Aaronson (STOC 2010) conjectured that almost k-wise independence fools constant-depth circuits; he called this the generalised Linial-Nisan conjecture. Aaronson himself later found a counterexample for depth-3 circuits. We give here an improved counterexample for depth-2 circuits (DNFs). This shows, for instance, that Bazzi's celebrated result (k-wise independence fools DNFs) cannot be generalised in a natural way. We also propose a way to circumvent our counterexample: We define a new notion of pseudorandomness called local couplings and show that it fools DNFs and even decision lists.
We address the following fundamental question: is there an efficient deterministic algorithm that, given 1(n), outputs a string of length n that has polynomial-time bounded Kolmogorov complexity (Omega) over tilde (n) or even n - o(n)? Under plausible complexity-theoretic assumptions, stating for example that there is an epsilon > 0 for which TIME[ T(n)]not subset of TIMENP[T(n)(epsilon)]/2(epsilon n) for appropriately chosen time-constructible T, we show that the answer to this question is positive (answering a question of [27]), and that the Range Avoidance problem [18, 20, 27] is efficiently solvable for uniform sequences of circuits with close to minimal stretch (answering a question of [14]). We obtain our results by giving efficient constructions of pseudo-random generators with almost optimal seed length against algorithms with small advice, under assumptions of the form mentioned above. We also apply our results to give the first complexity-theoretic evidence for explicit constructions of objects such as rigid matrices (in the sense of Valiant) and Ramsey graphs with near-optimal parameters.
We show that a generalization of the DAG-like query-to-communication lifting theorem, when proven using sunflowers over non-binary alphabets, yields lower bounds on the monotone circuit complexity and proof complexity of natural functions and formulas that are better than previously known results obtained using the approximation method. These include an n(Omega(k)) lower bound for the clique function up to k <= n(1/2-epsilon), and an exp(Omega(n(1/3-epsilon))) lower bound for a function in P.
Given an integer-valued function f : {0, 1}(n)-> {0, 1,..., m - 1} that is mildly hard to compute on instances drawn from some distribution D over {0, 1}(n), we show that the function g( x(1),..., x(t)) = f(x(1)) vertical bar center dot center dot center dot vertical bar f( xt) is strongly hard to compute on instances ( x(1),..., x(t)) drawn from the product distribution D-t. We also show the same for the task of approximately computing real-valued functions f : {0, 1} (n)-> [0, m). Our theorems immediately imply hardness self-amplification for several natural problems including Max-Clique and Max-SAT, Approximate #SAT, Entropy Estimation, etc..
We study connections between two fundamental questions from computer science theory. (1) Is witness encryption possible for NP [11]? That is, given an instance x of an NP-complete language L, can one encrypt a secret message with security contingent on the ability to provide a witness for x is an element of L? (2) Is computational learning (in the sense of [46, 30]) hard for NP? That is, is there a polynomial-time reduction from instances of L to instances of learning? Our main contribution is that certain formulations of NP-hardness of learning characterize the existence of witness encryption for NP. More specifically, we show: witness encryption for a language L is an element of NP is equivalent to a half-Levin reduction from L to the Computational Gap Learning problem (denoted CGL [2]), where a half-Levin reduction is the same as a Levin reduction but only required to preserve witnesses in one direction, and CGL formalizes agnostic learning as a decision problem. We show versions of the statement above for witness encryption secure against non-uniform and uniform adversaries. We also show that witness encryption for NP with ciphertexts of logarithmic length, along with a circuit lower bound for E, are together equivalent to NP-hardness of a generalized promise version of MCSP. We complement the above with a number of unconditional NP-hardness results for agnostic PAC learning. Extending a result of [16] to the standard setting of boolean circuits, we show NP-hardness of "semi-proper" learning. Namely: for some polynomial s, it is NP-hard to agnostically learn circuits of size s(n) by circuits of size s(n) center dot n(1/(log log n))(O(1)). Looking beyond the computational model of standard boolean circuits enables us to prove NP-hardness of improper learning (ie. without a restriction on the size of hypothesis returned by the learner). We obtain such results for: learning circuits with oracle access to a given randomly sampled string, and learning RAM programs. In particular, we show that a variant of MINLT [31] for RAM programs is NP-hard with parameters corresponding to the setting of improper learning. We view these results as partial progress toward the ultimate goal of showing NP-hardness of learning boolean circuits in an improper setting. Lastly, we give some consequences of NP-hardness of learning for private- and public-key cryptography. Improving a main result of [2], we show that if improper agnostic PAC learning is NP-hard under a randomized non-adaptive reduction (with some restrictions), then NP not subset of BPP implies the existence of i.o. one-way functions. In contrast, if CGL is NP-hard under a half-Levin reduction, then NP not subset of BPP implies the existence of i.o. public-key encryption.
