The celebrated result of Kabanets and Impagliazzo (Computational Complexity, 2004) showed that PIT algorithms imply circuit lower bounds, and vice versa. Since then it has been a major challenge to understand the precise connections between PIT and lower bounds. In particular, a main goal has been to understand which lower bounds suffice to obtain efficient PIT algorithms, and how close are they to lower bounds that are necessary for the conclusion. We construct polynomial-time PIT algorithms from lower bounds that are, up to relatively minor remaining gaps, necessary for the existence of such algorithms. That is, we prove that these lower bounds are, up to the mentioned minor gaps, both sufficient and necessary for polynomial-time PIT, over fields of characteristic zero. Over sufficiently large finite fields, we show a similar result wherein the PIT algorithm runs in time $n^{\log^{(c)}(n)}$, i.e. a power of $c$-iterated log for an arbitrarily large constant $c>1$. The key to these improvements is studying PIT versus lower bounds in the uniform setting, in which we focus on proving lower bounds for uniform arithmetic circuits and their variants (and on deducing algorithms from such lower bounds). Indeed, by working in this setting we obtain results that are significantly tighter than previously known results concerning polynomial-time PIT vs lower bounds, and are in fact also tighter than known hardness-vs-randomness connections in the Boolean setting. Our results are obtained by combining recent techniques from Boolean hardness vs randomness, and in particular the generator of Chen and Tell (FOCS 2021), with the algebraic hitting-set generator of Guo, Kumar, Saptharishi, and Solomon (SIAM J. Computing 2022) along with the bootstrapping ideas of Agrawal, Ghosh, and Saxena (STOC 2018) and of Kumar, Saptharishi, and Tengse (SODA 2019).

STOC ’25, Prague, Czechia

This paper revisits the study of two classical technical tools in theoretical computer science: Yao's trans-formation of distinguishers to next-bit predictors (FOCS 1982), and the “reconstruction paradigm” in pseudorandomness (e.g., as in Nisan and Wigderson, JCSS 1994). Recent works of Pyne, Raz, and Zhan (FOCS 2023) and Doron, Pyne, and Tell (STOC 2024) showed that both of these tools can be derandomized in the specific context of read-once branching programs (ROBPs), but left open the question of de randomizing them in more general settings. Our main contributions give appealing evidence that derandomization of the two tools is possible in general settings, show surprisingly strong consequences of such derandomization, and reveal several new settings where such derandomization is unconditionally possible for algorithms stronger than ROBPs (with useful consequences). Specifically: •We show that derandomizing these tools is equivalent to general derandomization. Specifically, we show that derandomizing distinguish - to- predict transformations is equivalent to prBPP=prP, and that derandomized reconstruction procedures (in a more general sense that we introduce) is equivalent to prBPP=prZPP. These statements hold even when scaled down to weak circuit classes and to algorithms that run in super-polynomial time. •Our main technical contributions are unconditional constructions of derandomized versions of Yao's transformation (or reductions of this task to other problems) for classes and for algorithms beyond ROBPs. Consequently, we deduce new results: A significant relaxation of the hypotheses required to derandomize the isolation lemma for logspace algorithms and deduce that NL=UL; and proofs that de-randomization necessitates targeted PRGs in catalytic logspace (unconditionally) and in logspace (conditionally). In addition, we introduce a natural subclass of prZPP that has been implicitly studied in recent works (Korten FOCS 2021, CCC 2022): The class of problems reducible to a problem called “Lossy Code”. We provide a structural characterization for this class in terms of derandomized reconstruction procedures, and show that this characterization is robust to several natural variations. Lastly, we present alternative proofs for classical results in the theory of pseudorandomness (such as two-sided derandomization reducing to one-sided), relying on the notion of deterministically transforming distinguishers to predictors as the main technical tool.

