11 Search Results for "Murray, Cody D."

Document
DOI: 10.4230/LIPIcs.STACS.2026.23
Simple Circuit Extensions for XOR in PTIME

Authors: Marco Carmosino, Ngu Dang, and Tim Jackman

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
The Minimum Circuit Size Problem for Partial Functions (MCSP^*) is hard assuming the Exponential Time Hypothesis (ETH) (Ilango, 2020). This breakthrough result leveraged a characterization of the optimal {∧, ∨, ¬} 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 replacing OR_n with a 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 ¬ gates count towards circuit size, optimal XOR_n circuits are binary trees of n-1 subcircuits computing (¬)XOR₂ (Kombarov, 2011). We extend this characterization when ¬ gates do not contribute the circuit size. Thus, the XOR-Simple Extension Problem is in polynomial time under both measures of circuit complexity. We conclude by discussing conjectures about the complexity of the f-Simple Extension problem for each explicit function f with known and unrestricted circuit lower bounds over the DeMorgan basis. Examining the conditions under which our Simple Extension Solver is efficient, we argue that multiplexer functions (MUX) are the most promising candidate for ETH-hardness of a Simple Extension Problem, towards proving ETH-hardness of total MCSP.
Cite as

Marco Carmosino, Ngu Dang, and Tim Jackman. Simple Circuit Extensions for XOR in PTIME. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{carmosino_et_al:LIPIcs.STACS.2026.23,
  author =	{Carmosino, Marco and Dang, Ngu and Jackman, Tim},
  title =	{{Simple Circuit Extensions for XOR in PTIME}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.23},
  URN =		{urn:nbn:de:0030-drops-255127},
  doi =		{10.4230/LIPIcs.STACS.2026.23},
  annote =	{Keywords: Minimum Circuit Size Problem, Circuit Lower Bounds, Exponential Time Hypothesis}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.37
New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String

Authors: Lijie Chen, Yang Hu, and Hanlin Ren

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The algebrization barrier, proposed by Aaronson and Wigderson (STOC '08, ToCT '09), captures the limitations of many complexity-theoretic techniques based on arithmetization. Notably, several circuit lower bounds that overcome the relativization barrier (Buhrman-Fortnow-Thierauf, CCC '98; Vinodchandran, TCS '05; Santhanam, STOC '07, SICOMP '09) remain subject to the algebrization barrier. In this work, we establish several new algebrization barriers to circuit lower bounds by studying the communication complexity of the following problem, called XOR-Missing-String: For m < 2^{n/2}, Alice gets a list of m strings x₁, … , x_m ∈ {0, 1}ⁿ, Bob gets a list of m strings y₁, … , y_m ∈ {0, 1}ⁿ, and the goal is to output a string s ∈ {0, 1}ⁿ that is not equal to x_i⊕ y_j for any i, j ∈ [m]. 1) We construct an oracle A₁ and its multilinear extension A₁̃ such that PostBPE^{A₁̃} has linear-size A₁-oracle circuits on infinitely many input lengths. That is, proving PostBPE ̸ ⊆ i.o.- SIZE[O(n)] requires non-algebrizing techniques. This barrier follows from a PostBPP communication lower bound for XOR-Missing-String. This is in contrast to the well-known algebrizing lower bound MA_E (⊆ PostBPE) ̸ ⊆ P/_poly. 2) We construct an oracle A₂ and its multilinear extension A₂̃ such that BPE^{A₂̃} has linear-size A₂-oracle circuits on all input lengths. Previously, a similar barrier was demonstrated by Aaronson and Wigderson, but in their result, A₂̃ is only a multiquadratic extension of A₂. Our results show that communication complexity is more useful than previously thought for proving algebrization barriers, as Aaronson and Wigderson wrote that communication-based barriers were "more contrived". This serves as an example of how XOR-Missing-String forms new connections between communication lower bounds and algebrization barriers. 3) Finally, we study algebrization barriers to circuit lower bounds for MA_E. Buhrman, Fortnow, and Thierauf proved a sub-half-exponential circuit lower bound for MA_E via algebrizing techniques. Toward understanding whether the half-exponential bound can be improved, we define a natural subclass of MA_E that includes their hard MA_E language, and prove the following result: For every super-half-exponential function h(n), we construct an oracle A₃ and its multilinear extension A₃̃ such that this natural subclass of MA_E^{A₃̃} has h(n)-size A₃-oracle circuits on all input lengths. This suggests that half-exponential might be the correct barrier for MA_E circuit lower bounds w.r.t. algebrizing techniques.
Cite as