This paper makes two primary contributions. First, we introduce the concept of counting martingales and use it to define counting measures and counting dimensions. Second, we apply these new tools to strengthen previous circuit lower bounds. Resource-bounded measure and dimension have traditionally focused on deterministic time and space bounds. We use counting complexity classes to develop resource-bounded counting measures and dimensions. Counting martingales are constructed using functions from the #P, SpanP, and GapP complexity classes. We show that counting martingales capture many martingale constructions in complexity theory. The resulting counting measures and dimensions are intermediate in power between the standard time-bounded and space-bounded notions, enabling finer-grained analysis where space-bounded measures are known, but time-bounded measures remain open. For example, we show that BPP has #P-dimension 0 and BQP has GapP-dimension 0, whereas the P-dimensions of these classes remain open. As our main application, we improve circuit-size lower bounds. Lutz (1992) strengthened Shannon's classic (1-epsilon) (2n)/ (n) lower bound (1949) to PSPACE-measure, showing that almost all problems require circuits of size 2(n)/ (n) (1 + (alpha log n)/(n) ), for any alpha < 1. We extend this result to SpanP-measure, with a proof that uses a connection through the Minimum Circuit Size Problem (MCSP) to construct a counting martingale. Our results imply that the stronger lower bound holds within the third level of the exponential-time hierarchy, whereas previously, it was only known in ESPACE. Under a derandomization hypothesis, this lower bound holds within the second level of the exponential-time hierarchy, specifically in the class E-NP. We also study the #P-dimension of classical circuit complexity classes and the GapP-dimension of quantum circuit complexity classes.
A proof system P is said to be automatable in time f(N) if there exists an algorithm that given as input an unsatisfiable formula F outputs a refutation of F in the proof system P in time f(N), where N is the size of the smallest P-refutation of F plus the size of F. Atserias and Bonet (ECCC 2002), observed that tree-like k-DNF resolution is automatable in time N-c center dot k logN for a universal constant c. We show that, under the randomized exponential-time hypothesis (rETH), this is tight up to a O(log k)-factor in the exponent, i.e., we prove that tree-like k-DNF resolution, for k at most logarithmic in the number of variables of F, is not automatable in time N-o(( k/ log k) center dot logN) unless rETH is false. Our proof builds on the non-automatability results for resolution by Atserias and Muller (FOCS 2019), for algebraic proof systems by de Rezende, Goos, Nordstrom, Pitassi, Robere and Sokolov (STOC 2021), and for tree-like resolution by de Rezende (LAGOS 2021).
TFNP studies the complexity of total, verifiable search problems, and represents the first layer of the total function polynomial hierarchy (TFPH). Recently, problems in higher levels of the TFPH have gained significant attention, partly due to their close connection to circuit lower bounds. However, very little is known about the relationships between problems in levels of the hierarchy beyond TFNP. Connections to proof complexity have had an outsized impact on our understanding of the relationships between subclasses of TFNP in the black-box model. Subclasses are characterized by provability in certain proof systems, which has allowed for tools from proof complexity to be applied in order to separate TFNP problems. In this work we begin a systematic study of the relationship between subclasses of total search problems in the polynomial hierarchy and proof systems. We show that, akin to TFNP, reductions to a problem in TF Sigma(d) are equivalent to proofs of the formulas expressing the totality of the problems in some Sigma(d)-proof system. Having established this general correspondence, we examine important subclasses of TFPH. We show that reductions to the StrongAvoid problem are equivalent to proofs in a Sigma(2)-variant of the (unary) Sherali-Adams proof system. As well, we explore the TFPH classes which result from well-studied proof systems, introducing a number of new TF Sigma(2) classes which characterize variants of DNF resolution, as well as TF Sigma(d) classes capturing levels of Sigma(d) -bounded-depth Frege.