We provide compelling evidence for the potential of hardness-vs.-randomness approaches to make progress on the long-standing problem of derandomizing space-bounded computation. Our first contribution is a derandomization of bounded-space machines from hardness assumptions for classes of uniform deterministic algorithms, for which strong (but non-matching) lower bounds can be unconditionally proved. We prove one such result for showing that BPL=L “on average”, and another similar result for showing that BPSPACE[O(n)]=DSPACE[O(n)]. Next, we significantly improve the main results of prior works on hardness-vs.-randomness for logspace. As one of our results, we relax the assumptions needed for derandomization with minimal memory footprint (i.e., showing BPSPACE[S]⊆ DSPACE[c · S] for a small constant c), by completely eliminating a cryptographic assumption that was needed in prior work. A key contribution underlying all of our results is non-black-box use of the descriptions of space-bounded Turing machines, when proving hardness-to-randomness results. That is, the crucial point allowing us to prove our results is that we use properties that are specific to space-bounded machines.

We establish an equivalence between two algorithmic tasks: derandomization, the deterministic simulation of probabilistic algorithms; and refutation, the deterministic construction of inputs on which a given probabilistic algorithm fails to compute a certain hard function. We prove that refuting low-space probabilistic streaming algorithms which attempt to compute functions $f \in \mathcal{F P}$ is equivalent to proving that $\operatorname{pr} \mathcal{B P} \mathcal{P}=\operatorname{pr} \mathcal{P}$, even in cases where a lower bound for f against such streaming algorithms (without a refuter) is already unconditionally known. We also demonstrate the generality of our connection between refutation and derandomization, by establishing connections between refuting classes of constant-depth circuits of sublinear size and derandomizing constant-depth circuits of polynomial size with threshold gates (i.e., $\mathcal{T C}^{0}$). Our connection generalizes and strengthens recent work on the characterization of derandomization. In particular, the refuter framework allows to directly compare several recent works to each other and to our work, as well as to chart a path for further progress. Along the way, we also improve the targeted hitting-set generator of Chen and Tell (FOCS 2021), showing that its translation of hardness to pseudorandomness scales down to $\mathcal{T C}^{0}$.

What is the actual cost of derandomization? And can we get it for free? These questions were recently raised by Doron et. al (FOCS 2020) and have been attracting considerable interest. In this work we extend the study of these questions to the setting of derandomizing interactive proofs systems.

For any n ∈ ℕ and d = o(loglog(n)), we prove that there is a Boolean function F on n bits and a value γ = 2−Θ(d) such that F can be computed by a uniform depth-(d + 1) AC0 circuit with O(n) wires, but F cannot be computed by any depth-d TC0 circuit with n1 + γ wires. This bound matches the current state-of-the-art lower bounds for computing explicit functions by threshold circuits of depth d > 2, which were previously known only for functions outside AC0 such as the parity function. Furthermore, in our result, the AC0 circuit computing F is a monotone *read-once formula* (i.e., an AND-OR tree), and the lower bound holds even in the average-case setting with respect to advantage n−γ.

The Exponential-Time Hypothesis (ETH) is a strengthening of the 𝒫 ≠ 𝒩𝒫 conjecture, stating that 3- SAT on n variables cannot be solved in (uniform) time 2 εċ n , for some ε > 0. In recent years, analogous hypotheses that are “exponentially strong” forms of other classical complexity conjectures (such as 𝒩𝒫⊈ ℬ𝒫𝒫 or co 𝒩𝒫⊈𝒩𝒫) have also been introduced and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely related questions of derandomization and circuit lower bounds . We show that even relatively mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: (1) The Randomized Exponential-Time Hypothesis (rETH) implies that ℬ𝒫𝒫 can be simulated on “average-case” in deterministic (nearly-)polynomial-time (i.e., in time 2 Õ(log( n )) = n loglog( n ) O(1) ). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length Õ(log ( n ))); this significantly improves the state-of-the-art in uniform “hardness-to-randomness” results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. (2) The Non-Deterministic Exponential-Time Hypothesis (NETH) implies that derandomization of ℬ𝒫𝒫 is completely equivalent to circuit lower bounds against ℰ, and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of NETH, and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Last, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if CircuitSAT for circuits over n bits of size poly(n) can be solved by probabilistic algorithms in time 2 n /polylog(n) , then ℬ𝒫ℰ does not have circuits of quasilinear size.