Lijie Chen, Yang Hu, and Hanlin Ren. New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2026.37,
  author =	{Chen, Lijie and Hu, Yang and Ren, Hanlin},
  title =	{{New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.37},
  URN =		{urn:nbn:de:0030-drops-253246},
  doi =		{10.4230/LIPIcs.ITCS.2026.37},
  annote =	{Keywords: circuit lower bound, algebrization barrier, missing string, communication complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.88
Lower Bounds and Separations for Torus Polynomials

Authors: Vaibhav Krishan and Sundar Vishwanathan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The class ACC⁰ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with AND, NOT and MOD_m gates, where m is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that MAJORITY does not belong to ACC⁰. A few years ago, Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach towards this conjecture. Torus polynomials approximate Boolean functions when the fractional part of their value on Boolean points is close to half the value of the function. They reduced the conjecture that MAJORITY ∉ ACC⁰ to a conjecture concerning the non-existence of low degree torus polynomials that approximate MAJORITY. We reduce the non-existence problem further, to a statement about finding feasible solutions for an infinite family of linear programs. The main advantage of this statement is that it allows for incremental progress, which means finding feasible solutions for successively larger collections of these programs. As an immediate first step, we find feasible solutions for a large class of these linear programs, leaving only a finite set for further consideration. Our method is inspired by the method of dual polynomials, which is used to study the approximate degree of Boolean functions. Using our method, we also propose a way to progress further. We prove several additional key results with the same method, which include: - A lower bound on the degree of symmetric torus polynomials that approximate the AND function. As a consequence, we get a separation that symmetric torus polynomials are weaker than their asymmetric counterparts. - An error-degree trade-off for symmetric torus polynomials approximating the MAJORITY function, strengthening the corresponding result of Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019). - The first lower bounds against torus polynomials approximating AND, showcasing the power of the machinery we develop. This lower bound nearly matches the corresponding upper bound. Hence, we get an almost complete characterization of the torus polynomial approximation degree of AND. - Lower bounds against asymmetric torus polynomials approximating MAJORITY, or AND, in the very low error regime. This partially answers a question posed in Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) about error-reduction for torus polynomials.
Cite as

Vaibhav Krishan and Sundar Vishwanathan. Lower Bounds and Separations for Torus Polynomials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 88:1-88:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{krishan_et_al:LIPIcs.ITCS.2026.88,
  author =	{Krishan, Vaibhav and Vishwanathan, Sundar},
  title =	{{Lower Bounds and Separations for Torus Polynomials}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{88:1--88:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.88},
  URN =		{urn:nbn:de:0030-drops-253751},
  doi =		{10.4230/LIPIcs.ITCS.2026.88},
  annote =	{Keywords: Circuit complexity, ACC, lower bounds, polynomials}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.67
#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank

Authors: Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
There is a large body of work that shows how to leverage lower bound techniques for circuit classes to obtain satisfiability algorithms that run in better than brute-force time [Ramamohan Paturi et al., 1997; Ryan Williams, 2014]. For circuits with threshold gates, there are several such algorithms based on either - Probabilistic Representations by low-degree polynomials, which allow for the use of fast polynomial evaluation algorithms, or - Low rank, which allows for an efficient reduction to rectangular matrix multiplication. In this paper, we use a related notion of probabilistic rank to obtain satisfiability algorithms for circuit classes contained in ACC⁰∘3-PTF, i.e. constant-depth circuits with modular counting gates and a single layer of degree-3 polynomial threshold functions. Even for the special case of a single 3-PTF, it is not clear how to use either of the above two strategies to get a non-trivial satisfiability algorithm. The best known algorithm in this case previously was based on memoization and yields worse guarantees than our algorithm.
Cite as

Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan. #SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{limaye_et_al:LIPIcs.MFCS.2025.67,
  author =	{Limaye, Nutan and Srinivasan, Adarsh and Srinivasan, Srikanth},
  title =	{{#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{67:1--67:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.67},
  URN =		{urn:nbn:de:0030-drops-241744},
  doi =		{10.4230/LIPIcs.MFCS.2025.67},
  annote =	{Keywords: probabilistic polynomials, probabilistic rank, circuit satisfiability, circuit lower bounds, polynomial method, threshold circuits}
}
Document
DOI: 10.4230/LIPIcs.WABI.2025.10
DiVerG: Scalable Distance Index for Validation of Paired-End Alignments in Sequence Graphs

Authors: Ali Ghaffaari, Alexander Schönhuth, and Tobias Marschall

Published in: LIPIcs, Volume 344, 25th International Conference on Algorithms for Bioinformatics (WABI 2025)