The notion of closure of a set of linear forms, first introduced by Efremenko, Garlik, and Itsykson [14], has proven instrumental in proving lower bounds on the sizes of regular and bounded-depth Res(circle plus) refutations [14, 3]. In this work, we present amortized closure, an enhancement that retains the properties of original closure [14] but offers tighter control on its growth. Specifically, adding a new linear form increases the amortized closure by at most one. We explore two applications that highlight the power of this new concept. Utilizing our newly defined amortized closure, we extend and provide a succinct and elegant proof of the recent lifting theorem by Chattopadhyay and Dvorak [10]. Namely we show that for an unsatisfiable CNF formula phi and a 1-stifling gadget g : {0, 1} (l)->{0, 1}, if the lifted formula phi o g has a tree-like Res(circle plus) refutation of size 2(d) and width w, then phi has a resolution refutation of depth d and width w. The original theorem by Chattopadhyay and Dvorak [10] applies only to the more restrictive class of strongly stifling gadgets. As a more significant application of amortized closure, we show improved lower bounds for bounded-depth Res(circle plus), extending the depth beyond that of Alekseev and Itsykson [3]. Our result establishes an exponential lower bound for depth-Omega( n log n) Res(circle plus) refutations of lifted Tseitin formulas, a notable improvement over the existing depth-Omega( n log log n) Res(circle plus) lower bound.
Information complexity is one of the most powerful techniques to prove information-theoretical lower bounds, in which Shannon entropy plays a central role. Though Shannon entropy has some convenient properties, such as the chain rule, it still has inherent limitations. One of the most notable barriers is the square-root loss, which appears in the square-root gap between entropy gaps and statistical distances, e.g., Pinsker's inequality. To bypass this barrier, we introduce a new method based on min-entropy analysis. Building on this new method, we prove the following results. An Omega(N Sigma i alpha i-maxi{alpha i}/k) randomized communication lower bound of the k-party set-intersection problem where the.. -th party holds a random set of size approximate to N1-alpha i. A tight Omega(n/k) randomized lower bound of the k-party Tree Pointer Jumping problems, improving an O(n/k(2)) lower bound by Chakrabarti, Cormode, and McGregor (STOC 08). An Omega(n/k+root n) lower bound of the Chained Index problem, improving an Omega(n/k(2)) lower bound by Cormode, Dark, and Konrad (ICALP 19). Since these problems served as hard problems for numerous applications in streaming lower bounds and cryptography, our new lower bounds directly improve these streaming lower bounds and cryptography lower bounds. On the technical side, min-entropy does not have nice properties such as the chain rule. To address this issue, we enhance the structure-vs-pseudorandomness decomposition used by Goos, Pitassi, and Watson (FOCS 17) and Yang and Zhang (STOC 24); both papers used this decomposition to prove communication lower bounds. In this paper, we give a new breath to this method in the multi-party setting, presenting a new toolkit for proving multi-party communication lower bounds.
We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1. We consider read-once group-products over a finite group G, i.e., tests of the form Pi (n) (i=1) g (xi) (i) where g(i) is an element of G, a special case of read-once permutation branching programs. We give generators with optimal seed length c(G) log( n/epsilon) over any p-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was >= log n log(1/epsilon), even for a fixed order. 2. We give a reduction that "lifts" suitable generators for group products over G to a generator that fools width- w block products, i.e., tests of the form Pi g(fi) (i) where the f(i) are arbitrary functions on disjoint blocks of w bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify. 3. Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed m with nearly optimal seed length. Previously this was known only for m = 2. 4. We give a new generator for products over "mixing groups." The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). This paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity - handling composite moduli - and points to several classes of groups to be attacked next.