The leading technical approach in uniform hardness-to-randomness in the last two decades faced several well-known barriers that caused results to rely on overly strong hardness assumptions, and yet still yield suboptimal conclusions. In this work we show uniform hardness-to-randomness results that simultaneously break through all of the known barriers. Specifically, consider any one of the following three assumptions:1)For some $\epsilon>0$ there exists a function f computable by uniform circuits of size $2^{O(n)}$ and depth $2^{o(n)}$ such that f is hard for probabilistic time $2^{\epsilon n}$.2)For every $c\in \mathbb{N}$ there exists a function f computable by logspace-uniform circuits of polynomial size and depth n2such that every probabilistic algorithm running in time ncfails to compute f on $\mathrm{a}(1/n)$-fraction of the inputs.3)For every $c\in \mathbb{N}$ there exists a logspace-uniform family of arithmetic formulas of degree n2over a field of size poly $(n)$ such that no algorithm running in probabilistic time nccan evaluate the family on a worst-case input. Assuming any of these hypotheses, where the hardness is for every sufficiently large input length $n\in \mathbb{N}$, we deduce that $\mathcal{R}\mathcal{P}$ can be derandomized in polynomial time and on all input lengths, on average. Furthermore, under the first assumption we also show that $\mathcal{B}\mathcal{P}\mathcal{P}$ can be derandomized in polynomial time, on average and on all input lengths, with logarithmically many advice bits. On the way to these results we also resolve two related open problems. First, we obtain an optimal worst-case to average-case reduction for computing problems in linear space by uniform probabilistic algorithms; this result builds on a new instance checker based on the doubly efficient proof system of Goldwasser, Kalai, and Rothblum (J. ACM, 2015). Secondly, we resolve the main open problem in the work of Carmosino, Impagliazzo and Sabin (ICALP 2018), by deducing derandomization from weak and general fine-grained hardness hypotheses. The full version of this paper is available online [5].

We propose a new approach to the hardness-to-randomness framework and to the$promise-\mathcal{BPP}\ = promise-\mathcal{P}$conjecture. Classical results rely on non-uniform hardness assumptions to construct derandomization algorithms that work in the worst-case, or rely on uniform hardness assumptions to construct derandomization algorithms that work only in the average-case. In both types of results, the derandomization algorithm is “black-box” and uses the standard PRG approach. In this work we present results that closely relate new and natural uniform hardness assumptions to worst-case derandomization of$promise-\mathcal{BPP}$, where the algorithms underlying the latter derandomization are non-black-box. In our main result, we show that$promise-\mathcal{BPP}\ = promise-\mathcal{P}$if the following holds: There exists a multi-output function computable by logspace-uniform circuits of polynomial size and depth$n^{2}$that cannot be computed by uniform probabilistic algorithms in time$n^{c}$, for some universal constant$c > 1$, on almost all inputs. The required failure on “almost all inputs” is stronger than the standard requirement of failing on one input of each length; however, the same assumption without the depth restriction on$f$is necessary for the conclusion. This suggests a potential equivalence between worst-case derandomization of$promise-\mathcal{BPP}$of any form (i.e., not necessarily by a black-box algorithm) and the existence of efficiently-computable functions that are hard for probabilistic algorithms on almost all inputs. In our second result, we introduce a new and uniform hardness-to-randomness tradeoff for the setting of superfast average-case derandomization: prior to this work, superfast average-case derandomization was known only under non-uniform hardness assumptions. In an extreme instantiation of our new tradeoff, under appealing uniform hardness assumptions, we show that for every polynomial$T(n)$and constant$\epsilon > 0$it holds that$\mathcal{BPTIME}[T]\subseteq \mathrm{heur}-\mathcal{DTIME}[T\cdot n^{\epsilon}]$, where the “heur” prefix means that no polynomial-time algorithm can find, with non-negligible probability, an input on which the deterministic simulation errs. Technically, our approach is to design targeted PRGs and HSGs, as introduced by Goldreich (LNCS, 2011). The targeted PRGs/HSGs “produce randomness from the input”, as sug-gested by Goldreich and Wigderson (RANDOM 2002); and their analysis relies on non-black-box versions of the reconstruction procedure of Impagliazzo and Wigderson (FOCS 1998). Our main reconstruction procedure crucially relies on the ideas underlying the proof system of Goldwasser, Kalai, and Rothblum (J. ACM 2015).