Abstract
Determining the distance between two loci within a genomic region is a recurrent operation in various tasks in computational genomics. A notable example of this task arises in paired-end read mapping as a form of validation of distances between multiple alignments. While straightforward for a single genome, graph-based reference structures render the operation considerably more involved. Given the sheer number of such queries in a typical read mapping experiment, an efficient algorithm for answering distance queries is crucial. In this paper, we introduce DiVerG, a compact data structure as well as a fast and scalable algorithm, for constructing distance indexes for general sequence graphs on multi-core CPU and many-core GPU architectures. DiVerG is based on PairG [Jain et al., 2019], but overcomes the limitations of PairG by exploiting the extensive potential for improvements in terms of scalability and space efficiency. As a consequence, DiVerG can process substantially larger datasets, such as whole human genomes, which are unmanageable by PairG. DiVerG offers faster index construction time and consistently faster query time with gains proportional to the size of the underlying compact data structure. We demonstrate that our method performs favorably on multiple real datasets at various scales. DiVerG achieves superior performance over PairG; e.g. resulting to 2.5-4x speed-up in query time, 44-340x smaller index size, and 3-50x faster construction time for the genome graph of the MHC region, as a particularly variable region of the human genome. The implementation is available at: https://github.com/cartoonist/diverg
Cite as

Ali Ghaffaari, Alexander Schönhuth, and Tobias Marschall. DiVerG: Scalable Distance Index for Validation of Paired-End Alignments in Sequence Graphs. In 25th International Conference on Algorithms for Bioinformatics (WABI 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 344, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ghaffaari_et_al:LIPIcs.WABI.2025.10,
  author =	{Ghaffaari, Ali and Sch\"{o}nhuth, Alexander and Marschall, Tobias},
  title =	{{DiVerG: Scalable Distance Index for Validation of Paired-End Alignments in Sequence Graphs}},
  booktitle =	{25th International Conference on Algorithms for Bioinformatics (WABI 2025)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-386-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{344},
  editor =	{Brejov\'{a}, Bro\v{n}a and Patro, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2025.10},
  URN =		{urn:nbn:de:0030-drops-239369},
  doi =		{10.4230/LIPIcs.WABI.2025.10},
  annote =	{Keywords: Sequence graph, distance index, read mapping, sparse matrix}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.20
Counting Martingales for Measure and Dimension in Complexity Classes

Authors: John M. Hitchcock, Adewale Sekoni, and Hadi Shafei

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
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-ε) 2ⁿ/n lower bound (1949) to PSPACE-measure, showing that almost all problems require circuits of size (2ⁿ/n)(1+(α log n)/n), for any α < 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.
Cite as

John M. Hitchcock, Adewale Sekoni, and Hadi Shafei. Counting Martingales for Measure and Dimension in Complexity Classes. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 20:1-20:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hitchcock_et_al:LIPIcs.CCC.2025.20,
  author =	{Hitchcock, John M. and Sekoni, Adewale and Shafei, Hadi},
  title =	{{Counting Martingales for Measure and Dimension in Complexity Classes}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{20:1--20:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.20},
  URN =		{urn:nbn:de:0030-drops-237145},
  doi =		{10.4230/LIPIcs.CCC.2025.20},
  annote =	{Keywords: resource-bounded measure, resource-bounded dimension, counting martingales, counting complexity, circuit complexity, Kolmogorov complexity, quantum complexity, Minimum Circuit Size Problem}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.54
Partial Minimum Branching Program Size Problem Is ETH-Hard

Authors: Ludmila Glinskih and Artur Riazanov

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem ({MBPSP}^{*}) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs. Combining these results with the recent unconditional lower bounds for {MCSP} [Ludmila Glinskih and Artur Riazanov, 2022], we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1- NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems. Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.
Cite as

Ludmila Glinskih and Artur Riazanov. Partial Minimum Branching Program Size Problem Is ETH-Hard. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{glinskih_et_al:LIPIcs.ITCS.2025.54,
  author =	{Glinskih, Ludmila and Riazanov, Artur},
  title =	{{Partial Minimum Branching Program Size Problem Is ETH-Hard}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{54:1--54:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.54},
  URN =		{urn:nbn:de:0030-drops-226822},
  doi =		{10.4230/LIPIcs.ITCS.2025.54},
  annote =	{Keywords: MCSP, branching programs, meta-complexity, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.30
Provability of the Circuit Size Hierarchy and Its Consequences

Authors: Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor C. Oliveira, and Dimitrios Tsintsilidas

