Kristoffer Arnsfelt Hansen | Aarhus University (original) (raw)

Papers by Kristoffer Arnsfelt Hansen

Abstract We prove that any AC0 circuit augmented with ϵ log2 n MODm gates and with a MAJORITY gat... more Abstract We prove that any AC0 circuit augmented with ϵ log2 n MODm gates and with a MAJORITY gate at the output, require size nΩ (log n) to compute MODl, when l has a prime factor not dividing m and ϵ is sufficiently small. We also obtain that the MOD2 function is hard on the average for AC0 circuits of size nϵ log n augmented with ϵ log2 n MODm gates, for every odd integer m and any sufficiently small ϵ.

We consider the computational power of constant width polynomial size cylindrical circuits and no... more We consider the computational power of constant width polynomial size cylindrical circuits and nondeterministic branching programs. We show that every function computed by a Õ 2° MOD° AC 0\ bf ∏ _2 ∘\ bf MOD ∘\ bf AC^ 0 circuit can also be computed by a constant width polynomial size cylindrical nondeterministic branching program (or cylindrical circuit) and that every function computed by a constant width polynomial size cylindrical circuit belongs to ACC 0.

Abstract For any integers m and l, where m has r sufficiently large (depending on l) factors, tha... more Abstract For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z m of degree O (r radicn) that is a generalized representation (commonly also called weak representation) of the MOD l function. We give a detailed study of the case when m has exactly two distinct prime factors, and classify the minimum possible degree for a symmetric representing polynomial

We observe that a combination of known top-down and bottom-up lower bound techniques of circuit c... more We observe that a combination of known top-down and bottom-up lower bound techniques of circuit complexity may yield new circuit lower bounds. An important example is this: Razborov and Wigderson showed that a certain function f in ACC 0 cannot be computed by polynomial size circuits consisting of two layers of MAJORITY gates at the top and a layer of AND gates at the bottom.

We consider constant depth circuits augmented with few modular, or more generally, arbitrary symm... more We consider constant depth circuits augmented with few modular, or more generally, arbitrary symmetric gates. We prove that circuits augmented with o (log 2 n) symmetric gates must have size n W (log n)^ Ω (\ rm log {\ it n) to compute a certain (complicated) function in ACC 0.

We obtain a characterization of ACC 0 in terms of a natural class of constant width circuits, nam... more We obtain a characterization of ACC 0 in terms of a natural class of constant width circuits, namely in terms of constant width polynomial size planar circuits. This is shown via a characterization of the class of acyclic digraphs which can be embedded on a cylinder surface in such a way that all arcs flow along the same direction of the axis of the cylinder.

We consider the problem of maintaining a maximum matching in a convex bipartite graph G=(V, E) un... more We consider the problem of maintaining a maximum matching in a convex bipartite graph G=(V, E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense.

Abstract We bound the minimum number w of wires needed to compute any (asymptotically good) error... more Abstract We bound the minimum number w of wires needed to compute any (asymptotically good) error-correcting code C:{0, 1} Ω (n)->{0, 1} n with minimum distance Ω (n), using unbounded fan-in circuits of depth d with arbitrary gates.

Abstract. Barrington, Straubing & Thérien (1990) conjectured that the Boolean A nd function can n... more Abstract. Barrington, Straubing & Thérien (1990) conjectured that the Boolean A nd function can not be computed by polynomial size constant depth circuits built from modular counting gates, ie, by CC 0 circuits. In this work we show that the A nd function can be computed by uniform probabilistic CC 0 circuits that use only O (log n) random bits. This may be viewed as evidence contrary to the conjecture.

Abstract. We study the following question, communicated to us by Miklós Ajtai: Can all explicit (... more Abstract. We study the following question, communicated to us by Miklós Ajtai: Can all explicit (eg, polynomial time computable) functions f:({0, 1} w) 3→{0, 1} w be computed by word circuits of constant size? A word circuit is an acyclic circuit where each wire holds a word (ie, an element of {0, 1} w) and each gate G computes some binary operation gG:({0, 1} w) 2→{0, 1} w, defined for all word lengths w. We present an explicit function so that its w'th slice for any w≥ 8 cannot be computed by word circuits with at most 4 gates.