We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are computable in VNC^0, i.e., can be computed by arithmetic formulas of constant size. Unconditionally, we construct a VNC^0-computable generator that hits arithmetic circuits of constant depth and polynomial size. We also give conditional constructions, under strong but plausible hardness assumptions, of VNC^0-computable generators that hit arithmetic formulas and arithmetic branching programs of polynomial size, respectively. As a corollary of our constructions, we derive lower bounds for subsystems of the Geometric Ideal Proof System of Grochow and Pitassi. Constructions of such generators are implicit in prior work of Kayal on lower bounds for the degree of annihilating polynomials. Our main contribution is a construction whose correctness relies on circuit complexity lower bounds rather than degree lower bounds.
The question of optimal derandomization, introduced by Doron et. al (JACM 2022), garnered significant recent attention. Works in recent years showed conditional superfast derandomization algorithms, as well as conditional impossibility results, and barriers for obtaining superfast derandomization using certain black-box techniques. Of particular interest is the extreme high-end, which focuses on "free lunch" derandomization, as suggested by Chen and Tell (FOCS 2021). This is derandomization that incurs essentially no time overhead, and errs only on inputs that are infeasible to find. Constructing such algorithms is challenging, and so far there have not been any results following the one in their initial work. In their result, their algorithm is essentially the classical Nisan-Wigderson generator, and they relied on an ad-hoc assumption asserting the existence of a function that is non-batch-computable over all polynomial-time samplable distributions. In this work we deduce free lunch derandomization from a variety of natural hardness assumptions. In particular, we do not resort to non-batch-computability, and the common denominator for all of our assumptions is hardness over all polynomial-time samplable distributions, which is necessary for the conclusion. The main technical components in our proofs are constructions of new and superfast targeted generators, which completely eliminate the time overheads that are inherent to all previously known constructions. In particular, we present an alternative construction for the targeted generator by Chen and Tell (FOCS 2021), which is faster than the original construction, and also more natural and technically intuitive. These contributions significantly strengthen the evidence for the possibility of free lunch derandomization, distill the required assumptions for such a result, and provide the first set of dedicated technical tools that are useful for studying the question.
In this paper, we give the first subexponential (and in fact quasi-polynomial time) reconstruction algorithm for depth 3 circuits of top fan-in 3 (Sigma Pi Sigma(3) circuits) over the fields R and C. Concretely, we show that given blackbox access to an n-variate polynomial f computed by a Sigma Pi Sigma(3) circuit of size s, there is a randomized algorithm that runs in time quasi-poly( n, s) and outputs a generalized Sigma Pi Sigma(3) circuit computing f. The size s includes the bit complexity of coefficients appearing in the circuit. Depth 3 circuits of constant fan-in (Sigma Pi Sigma(k) circuits) and closely related models have been extensively studied in the context of polynomial identity testing (PIT). The study of PIT for these models led to an understanding of the structure of identically zero Sigma Pi Sigma(3) circuits and Sigma Pi Sigma(k) circuits using some very elegant connections to discrete geometry, specifically the Sylvester-Gallai Theorem, and colorful and high dimensional variants of them. Despite a lot of progress on PIT for Sigma Pi Sigma(k) circuits and more recently on PIT for depth 4 circuits of bounded top and bottom fan-in, reconstruction algorithms for Sigma Pi Sigma(k) circuits has proven to be extremely challenging. In this paper, we build upon the structural results for identically zero Sigma Pi Sigma(3) circuits that bound their rank, and prove stronger structural properties of Sigma Pi Sigma(3) circuits (again using connections to discrete geometry). One such result is a bound on the number of codimension 3 subspaces on which a polynomial computed by an Sigma Pi Sigma(3) circuit can vanish on. Armed with the new structural results, we provide the first reconstruction algorithms for Sigma Pi Sigma(3) circuits over R and C. Our work extends the work of [Sinha, CCC 2016] who provided a reconstruction algorithm for Sigma Pi Sigma(2) circuits over R and C as well as the works of [Shpilka, STOC 2007] who provided a reconstruction algorithms for Sigma Pi Sigma(2) circuits in the setting of small finite fields, and [Karnin-Shpilka, CCC 2009] who provided reconstruction algorithms for Sigma Pi Sigma(k) circuits in the setting of small finite fields.
