided the number of work tapes and tape alphabet size are held fixed. It is also shown that Turing machines with k+1 work tapes are more powerful than those with k work tapes if the tape alphabet size and the amount of work space are held constant

Translational lemmas, polynomial time, and (log n)j-space

AbstractTranslational lemmas are stated in a general framework and then applied to specific complexity classes. Necessary and sufficient conditions are given for every set accepted by a Turing acceptor which operates in linear or polynomial time to be accepted by a Turing acceptor which operates in space (log n)j for some j ⩾ 1

Easiness Amplification and Uniform Circuit Lower Bounds

We present new consequences of the assumption that time-bounded algorithms can be "compressed" with non-uniform circuits. Our main contribution is an "easiness amplification" lemma for circuits. One instantiation of the lemma says: if n^{1+e}-time, tilde{O}(n)-space computations have n^{1+o(1)} size (non-uniform) circuits for some e > 0, then every problem solvable in polynomial time and tilde{O}(n) space has n^{1+o(1)} size (non-uniform) circuits as well. This amplification has several consequences: * An easy problem without small LOGSPACE-uniform circuits. For all e > 0, we give a natural decision problem, General Circuit n^e-Composition, that is solvable in about n^{1+e} time, but we prove that polynomial-time and logarithmic-space preprocessing cannot produce n^{1+o(1)}-size circuits for the problem. This shows that there are problems solvable in n^{1+e} time which are not in LOGSPACE-uniform n^{1+o(1)} size, the first result of its kind. We show that our lower bound is non-relativizing, by exhibiting an oracle relative to which the result is false. * Problems without low-depth LOGSPACE-uniform circuits. For all e > 0, 1 < d < 2, and e < d we give another natural circuit composition problem computable in tilde{O}(n^{1+e}) time, or in O((log n)^d) space (though not necessarily simultaneously) that we prove does not have SPACE[(log n)^e]-uniform circuits of tilde{O}(n) size and O((log n)^e) depth. We also show SAT does not have circuits of tilde{O}(n) size and log^{2-o(1)}(n) depth that can be constructed in log^{2-o(1)}(n) space. * A strong circuit complexity amplification. For every e > 0, we give a natural circuit composition problem and show that if it has tilde{O}(n)-size circuits (uniform or not), then every problem solvable in 2^{O(n)} time and 2^{O(sqrt{n log n})} space (simultaneously) has 2^{O(sqrt{n log n})}-size circuits (uniform or not). We also show the same consequence holds assuming SAT has tilde{O}(n)-size circuits. As a corollary, if n^{1.1} time computations (or O(n) nondeterministic time computations) have tilde{O}(n)-size circuits, then all problems in exponential time and subexponential space (such as quantified Boolean formulas) have significantly subexponential-size circuits. This is a new connection between the relative circuit complexities of easy and hard problems

Effective guessing has unlikely consequences

Funding: EPSRC Grant number EP/P015638/1.A classic result of Paul, Pippenger, Szemeredi and Trotter states that DTIME(n) ⊊ NTIME(n). The natural question then arises: could the inclusion DTIME(t (n)) ⊆ NTIME(n) hold for some superlinear time-constructible function t (n)? If such a function t (n) does exist, then there also exist effective nondeterministic guessing strategies to speed up deterministic computations. In this work, we prove limitations on the effectiveness of nondeterministic guessing to speed up deterministic computations by showing that the existence of effective nondeterministic guessing strategies would have unlikely consequences. In particular, we show that if a subpolynomial amount of nondeterministic guessing could be used to speed up deterministic computation by a polynomial factor, then P ⊊ NTIME(n). Furthermore, even achieving a logarithmic speedup at the cost of making every step nondeterministic would show that SAT ∈ NTIME(n) under appropriate encodings. Of possibly independent interest, under such encodings we also show that SAT can be decided in O(n log n) steps on a nondeterministic multitape Turing machine, improving on the well-known O(n(log n)c) bound for some constant but undetermined exponent c ≥ 1.Publisher PDFPeer reviewe

Reversibility and Adiabatic Computation: Trading Time and Space for Energy

Future miniaturization and mobilization of computing devices requires energy parsimonious `adiabatic' computation. This is contingent on logical reversibility of computation. An example is the idea of quantum computations which are reversible except for the irreversible observation steps. We propose to study quantitatively the exchange of computational resources like time and space for irreversibility in computations. Reversible simulations of irreversible computations are memory intensive. Such (polynomial time) simulations are analysed here in terms of `reversible' pebble games. We show that Bennett's pebbling strategy uses least additional space for the greatest number of simulated steps. We derive a trade-off for storage space versus irreversible erasure. Next we consider reversible computation itself. An alternative proof is provided for the precise expression of the ultimate irreversibility cost of an otherwise reversible computation without restrictions on time and space use. A time-irreversibility trade-off hierarchy in the exponential time region is exhibited. Finally, extreme time-irreversibility trade-offs for reversible computations in the thoroughly unrealistic range of computable versus noncomputable time-bounds are given.Comment: 30 pages, Latex. Lemma 2.3 should be replaced by the slightly better ``There is a winning strategy with $n+2$ pebbles and $m-1$ erasures for pebble games $G$ with $T_G= m2^n$, for all $m \geq 1$'' with appropriate further changes (as pointed out by Wim van Dam). This and further work on reversible simulations as in Section 2 appears in quant-ph/970300

On efficient simulations of multicounter machines

An oblivious 1-tape Turing machine can simulate a multicounter machine on-line in linear time and logarithmic space. This leads to a linear cost combinational logic network implementing the first n steps of a multicounter machine and also to a linear time/logarithmic space on-line simulation by an oblivious logarithmic cost RAM. An oblivious log*n-head tape unit can simulate the first n steps of a multicounter machine in real-time, which leads to a linear cost combinational logic network with a constant data rate