Starting from Zermelo's classical formal treatment of chess, we trace through history the analysi... more Starting from Zermelo's classical formal treatment of chess, we trace through history the analysis of two-player win/lose/draw games with perfect information and potentially infinite play. Such chess-like games have appeared in many different research communities, and methods for solving them, such as retrograde analysis, have been rediscovered independently. We then revisit Washburn's deterministic graphical games (DGGs), a natural generalization of chess-like games to arbitrary zero-sum payoffs.

We consider Boolean exact threshold functions defined by linear equations, and in general degree ... more We consider Boolean exact threshold functions defined by linear equations, and in general degree d polynomials. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close and in the linear case in particular they are almost matching.

Abstract We exhibit a deterministic concurrent reachability game PURGATORY n with n non-terminal ... more Abstract We exhibit a deterministic concurrent reachability game PURGATORY n with n non-terminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given isin< 1/2 must use non-zero behavior probabilities that are less than (isin 2/(1-isin)) 2n-2. Also, even to achieve the value within say 1-2-n/2, doubly exponentially small behavior probabilities in the number of positions must be used.

Abstract We initiate a systematic study of constant depth Boolean circuits built using exact thre... more Abstract We initiate a systematic study of constant depth Boolean circuits built using exact threshold gates. We consider both unweighted and weighted exact threshold gates and introduce corresponding circuit classes. We next show that this gives a hierarchy of classes that seamlessly interleave with the well-studied corresponding hierarchies defined using ordinary threshold gates. A major open problem in Boolean circuit complexity is to provide an explicit super-polynomial lower bound for depth two threshold circuits.

Abstract We revisit the computational power of constant width polynomial size planar nondetermini... more Abstract We revisit the computational power of constant width polynomial size planar nondeterministic branching programs. We show that they are capable of computing any function computed by a Pi 2 o CC 0 o AC 0 circuit in polynomial size. In the quasipolynomial size setting we obtain a characterization of ACC 0 by constant width planar non-deterministic branching programs.

We consider finding maximin strategies and equilibria of explicitly given extensive form games wi... more We consider finding maximin strategies and equilibria of explicitly given extensive form games with imperfect information but with no moves of chance. We show that a maximin pure strategy for a two-player game with perfect recall and no moves of chance can be found in time linear in the size of the game tree and that all pure Nash equilibrium outcomes of a two-player general-sum game with perfect recall and no moves of chance can be enumerated in time linear in the size of the game tree.

The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NP-h... more The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NP-hard and Sqrt-Sum-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect. Analogous results are shown for a number of other solution concepts, including proper equilibrium,(the strategy part of) sequential equilibrium, quasi-perfect equilibrium and CURB.

We consider the class of constant depth AND/OR circuits augmented with a layer of modular countin... more We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, ie AC 0° MOD m\ bf AC^ 0 ∘\ bf MOD _m circuits. We show that the following holds for several types of gates GG: by adding a gate of type GG at the output, it is possible to obtain an equivalent probabilistic depth 2 circuit of quasipolynomial size consisting of a gate of type GG at the output and a layer of modular counting gates, ie G° MOD m G ∘\ bf MOD _m circuits.

Abstract: For matrix games we study how small nonzero probability must be used in optimal strateg... more Abstract: For matrix games we study how small nonzero probability must be used in optimal strategies. We show that for nxn win-lose-draw games (ie (-1, 0, 1) matrix games) nonzero probabilities smaller than n^{-O (n)} are never needed. We also construct an explicit nxn win-lose game such that the unique optimal strategy uses a nonzero probability as small as n^{-Omega (n)}.

Two standard algorithms for approximately solving two-player zero-sum concurrent reachability gam... more Two standard algorithms for approximately solving two-player zero-sum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of 2 m Q (N) 2^{m^{\ Theta (N)}} on the worst case number of iterations needed for both of these algorithms to provide non-trivial approximations to the value of a game with N non-terminal positions and m actions for each player in each position.