Sublogarithmic uniform Boolean proof nets

Using a proofs-as-programs correspondence, Terui was able to compare two models of parallel computation: Boolean circuits and proof nets for multiplicative linear logic. Mogbil et. al. gave a logspace translation allowing us to compare their computational power as uniform complexity classes. This paper presents a novel translation in AC0 and focuses on a simpler restricted notion of uniform Boolean proof nets. We can then encode constant-depth circuits and compare complexity classes below logspace, which were out of reach with the previous translations.Comment: In Proceedings DICE 2011, arXiv:1201.034

Branching-time model checking of one-counter processes

One-counter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from the modal mu-calculus for this problem. First, we analyze the periodic behaviour of CTL over OCPs and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. Thus, model checking fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This generalizes a result of the first author, Mayr, and To for the expression complexity of CTL's fragment EF. Second, we prove that already over some fixed OCP, CTL model checking is PSPACE-hard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To obtain the latter result, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable. We demonstrate that our approach can be used to obtain further results. We show that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus establishing a matching lower bound and answering an open question of the first author, Mayr, and To. We moreover show that the following problem is hard for PSPACE: Given a one-counter Markov decision process, a set of target states with counter value zero each, and an initial state, to decide whether the probability that the initial state will eventually reach one of the target states is arbitrarily close to 1. This improves a previously known lower bound for every level of the Boolean hierarchy by Brazdil et al

Non-deterministic Boolean Proof Nets

16 pagesInternational audienceWe introduce Non-deterministic Boolean proof nets to study the correspondence with Boolean circuits, a parallel model of computation. We extend the cut elimination of Non-deterministic Multiplicative Linear logic to a parallel procedure in proof nets. With the restriction of proof nets to Boolean types, we prove that the cut-elimination procedure corresponds to Non-deterministic Boolean circuit evaluation and reciprocally. We obtain implicit characterization of the complexity classes NP and NC (the efficiently parallelizable functions)

Programmability of Chemical Reaction Networks

Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a well-stirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and Boolean Logic Circuits, Vector Addition Systems, Petri Nets, Gate Implementability, Primitive Recursive Functions, Register Machines, Fractran, and Turing Machines. A theme to these investigations is the thin line between decidable and undecidable questions about SCRN behavior

Learning circuits with few negations

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A. A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions)

MALL proof equivalence is Logspace-complete, via binary decision diagrams

Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the question of proofnets: finding canonical representatives of equivalence classes of proofs that have good computational properties. It can also be seen as the word problem for the notion of free category corresponding to the logic. It has been recently shown that proof equivalence in MLL (the multiplicative with units fragment of linear logic) is PSPACE-complete, which rules out any low-complexity notion of proofnet for this particular logic. Since it is another fragment of linear logic for which attempts to define a fully satisfactory low-complexity notion of proofnet have not been successful so far, we study proof equivalence in MALL- (multiplicative-additive without units fragment of linear logic) and discover a situation that is totally different from the MLL case. Indeed, we show that proof equivalence in MALL- corresponds (under AC0 reductions) to equivalence of binary decision diagrams, a data structure widely used to represent and analyze Boolean functions efficiently. We show these two equivalent problems to be LOGSPACE-complete. If this technically leaves open the possibility for a complete solution to the question of proofnets for MALL-, the established relation with binary decision diagrams actually suggests a negative solution to this problem.Comment: in TLCA 201

Safe Recursion on Notation into a Light Logic by Levels

We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels (LALL), derived from the logic L4. LALL is an intuitionistic deductive system, with a polynomial time cut elimination strategy. The embedding allows to represent every term t of SRN as a family of proof nets |t|^l in LALL. Every proof net |t|^l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as proof nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the "fuzzy" borders of paragraph boxes that LALL inherits from L4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of |t|^l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of |t|^l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principles SRN relies on, and the proof theoretic one, called /stratification/, at the base of Light Linear Logic

Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for depth-two linear threshold circuits with arbitrary weights, and depth-three majority circuits computing an explicit function. $\bullet$ We prove that for all $\epsilon\gg \sqrt{\log(n)/n}$, the linear-time computable Andreev's function cannot be computed on a $(1/2+\epsilon)$-fraction of $n$-bit inputs by depth-two linear threshold circuits of $o(\epsilon^3 n^{3/2}/\log^3 n)$ gates, nor can it be computed with $o(\epsilon^{3} n^{5/2}/\log^{7/2} n)$ wires. This establishes an average-case ``size hierarchy'' for threshold circuits, as Andreev's function is computable by uniform depth-two circuits of $o(n^3)$ linear threshold gates, and by uniform depth-three circuits of $O(n)$ majority gates. $\bullet$ We present a new function in $P$ based on small-biased sets, which we prove cannot be computed by a majority vote of depth-two linear threshold circuits with $o(n^{3/2}/\log^3 n)$ gates, nor with $o(n^{5/2}/\log^{7/2}n)$ wires. $\bullet$ We give tight average-case (gate and wire) complexity results for computing PARITY with depth-two threshold circuits; the answer turns out to be the same as for depth-two majority circuits. The key is a new random restriction lemma for linear threshold functions. Our main analytical tool is the Littlewood-Offord Lemma from additive combinatorics