arXiv subjects

Marco Carmosino

Publications and source records attributed to Marco Carmosino.

9 recordsLinked to original sources

arXiv2026

Constructive Separations from Gate Elimination

Gate elimination is the primary technique for proving explicit lower bounds against general Boolean circuits, including Li and Yang's state-of-the-art $3.1n - o(n)$ bound for affine dispersers (STOC 2022). Every circuit lower bound is implicitly existential: every circuit that is too small to compute $f$ must err on some input. This raises a natural question: are these lower bounds \emph{constructive}? That is, can we efficiently produce such errors? Chen, Jin, Santhanam, and Williams showed that constructivity plays a central role in many longstanding open problems in complexity theory, and explicitly raised the question of which circuit lower bound techniques can be made constructive (FOCS 2021). We show that a variety of gate elimination arguments yield refuters -- efficient algorithms that, when given a circuit that is too small to compute a function $f$, produce an input on which the circuit errs. Our results range from elementary lower bounds for $XOR$ and the multiplexer to more sophisticated arguments for affine dispersers. Underlying these results is a shift in perspective: gate elimination arguments \emph{are} algorithms. Each step either simplifies the circuit or reveals a violation of some structural or functional property, from which, with a little additional work, explicit counterexamples can be extracted. We further strengthen the $XOR$ result to handle circuits that \emph{match} the lower bound: given any DeMorgan circuit of size $3(n-1)$ that fails to compute $XOR_n$, we can efficiently produce a counterexample. While refuters follow from the gate elimination arguments themselves, this refinement requires a complete characterization of the set of optimal circuits computing $XOR$ -- a requirement rarely met by other explicit functions.

cs.CC↗

arXiv2026

Convergent Gate Elimination and Constructive Circuit Lower Bounds

Towards better understanding of gate elimination, the only method known that can prove complexity lower bounds for explicit functions against unrestricted Boolean circuits, this work contributes: (1) formalizing circuit simplifications as a convergent term graph rewriting system and (2) giving a simple and constructive proof of a classical lower bound using this system. First, we show that circuit simplification is a convergent term graph rewriting system over the DeMorgan and $\{\land, \lor, \oplus\}$ bases. We define local rewriting rules from Boolean identities such that every simplification sequence yields an identical final result (up to circuit isomorphism or bisimulation). Convergence enables rigorous reasoning about structural properties of simplified circuits without dependence on the order of simplification. Then, we show that there is \emph{no similar} convergent formalization of circuit simplification over the $U_2$ and $B_2$ bases. Then, we use our simplification system to give a constructive circuit lower bound, generalizing Schnorr's classical result that the XOR function requires $3(n - 1)$ gates to compute in the DeMorgan basis. A constructive lower bound $f \not\in C$ gives an algorithm (called a "refuter") that efficiently finds counter-examples for every $C$-circuit trying to compute the function $f$. Chen, Jin, Santhanam, and Williams showed that constructivity plays a central role in many longstanding open problems about complexity theory (FOCS 2021), so it is natural to ask for constructive circuit lower bounds from gate elimination arguments. This demonstrates how using convergent simplification can lead to shorter and more modular proofs of circuit lower bounds. Furthermore, until this work, no constructive lower bound had been proved via gate elimination.

cs.CC↗

arXiv2025

Simple Circuit Extensions for XOR in PTIME

The Minimum Circuit Size Problem for Partial Functions ($MCSP^*$) is hard assuming the Exponential Time Hypothesis (ETH) (Ilango, 2020). This breakthrough hardness result leveraged a characterization of the optimal $\{\land, \lor, \neg\}$ circuits for $n$-bit $OR$ ($OR_n$) and a reduction from the partial $f$-Simple Extension Problem where $f = OR_n$. It remains open to extend that reduction to show ETH-hardness of total $MCSP$. However, Ilango observed that the total $f$-Simple Extension Problem is easy whenever $f$ is computed by read-once formulas (like $OR_n$). Therefore, extending Ilango's proof to total $MCSP$ would require one to replace $OR_n$ with a slightly more complex but similarly well-understood Boolean function. This work shows that the $f$-Simple Extension problem remains easy when $f$ is the next natural candidate: $XOR_n$. We first develop a fixed-parameter tractable algorithm for the $f$-Simple Extension Problem that is efficient whenever the optimal circuits for $f$ are (1) linear in size, (2) polynomially "few" and efficiently enumerable in the truth-table size (up to isomorphism and permutation of inputs), and (3) all have constant bounded fan-out. $XOR_n$ satisfies all three of these conditions. When $\neg$ gates count towards circuit size, optimal $XOR_n$ circuits are binary trees of $n-1$ subcircuits computing $(\neg)XOR_2$ (Kombarov, 2011). We extend this characterization when $\neg$ gates do not contribute the circuit size. Thus, the $XOR$-Simple Extension Problem is in polynomial time under both measures of circuit complexity.

cs.CC↗

arXiv2025

On the Number of Quantifiers Needed to Define Boolean Functions

The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight upper bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov's upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on $n$-bit inputs can be defined by a FO sentence having $(1 + \varepsilon)n\log(n) + O(1)$ quantifiers, and that this is essentially tight. We reduce this number to $(1 + \varepsilon)\log(n) + O(1)$ when the Boolean function in question is sparse.

cs.LO↗

arXiv2025

Enforcing Demographic Coherence: A Harms Aware Framework for Reasoning about Private Data Release