Reed-Muller codes consist of evaluations of n-variate polynomials over a finite field F with degree at most d. Much like every linear code, Reed-Muller codes can be characterized by constraints, where a codeword is valid if and only if it satisfies all degree-d constraints. For a subset (X) over tilde subset of F-n, we introduce the notion of (X) over tilde -quotient Reed-Muller code. A function F : (X) over tilde -> F is a valid codeword in the quotient code if it satisfies all the constraints of degree- d polynomials lying in (X) over tilde. This gives rise to a novel phenomenon: a quotient codeword may have many extensions to original codewords. This weakens the connection between original codewords and quotient codewords which introduces a richer range of behaviors along with substantial new challenges. Our goal is to answer the following question: what properties of (X) over tilde will imply that the quotient code inherits its distance and list-decoding radius from the original code? We address this question using techniques developed by Bhowmick and Lovett [8], identifying key properties of F-n used in their proof and extending them to general subsets (X) over tilde subset of F-n. By introducing a new tool, we overcome the novel challenge in analyzing the quotient code that arises from the weak connection between original and quotient codewords. This enables us to apply known results from additive combinatorics and algebraic geometry [34, 35, 37] to show that when (X) over tilde is a high rank variety, (X) over tilde -quotient Reed-Muller codes inherit the distance and list-decoding parameters from the original Reed-Muller codes.
We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity. For any delta < epsilon and any constant alpha > 0, our sampler uses m+ O(log(1/delta)) random bits to output t = O(( (1)/(epsilon)2 log (1)/(delta))1+alpha) samples Z(1),..., Z(t) is an element of {0, 1}(m) such that for any function f : {0, 1}(m)->[0, 1], Pr [vertical bar(1)/(t) Sigma(t) (i=1) f( Z(i)) - E[f]vertical bar <=epsilon] >= 1 - delta. The randomness complexity is optimal up to a constant factor, and the sample complexity is optimal up to the O(( (1)/(epsilon)2 log (1)/(delta))(alpha)) factor. Our technique generalizes to matrix samplers. A matrix sampler is defined similarly, except that f : {0, 1} (m)-> C-dxd and the absolute value is replaced by the spectral norm. Our matrix sampler achieves randomness complexity m + O-similar to(log( d/delta)) and sample complexity O(( (1)/(epsilon)2 log (d)/(delta))(1+alpha)) for any constant alpha > 0, both near-optimal with only a logarithmic factor in randomness complexity and an additional alpha exponent on the sample complexity. We use known connections with randomness extractors and list-decodable codes to give applications to these objects. Specifically, we give the first extractor construction with optimal seed length up to an arbitrarily small constant factor above 1, when the min-entropy k = beta n for a large enough constant beta < 1. Finally, we generalize the definition of averaging sampler to any normed vector space.
is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for , and consequently is an FP^NP vs. #P dichotomy as well, stating that each problem is either in FP^NP or #P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in FP^NP. We believe that this result is a powerful method for building reductions in complex-valued , as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.
We study linearity testing over the p-biased hypercube ({0, 1}(n), mu(circle times n) (p)) in the 1% regime. For a distribution upsilon supported over {x is an element of{0, 1}(k) : Sigma (k) (i=1) x(i) = 0 (mod 2)}, with marginal distribution mu(p) in each coordinate, the corresponding k-query linearity test Lin(upsilon) proceeds as follows: Given query access to a function f : {0, 1} (n)-> {-1, 1}, sample ( x(1),..., x(k)) similar to upsilon(circle times n), query f on x(1),..., x(k), and accept if and only if Pi(i is an element of[k]) f( x(i)) = 1. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for 0 < p <=(1)/(2), that if k >= 1 + (1)/(p), then there exists a distribution upsilon such that the test Lin(upsilon) works in the 1% regime; that is, any function f : {0, 1} (n)-> {-1, 1} passing the test Lin(upsilon) with probability >=(1)/(2) +epsilon, for some constant epsilon > 0, satisfies Pr (circle times n)(x similar to mu p) [f(x) = g(x)]>=(1)/(2) + delta, for some linear function g, and a constant delta = delta(epsilon) > 0. Conversely, we show that if k < 1 + (1)/(p), then no such test Lin(upsilon) works in the 1% regime. Our key observation is that the linearity test Lin(upsilon) works if and only if the distribution. satisfies a certain pairwise independence property.