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
The Circuit Size Hierarchy (CSH^a_b) states that if a > b ≥ 1 then the set of functions on n variables computed by Boolean circuits of size n^a is strictly larger than the set of functions computed by circuits of size n^b. This result, which is a cornerstone of circuit complexity theory, follows from the non-constructive proof of the existence of functions of large circuit complexity obtained by Shannon in 1949. Are there more "constructive" proofs of the Circuit Size Hierarchy? Can we quantify this? Motivated by these questions, we investigate the provability of CSH^a_b in theories of bounded arithmetic. Among other contributions, we establish the following results: i) Given any a > b > 1, CSH^a_b is provable in Buss’s theory 𝖳²₂. ii) In contrast, if there are constants a > b > 1 such that CSH^a_b is provable in the theory 𝖳¹₂, then there is a constant ε > 0 such that 𝖯^NP requires non-uniform circuits of size at least n^{1 + ε}. In other words, an improved upper bound on the proof complexity of CSH^a_b would lead to new lower bounds in complexity theory. We complement these results with a proof of the Formula Size Hierarchy (FSH^a_b) in PV₁ with parameters a > 2 and b = 3/2. This is in contrast with typical formalizations of complexity lower bounds in bounded arithmetic, which require APC₁ or stronger theories and are not known to hold even in 𝖳¹₂.
Cite as

Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor C. Oliveira, and Dimitrios Tsintsilidas. Provability of the Circuit Size Hierarchy and Its Consequences. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carmosino_et_al:LIPIcs.ITCS.2025.30,
  author =	{Carmosino, Marco and Kabanets, Valentine and Kolokolova, Antonina and C. Oliveira, Igor and Tsintsilidas, Dimitrios},
  title =	{{Provability of the Circuit Size Hierarchy and Its Consequences}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{30:1--30:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.30},
  URN =		{urn:nbn:de:0030-drops-226586},
  doi =		{10.4230/LIPIcs.ITCS.2025.30},
  annote =	{Keywords: Bounded Arithmetic, Circuit Complexity, Hierarchy Theorems}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.30
Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems

Authors: Lijie Chen, Dylan M. McKay, Cody D. Murray, and R. Ryan Williams

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
A frontier open problem in circuit complexity is to prove P^{NP} is not in SIZE[n^k] for all k; this is a necessary intermediate step towards NP is not in P_{/poly}. Previously, for several classes containing P^{NP}, including NP^{NP}, ZPP^{NP}, and S_2 P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove C is not in SIZE[n^k] for all k, we show that C subset P_{/poly} implies a "collapse" D = C for some larger class D, where we already know D is not in SIZE[n^k] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for P^{NP} that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for P^{NP} are equivalent to fixed-polynomial size circuit lower bounds for P^{NP}. That is, P^{NP} is not in SIZE[n^k] for all k if and only if (NP subset P_{/poly} implies PH subset i.o.- P^{NP}_{/n}). Next, we present new consequences of the assumption NP subset P_{/poly}, towards proving similar results for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. That is, if NP is not in SIZE[n^k] for all k, then for all C in {Parity-P, PP, PSPACE, EXP}, C subset P_{/poly} implies C subset i.o.-NP_{/n^epsilon} for all epsilon > 0. Note that unconditionally, the collapses are only to MA and not NP. We also explore consequences of circuit lower bounds for a sparse language in NP. Among other results, we show if a polynomially-sparse NP language does not have n^{1+epsilon}-size circuits, then MA subset i.o.-NP_{/O(log n)}, MA subset i.o.-P^{NP[O(log n)]}, and NEXP is not in SIZE[2^{o(m)}]. Finally, we observe connections between these results and the "hardness magnification" phenomena described in recent works.
Cite as

Lijie Chen, Dylan M. McKay, Cody D. Murray, and R. Ryan Williams. Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.CCC.2019.30,
  author =	{Chen, Lijie and McKay, Dylan M. and Murray, Cody D. and Williams, R. Ryan},
  title =	{{Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{30:1--30:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.30},
  URN =		{urn:nbn:de:0030-drops-108525},
  doi =		{10.4230/LIPIcs.CCC.2019.30},
  annote =	{Keywords: Karp-Lipton Theorems, Circuit Lower Bounds, Derandomization, Hardness Magnification}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.8
Easiness Amplification and Uniform Circuit Lower Bounds