We present new constructions of pseudorandom generators (PRGs) for two of the most widely studied non-uniform circuit classes in complexity theory. Our main result is a construction of the first non-trivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth$d\in\mathbb{N}$and$n^{1+\delta}$wires, where$\delta=2^{-O(d)}$, using seed length$O(n^{1-\delta})$and with error$2^{-n^{\delta}}$. This tightly matches the best known lower bounds for this circuit class. As a consequence of our result, all the known hardness for LTF circuits has now effectively been translated into pseudorandomness. This brings the extensive effort in the last decade to construct PRGs and deterministic circuit-analysis algorithms for this class to the point where any subsequent improvement would yield breakthrough lower bounds. Our second contribution is a PRG for De Morgan formulas of size$s$whose seed length is$s^{1/3+o(1)}\cdot\text{polylog}(1/\epsilon)$for error$\epsilon$. In particular, our PRG can fool formulas of sub-cubic size$s=n^{3-\Omega(1)}$with an exponentially small error$\epsilon=\exp(-n^{\Omega(1)})$. This significantly improves the inverse-polynomial error of the previous state-of-the-art for such formulas by Impagliazzo, Meka, and Zuckerman (FOCS 2012, JACM 2019), and again tightly matches the best currently-known lower bounds for this class. In both settings, a key ingredient in our constructions is a pseudorandom restriction procedure that has tiny failure probability, but simplifies the function to a non-natural “hybrid computational model” that combines several computational models. As part of our proofs we also construct “extremely low-error” PRGs for related circuit classes; for example, we construct a PRG for arbitrary functions of$s$LTFs that can handle even the extreme setting of parameters$s=n/\text{polylog}(n)$and$\epsilon=2^{-n/\text{polylog}(n)}$.

Extending the classical “hardness-to-randomness” line-of-works, Doron, Moshkovitz, Oh, and Zuckerman (FOCS 2020) recently proved that derandomization with near-quadratic time overhead is possible, under the assumption that there exists a function in DTIME[2n] that cannot be computed by randomized SVN circuits of size 2(1−є)· n for a small є.

The problem of constructing hitting-set generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct a hitting-set generator for polynomials p: 𝔽ⁿ → 𝔽 of degree d if we are guaranteed that the polynomial vanishes on at most an ε > 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/|𝔽|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). In this work our main interest is a systematic study of the relaxed problem, in its general form, and we prove results that significantly improve and extend the two previously-known results. Our contributions are of two types: - Over fields of size 2 ≤ |𝔽| ≤ poly(n), we show that the seed length of any hitting-set generator for polynomials of degree d ≤ n^{.49} that vanish on at most ε = |𝔽|^{-t} of their inputs is at least Ω((d/t)⋅log(n)). - Over 𝔽₂, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d ≤ n^{.99} that vanish on at most ε = |𝔽|^{-t} of their inputs with seed length O((d-t)⋅log(n)). We also show a polynomial-time computable hitting-set generator with seed length O((d-t)⋅(2^{d-t}+log(n))). In addition, we prove that the problem we study is closely related to the following question: "Does there exist a small set S ⊆ 𝔽ⁿ whose degree-d closure is very large?", where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.