The technical literature about data privacy largely consists of two complementary approaches: formal definitions of conditions sufficient for privacy preservation and attacks that demonstrate privacy breaches. Differential privacy is an accepted standard in the former sphere. However, differential privacy's powerful adversarial model and worst-case guarantees may make it too stringent in some situations, especially when achieving it comes at a significant cost to data utility. Meanwhile, privacy attacks aim to expose real and worrying privacy risks associated with existing data release processes but often face criticism for being unrealistic. Moreover, the literature on attacks generally does not identify what properties are necessary to defend against them. We address the gap between these approaches by introducing demographic coherence, a condition inspired by privacy attacks that we argue is necessary for data privacy. This condition captures privacy violations arising from inferences about individuals that are incoherent with respect to the demographic patterns in the data. Our framework focuses on confidence rated predictors, which can in turn be distilled from almost any data-informed process. Thus, we capture privacy threats that exist even when no attack is explicitly being carried out. Our framework not only provides a condition with respect to which data release algorithms can be analysed but suggests natural experimental evaluation methodologies that could be used to build practical intuition and make tangible assessment of risks. Finally, we argue that demographic coherence is weaker than differential privacy: we prove that every differentially private data release is also demographically coherent, and that there are demographically coherent algorithms which are not differentially private.

cs.CR↗

arXiv2024

Multi-Structural Games and Beyond

Multi-structural (MS) games are combinatorial games that capture the number of quantifiers of first-order sentences. On the face of their definition, MS games differ from Ehrenfeucht-Fraisse (EF) games in two ways: first, MS games are played on two sets of structures, while EF games are played on a pair of structures; second, in MS games, Duplicator can make any number of copies of structures. In the first part of this paper, we perform a finer analysis of MS games and develop a closer comparison of MS games with EF games. In particular, we point out that the use of sets of structures is of the essence and that when MS games are played on pairs of structures, they capture Boolean combinations of first-order sentences with a fixed number of quantifiers. After this, we focus on another important difference between MS games and EF games, namely, the necessity for Spoiler to play on top of a previous move in order to win some MS games. Via an analysis of the types realized during MS games, we delineate the expressive power of the variant of MS games in which Spoiler never plays on top of a previous move. In the second part we focus on simultaneously capturing number of quantifiers and number of variables in first-order logic. We show that natural variants of the MS game do *not* achieve this. We then introduce a new game, the quantifier-variable tree game, and show that it simultaneously captures the number of quantifiers and number of variables. We conclude by generalizing this game to a family of games, the *syntactic games*, that simultaneously capture reasonable syntactic measures and the number of variables.

cs.LO↗

arXiv2024

Parallel Play Saves Quantifiers

The number of quantifiers needed to express first-order properties is captured by two-player combinatorial games called multi-structural (MS) games. We play these games on linear orders and strings, and introduce a technique we call "parallel play", that dramatically reduces the number of quantifiers needed in many cases. Linear orders and strings are the most basic representatives of ordered structures -- a class of structures that has historically been notoriously difficult to analyze. Yet, in this paper, we provide upper bounds on the number of quantifiers needed to characterize different-sized subsets of these structures, and prove that they are tight up to constant factors, including, in some cases, up to a factor of $1+\varepsilon$, for arbitrarily small $\varepsilon$.

cs.LO↗

arXiv2022

Creating quantum-resistant classical-classical OWFs from quantum-classical OWFs

One-way functions (OWF) are one of the most essential cryptographic primitives, the existence of which results in wide-ranging ramifications such as private-key encryption and proving $P \neq NP$. These OWFs are often thought of as having classical input and output (i.e. binary strings), however, recent work proposes OWF constructions where the input and/or the output can be quantum. In this paper, we demonstrate that quantum-classical (i.e. quantum input, classical output) OWFs can be used to produce classical-classical (i.e. classical input, classical output) OWFs that retain the one-wayness property against any quantum polynomial adversary (i.e. quantum-resistant). We demonstrate this in two ways. Firstly, we propose a definition of quantum-classical OWFs and show that the existence of such a quantum-classical OWF would imply the existence of a classical-classical OWF. Secondly, we take a proposed quantum-classical OWF and demonstrate how to turn it into a classical-classical OWF. In summary, this paper showcases another possible route into proving the existence of classical-classical OWFs (assuming intermediate quantum computations are allowed) using a "domain-shifting" technique between classical and quantum information, with the added bonus that such OWFs are also going to be quantum-resistant.

quant-ph↗

arXiv2021

Learning with distributional inverters

We generalize the "indirect learning" technique of Furst et. al., 1991 to reduce from learning a concept class over a samplable distribution $μ$ to learning the same concept class over the uniform distribution. The reduction succeeds when the sampler for $μ$ is both contained in the target concept class and efficiently invertible in the sense of Impagliazzo & Luby, 1989. We give two applications. - We show that AC0[q] is learnable over any succinctly-described product distribution. AC0[q] is the class of constant-depth Boolean circuits of polynomial size with AND, OR, NOT, and counting modulo $q$ gates of unbounded fanins. Our algorithm runs in randomized quasi-polynomial time and uses membership queries. - If there is a strongly useful natural property in the sense of Razborov & Rudich 1997 -- an efficient algorithm that can distinguish between random strings and strings of non-trivial circuit complexity -- then general polynomial-sized Boolean circuits are learnable over any efficiently samplable distribution in randomized polynomial time, given membership queries to the target function

cs.LG↗