We consider a problem of approximating the size of the largest clique in a graph, using a monotone circuit. Concretely, we focus on distinguishing a random Erdos-Renyi graph G(n, p), with p = n- (2)/ (alpha-1) chosen st. with high probability it does not even contain an a-clique, from a random clique on beta vertices (where alpha <= beta). Using the approximation method of Razborov, Alon and Boppana showed in their influential work in 1987 that as long as root alpha beta < n(1- delta)/log n, this problem requires a monotone circuit of size n(Omega(delta root alpha)), implying a lower bound of 2O(similar to)( n(1/3)) for the exact version of the problem Clique(k) when k approximate to n(2/3). Recently, Cavalar, Kumar, and Rossman improved their result by showing a tight lower bound n(Omega(k)), in a limited range k <= n(1/3), implying a comparable 2 O-similar to( n(1/3)) lower bound after choosing the largest admissible k. We combine the ideas of Cavalar, Kumar and Rossman with recent breakthrough results on sunflower conjecture by Alweiss, Lovett, Wu, and Zhang to show that as long as alpha beta < n(1- delta) /log n, any monotone circuit rejecting G(n, p) graph while accepting a beta-clique needs to have size at least n Omega( delta(2) alpha); this implies a stronger 2 O-similar to(root n) lower bound for the unrestricted version of the problem. We complement this result with a construction of an explicit monotone circuit of size O( n(delta 2 alpha/2)) which rejects G(n, p), and accepts any graph containing beta-clique whenever beta > n(1-delta). In particular, those two theorems give a precise characterization of the smallest beta-clique that can be distinguished from G(n,1/2): when beta > n/2(C root log n), there is a polynomial-size circuit that solves it, while for beta < n/2(omega(root log n)) every circuit needs size n(omega(1)).
A natural model of a source of randomness consists of a long stream of symbols X = X-1 omicron...omicron X-t, with some guarantee on the entropy of X-i conditioned on the outcome of the prefix x(1),..., x(i-1). We study unpredictable sources, a generalization of the almost Chor-Goldreich (CG) sources considered in [9]. In an unpredictable source X, for a typical draw of x similar to X, for most i-s, the element x(i) has a low probability of occurring given x(1),..., x(i-1). Such a model relaxes the often unrealistic assumption of a CG source that for every i, and every x(1),..., x(i-1), the next symbol X-i has sufficiently large entropy. Unpredictable sources subsume all previously considered notions of almost CG sources, including notions that [9] failed to analyze, and including those that are equivalent to general sources with high min entropy. For a lossless expander G = ( V, E) with m = log |V|, we consider a random walk V-0, V-1,..., V-t on G using unpredictable instructions that have sufficient entropy with respect to m. Our main theorem is that for almost all the steps t/2 <= i <= t in the walk, the vertex V-i is close to a distribution with min-entropy at least m - O(1). As a result, we obtain seeded online condensers with constant entropy gap, and seedless (deterministic) condensers outputting a constant fraction of the entropy. In particular, our condensers run in space comparable to the output entropy, as opposed to the size of the stream, and even when the length t of the stream is not known ahead of time. As another corollary, we obtain a new extractor based on expander random walks handling lower entropy than the classic expander based construction relying on spectral techniques [11]. As our main technical tool, we provide a novel analysis covering a key case of adversarial random walks on lossless expanders that [9] fails to address. As part of the analysis, we provide a "chain rule for vertex probabilities". The standard chain rule states that for every x similar to X and i, Pr( x(1),..., x(i)) = Pr[ X-i = x(i)|X-[1,X- i-1] - x(1),..., x(i-1)] center dot Pr( x(1),..., x(i-1)). If W( x(1),..., x(i)) is the vertex reached using x(1),..., x(i), then the chain rule for vertex probabilities essentially states that the same phenomena occurs for a typical x: Pr[V-i = W( x(1),..., x(i))]less than or similar to Pr[ X-i = x(i)|X-[1,X- i-1] = x(1),..., xi-1] center dot Pr[ Vi-1 = W( x(1),..., x(i-1))], where V-i is the vertex distribution of the random walk at step i using X.