The Exponential-Time Hypothesis (ETH) is a strengthening of the P ≠ NP conjecture, stating that 3-SAT on n variables cannot be solved in (uniform) time 2ε·n, for some . In recent years, analogous hypotheses that are “exponentially-strong” forms of other classical complexity conjectures (such as NP ⊄ eq BPP or coNP ⊄ eq NP) have also been introduced, and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely-related questions of derandomization and circuit lower bounds. We show that even relatively-mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: 1) The Randomized Exponential-Time Hypothesis (rETH) implies that BPP can be simulated on “average-case” in deterministic (nearly-)polynomial-time (i.e., in time 2~O(log(n))=nloglog(n)O(1)). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length ~O(log(n))); this significantly improves the state-of-the-art in uniform “hardness-to-randomness” results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. 2) The Non-Deterministic Exponential-Time Hypothesis (NETH) implies that derandomization of BPP is completely equivalent to circuit lower bounds against E, and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of NETH, and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Lastly, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if CireuitSAT for circuits over n bits of size poly(n) can be solved by probabilistic algorithms in time 2n/polylog(n), then BPε does not have circuits of quasilinear size.

We introduce new forms of attack on expander-based cryptography, and in particular on Goldreich's pseudorandom generator and one-way function. Our attacks exploit low circuit complexity of the underlying expander's neighbor function and/or of the local predicate. Our two key conceptual contributions are: 1) We put forward the possibility that the choice of expander matters in expander-based cryptography. In particular, using expanders whose neighbour function has low circuit complexity might compromise the security of Goldreich's PRG and OWF in certain settings. 2) We show that the security of Goldreich's PRG and OWF is closely related to two other long-standing problems: Specifically, to the existence of unbalanced lossless expanders with low-complexity neighbor function, and to limitations on circuit lower bounds (i.e., natural proofs). In particular, our results further motivate the investigation of affine/local unbalanced lossless expanders and of average-case lower bounds against DNF-XOR circuits. We prove two types of technical results that support the above conceptual messages. First, we unconditionally break Goldreich's PRG when instantiated with a specific expander (whose existence we prove), for a class of predicates that match the parameters of the currently-best "hard" candidates, in the regime of quasi-polynomial stretch. Secondly, conditioned on the existence of expanders whose neighbor functions have extremely low circuit complexity, we present attacks on Goldreich's generator in the regime of polynomial stretch. As one corollary, conditioned on the existence of the foregoing expanders, we show that either the parameters of natural properties for several constant-depth circuit classes cannot be improved, even mildly; or Goldreich's generator is insecure in the regime of a large polynomial stretch, regardless of the predicate used.

The best known lower bounds for the circuit class TC0 are only slightly super-linear. Similarly, the best known algorithm for derandomization of this class is an algorithm for quantified derandomization (i.e., a weak type of derandomization) of circuits of slightly super-linear size. In this paper we show that even very mild quantitative improvements of either of the two foregoing results would already imply super-polynomial lower bounds for TC0. Specifically:

Consider a deterministic algorithm that tries to find a string in an unknown set S\subseteq{0,1}^n, under the promise that S has large density. The only information that the algorithm can obtain about S is estimates of the density of S in adaptively chosen subsets of {0,1}^n, up to an additive error of mu>0. This problem is appealing as a derandomization problem, when S is the set of satisfying inputs for a circuit C:{0,1}^n->{0,1} that accepts many inputs: In this context, an algorithm as above constitutes a deterministic black-box reduction of the problem of hitting C (i.e., finding a satisfying input for C) to the problem of approximately counting the number of satisfying inputs for C on subsets of {0,1}^n. We prove tight lower bounds for this problem, demonstrating that naive approaches to solve the problem cannot be improved upon, in general. First, we show a tight trade-off between the estimation error mu and the required number of queries to solve the problem: When mu=O(log(n)/n) a polynomial number of queries suffices, and when mu>=(4log(n)/n) the required number of queries is 2^{Theta(mu \cdot n)}. Secondly, we show that the problem "resists" parallelization: Any algorithm that works in iterations, and can obtain p=p(n) density estimates "in parallel" in each iteration, still requires Omega( frac{n}{log(p)+log(1/mu)} ) iterations to solve the problem. This work extends the well-known work of Karp, Upfal, and Wigderson (1988), who studied the setting in which S is only guaranteed to be non-empty (rather than dense), and the algorithm can only probe subsets for the existence of a solution in them. In addition, our lower bound on parallel algorithms affirms a weak version of a conjecture of Motwani, Naor, and Naor (1994); we also make progress on a stronger version of their conjecture.

