Rice-Like Theorems for Automata Networks

2012

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.

Theory of Computing Systems / Mathematical Systems Theory, 1991

We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic-space bounded) reducibilities turn out to be different for any class containingNE. For space classes the completeness notions under logspace reducibilities can be separated for any class properly containingLOGSPACE. Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes.

We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k > 0, we give an explicit Σ p 2 language, acceptable by a Σ p 2 -machine with running time O(n k 2 +k ), that requires circuit size > n k . This provides a constructive version of an existence theorem of Kannan [Kan82]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomialtime hierarchy requires super polynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and Nisan-Wigderson pseudorandom generator.

We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k > 0, we give an explicit Σ p 2 language, acceptable by a Σ p 2 -machine with running time O(n k 2 +k ), that requires circuit size > n k . This provides a constructive version of an existence theorem of Kannan [Kan82]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomialtime hierarchy requires super polynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and Nisan-Wigderson pseudorandom generator.

Siam Journal on Computing, 2000

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all Turing-complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing-complete set for doubly exponential space that is not.

Lecture Notes in Computer Science, 2005

We consider a scenario in which anonymous, finite-state sensing devices are deployed in an ad-hoc communication network of arbitrary size and unknown topology, and explore what properties of the network graph can be stably computed by the devices. We show that they can detect whether the network has degree bounded by a constant d, and, if so, organize a computation that achieves asymptotically optimal linear memory use. We define a model of stabilizing inputs to such devices and show that a large class of predicates of the multiset of final input values are stably computable in any weakly-connected network. We also show that nondeterminism in the transition function does not increase the class of stably computable predicates.

Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., 2004

We study the robustness of complete sets for various complexity classes. A complete set A is robust if for any f (n)-dense set S ∈ P, A − S is still complete, where f (n) ranges from log(n), polynomial, to subexponential. We show that robustness can be used to separate complexity classes: • for every ≤ p m-complete set A for EXP and any subexponential dense sets S ∈ P, A − S is still Turing complete and under a reasonable hardness assumption even ≤ p m-complete. • For EXP and the delta levels of the exponential hierarchy we show that for every Turing complete set A and any log-dense set S ∈ P, A − S is still Turing complete.

Theoretical Computer Science, 2000

A primitive product is a composition of a ÿnite sequence of ÿnite automata such that feedback is limited to no further than the previous factor. Furthermore, the input to each factor depends only on the global input to the system and the states of at most three factors (including the factor itself). Conversely, the state of a factor may directly in uence only at most three factors (including the factor itself). Additional conditions guarantee a strong planarity property known as outerplanarity. Any primitive product of automata can be realized in an outerplanar layout. This is desirable from the engineering point of view of simple circuit wiring as a circuit whose components and wires comprise the nodes and edges of an outerplanar graph may be realized on a two-dimensional surface, and moreover, new wires can be run from a point outside the circuit to any or all nodes of the circuit without crossing each other or any existing wires. We constructively show that if A is a ÿnite automaton satisfying Letichevsky's criterion, then any ÿnite automaton can be homomorphically represented by (i.e. is a homomorphic image of a subautomaton of, or equivalently, is a letter-to-letter [length-preserving] divisor of) a primitive product of copies of A. A class K of ÿnite automata is homomorphically complete under a given product , by deÿnition, if every ÿnite automaton can be homomorphically represented as a -product of automata from K. By Letichevsky's characterization of homomorphically complete classes under the general product (unrestricted ÿnite composition), our results imply that a class of ÿnite automata is homomorphically complete under the general product if and only if it is homomorphically complete under the primitive product. : S 0 3 0 4 -3 9 7 5 ( 9 9 ) 0 0 2 7 4 -1 An alphabet is a ÿnite nonvoid set X , the elements of X are called letters. For any alphabet X , let X * denote the free monoid of all words over X (including the empty word ). Moreover, denote by X + ( = X * \{ }) the free semigroup of all nonempty words over X .

2006

Analysis by reduction is a method used in linguistics for checking the correctness of sentences of natural languages. This method is modelled by restarting automata. Here we introduce and study a new type of restarting automaton, the socalled t-sRL-automaton, which is an RL-automaton that is rather restricted in that it has a window of size 1 only, and that it works under a minimal acceptance condition. On the other hand, it is allowed to perform up to t rewrite (that is, delete) steps per cycle. Here we study the gap-complexity of these automata. The membership problem for a language that is accepted by a t-sRL-automaton with a bounded number of gaps can be solved in polynomial time. On the other hand, t-sRL-automata with an unbounded number of gaps accept NP-complete languages.