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1. Lifting with simple gadgets and applications to circuit and proof complexity

   Meir, O; Nördstrom, J; Pitassi, T; Robere, R; de Rezende, Susanna F; Vinyals, M. (February 2020, Electronic colloquium on computational complexity)

   We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: • We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. • We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal. 

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2. Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization

   https://doi.org/10.4230/LIPIcs.ICALP.2023.12  

   Amireddy, Prashanth; Garg, Ankit; Kayal, Neeraj; Saha, Chandan; Thankey, Bhargav (January 2023, 50th International Colloquium on Automata, Languages, and Programming)

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   A recent breakthrough work of Limaye, Srinivasan and Tavenas [Nutan Limaye et al., 2021] proved superpolynomial lower bounds for low-depth arithmetic circuits via a "hardness escalation" approach: they proved lower bounds for low-depth set-multilinear circuits and then lifted the bounds to low-depth general circuits. In this work, we prove superpolynomial lower bounds for low-depth circuits by bypassing the hardness escalation, i.e., the set-multilinearization, step. As set-multilinearization comes with an exponential blow-up in circuit size, our direct proof opens up the possibility of proving an exponential lower bound for low-depth homogeneous circuits by evading a crucial bottleneck. Our bounds hold for the iterated matrix multiplication and the Nisan-Wigderson design polynomials. We also define a subclass of unrestricted depth homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This significantly generalizes the superpolynomial lower bounds for regular formulas [Neeraj Kayal et al., 2014; Hervé Fournier et al., 2015]. 

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3. Linear Sketching over F_2

   https://doi.org/10.4230/LIPIcs.CCC.2018.8  

   Kannan, Sampath; Mossel, Elchanan; Sanyal, Swagato; Yaroslavtsev, Grigory (August 2018, Leibniz international proceedings in informatics)

   We initiate a systematic study of linear sketching over F_2. For a given Boolean function treated as f : F_2^n -> F_2 a randomized F_2-sketch is a distribution M over d x n matrices with elements over F_2 such that Mx suffices for computing f(x) with high probability. Such sketches for d << n can be used to design small-space distributed and streaming algorithms. Motivated by these applications we study a connection between F_2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F_2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F_2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F_2-sketching is optimal (up to constant factors) for uniformly distributed inputs. Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F_2 can be constructed as F_2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length O(n) constructed through uniformly random updates. 

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4. Quantified Linear Arithmetic Satisfiability via Fine-Grained Strategy Improvement

   https://doi.org/10.1007/978-3-031-65627-9_5  

   Murphy, Charlie; Kincaid, Zachary (January 2024, Springer Nature Switzerland)

   Abstract Checking satisfiability of formulae in the theory of linear arithmetic has far reaching applications, including program verification and synthesis. Many satisfiability solvers excel at proving and disproving satisfiability of quantifier-free linear arithmetic formulas and have recently begun to support quantified formulas. Beyond simply checking satisfiability of formulas, fine-grained strategies for satisfiability games enables solving additional program verification and synthesis tasks. Quantified satisfiability games are played between two players—SAT and UNSAT—who take turns instantiating quantifiers and choosing branches of boolean connectives to evaluate the given formula. A winning strategy for SAT (resp. UNSAT) determines the choices of SAT (resp. UNSAT) as a function of UNSAT ’s (resp. SAT ’s) choices such that the given formula evaluates to true (resp. false) no matter what choices UNSAT (resp. SAT) may make. As we are interested in both checking satisfiabilityandsynthesizing winning strategies, we must avoid conversion to normal-forms that alter the game semantics of the formula (e.g. prenex normal form). We present fine-grained strategy improvement and strategy synthesis, the first technique capable of synthesizing winning fine-grained strategies for linear arithmetic satisfiability games, which may be used in higher-level applications. We experimentally evaluate our technique and find it performs favorably compared with state-of-the-art solvers. 

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5. SMC: Satisfiability Modulo Convex Programming

   https://doi.org/10.1109/JPROC.2018.2849003  

   Shoukry, Yasser; Nuzzo, Pierluigi; Sangiovanni-Vincentelli, Alberto L.; Seshia, Sanjit A.; Pappas, George J.; Tabuada, Paulo (January 2018, Proceedings of the IEEE)

   The design of cyber-physical systems (CPSs) requires methods and tools that can efficiently reason about the interaction between discrete models, e.g., representing the behaviors of ``cyber'' components, and continuous models of physical processes. Boolean methods such as satisfiability (SAT) solving are successful in tackling large combinatorial search problems for the design and verification of hardware and software components. On the other hand, problems in control, communications, signal processing, and machine learning often rely on convex programming as a powerful solution engine. However, despite their strengths, neither approach would work in isolation for CPSs. In this paper, we present a new satisfiability modulo convex programming (SMC) framework that integrates SAT solving and convex optimization to efficiently reason about Boolean and convex constraints at the same time. We exploit the properties of a class of logic formulas over Boolean and nonlinear real predicates, termed monotone satisfiability modulo convex formulas, whose satisfiability can be checked via a finite number of convex programs. Following the lazy satisfiability modulo theory (SMT) paradigm, we develop a new decision procedure for monotone SMC formulas, which coordinates SAT solving and convex programming to provide a satisfying assignment or determine that the formula is unsatisfiable. A key step in our coordination scheme is the efficient generation of succinct infeasibility proofs for inconsistent constraints that can support conflict-driven learning and accelerate the search. We demonstrate our approach on different CPS design problems, including spacecraft docking mission control, robotic motion planning, and secure state estimation. We show that SMC can handle more complex problem instances than state-of-the-art alternative techniques based on SMT solving and mixed integer convex programming. 

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