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for TC0, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for TC0. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of TC0 circuits of depth d>2.

This work studies the question of quantified derandomization, which was introduced by Goldreich and Wigderson (STOC 2014). The generic quantified derandomization problem is the following: For a circuit class cal{C} and a parameter B=B(n), given a circuit C in cal{C} with n input bits, decide whether C rejects all of its inputs, or accepts all but B(n) of its inputs. In the current work we consider three settings for this question. In each setting, we bring closer the parameter setting for which we can unconditionally construct relatively fast quantified derandomization algorithms, and the "threshold" values (for the parameters) for which any quantified derandomization algorithm implies a similar algorithm for standard derandomization. For constant-depth circuits, we construct an algorithm for quantified derandomization that works for a parameter B(n) that is only slightly smaller than a "threshold" parameter, and is significantly faster than the best currently-known algorithms for standard derandomization. On the way to this result we establish a new derandomization of the switching lemma, which significantly improves on previous results when the width of the formula is small. For constant-depth circuits with parity gates, we lower a "threshold" of Goldreich and Wigderson from depth five to depth four, and construct algorithms for quantified derandomization of a remaining type of layered depth-3 circuit that they left as an open problem. We also consider the question of constructing hitting-set generators for multivariate polynomials over large fields that vanish rarely, and prove two lower bounds on the seed length of such generators. Several of our proofs rely on an interesting technique, which we call the randomized tests technique. Intuitively, a standard technique to deterministically find a "good" object is to construct a simple deterministic test that decides the set of good objects, and then "fool" that test using a pseudorandom generator. We show that a similar approach works also if the simple deterministic test is replaced with a distribution over simple tests, and demonstrate the benefits in using a distribution instead of a single test.

This work studies a new type of problems in property testing, called dual problems. For a set Ϊ in a metric space and δ > 0, denote by Fδ(Ϊ) the set of elements that are δ-far from Ϊ. Then, in property testing, a δtester for π is required to accept inputs from Ϊ and reject inputs from Fδ(Ϊ). A natural dual problem is the problem of δ-testing the set of "no" instances, that is Fδ(Ϊ): A δ-tester for Fδ(Ϊ) needs to accept inputs from Fδ(Ϊ) and reject inputs that are δ-far from Fδ(Ϊ) that is, it rejects inputs from Fδ(Fδ(Ϊ)). When #938;=Fδ(Fδ(Ϊ)) the dual problem is essentially equivalent to the original one, but this equality does not hold in general.

Enterprises today are struggling with exponential growth in data volumes, leading to an increased demand for scale-out solutions for applications. The resiliency of such solutions to faults might be compromised, as the risk of failure increases with the number of hosts participating in the system. In this paper we present a transparent memory scale-out solution based on memory mirroring, guaranteeing zero-downtime failover of remote hosts. Mirroring-based approaches have been avoided in the past due to limitations of older-generation networking fabrics, yet progress achieved by newer generations has significantly mitigated these limitations. We demonstrate high performance and fault resiliency of our solution using synthetic and commercial benchmarks. Improving fault resiliency with memory mirroring enhances the practicality of memory scale-out solutions.