Authors: Cody D. Murray and R. Ryan Williams

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
We present new consequences of the assumption that time-bounded algorithms can be "compressed" with non-uniform circuits. Our main contribution is an "easiness amplification" lemma for circuits. One instantiation of the lemma says: if n^{1+e}-time, tilde{O}(n)-space computations have n^{1+o(1)} size (non-uniform) circuits for some e > 0, then every problem solvable in polynomial time and tilde{O}(n) space has n^{1+o(1)} size (non-uniform) circuits as well. This amplification has several consequences: * An easy problem without small LOGSPACE-uniform circuits. For all e > 0, we give a natural decision problem, General Circuit n^e-Composition, that is solvable in about n^{1+e} time, but we prove that polynomial-time and logarithmic-space preprocessing cannot produce n^{1+o(1)}-size circuits for the problem. This shows that there are problems solvable in n^{1+e} time which are not in LOGSPACE-uniform n^{1+o(1)} size, the first result of its kind. We show that our lower bound is non-relativizing, by exhibiting an oracle relative to which the result is false. * Problems without low-depth LOGSPACE-uniform circuits. For all e > 0, 1 < d < 2, and e < d we give another natural circuit composition problem computable in tilde{O}(n^{1+e}) time, or in O((log n)^d) space (though not necessarily simultaneously) that we prove does not have SPACE[(log n)^e]-uniform circuits of tilde{O}(n) size and O((log n)^e) depth. We also show SAT does not have circuits of tilde{O}(n) size and log^{2-o(1)}(n) depth that can be constructed in log^{2-o(1)}(n) space. * A strong circuit complexity amplification. For every e > 0, we give a natural circuit composition problem and show that if it has tilde{O}(n)-size circuits (uniform or not), then every problem solvable in 2^{O(n)} time and 2^{O(sqrt{n log n})} space (simultaneously) has 2^{O(sqrt{n log n})}-size circuits (uniform or not). We also show the same consequence holds assuming SAT has tilde{O}(n)-size circuits. As a corollary, if n^{1.1} time computations (or O(n) nondeterministic time computations) have tilde{O}(n)-size circuits, then all problems in exponential time and subexponential space (such as quantified Boolean formulas) have significantly subexponential-size circuits. This is a new connection between the relative circuit complexities of easy and hard problems.
Cite as

Cody D. Murray and R. Ryan Williams. Easiness Amplification and Uniform Circuit Lower Bounds. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{murray_et_al:LIPIcs.CCC.2017.8,
  author =	{Murray, Cody D. and Williams, R. Ryan},
  title =	{{Easiness Amplification and Uniform Circuit Lower Bounds}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{8:1--8:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.8},
  URN =		{urn:nbn:de:0030-drops-75421},
  doi =		{10.4230/LIPIcs.CCC.2017.8},
  annote =	{Keywords: uniform circuit complexity, time complexity, space complexity, non-relativizing, amplification}
}
Document
DOI: 10.4230/LIPIcs.CCC.2015.365
On the (Non) NP-Hardness of Computing Circuit Complexity

Authors: Cody D. Murray and R. Ryan Williams

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Abstract
The Minimum Circuit Size Problem (MCSP) is: given the truth table of a Boolean function f and a size parameter k, is the circuit complexity of f at most k? This is the definitive problem of circuit synthesis, and it has been studied since the 1950s. Unlike many problems of its kind, MCSP is not known to be NP-hard, yet an efficient algorithm for this problem also seems very unlikely: for example, MCSP in P would imply there are no pseudorandom functions. Although most NP-complete problems are complete under strong "local" reduction notions such as poly-logarithmic time projections, we show that MCSP is provably not NP-hard under O(n^(1/2-epsilon))-time projections, for every epsilon > 0. We prove that the NP-hardness of MCSP under (logtime-uniform) AC0 reductions would imply extremely strong lower bounds: NP \not\subset P/poly and E \not\subset i.o.-SIZE(2^(delta * n)) for some delta > 0 (hence P = BPP also follows). We show that even the NP-hardness of MCSP under general polynomial-time reductions would separate complexity classes: EXP != NP \cap P/poly, which implies EXP != ZPP. These results help explain why it has been so difficult to prove that MCSP is NP-hard. We also consider the nondeterministic generalization of MCSP: the Nondeterministic Minimum Circuit Size Problem (NMCSP), where one wishes to compute the nondeterministic circuit complexity of a given function. We prove that the Sigma_2 P-hardness of NMCSP, even under arbitrary polynomial-time reductions, would imply EXP \not\subset P/poly.
Cite as

Cody D. Murray and R. Ryan Williams. On the (Non) NP-Hardness of Computing Circuit Complexity. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 365-380, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{murray_et_al:LIPIcs.CCC.2015.365,
  author =	{Murray, Cody D. and Williams, R. Ryan},
  title =	{{On the (Non) NP-Hardness of Computing Circuit Complexity}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{365--380},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.365},
  URN =		{urn:nbn:de:0030-drops-50745},
  doi =		{10.4230/LIPIcs.CCC.2015.365},
  annote =	{Keywords: circuit lower bounds, Minimum Circuit Size Problem, NP-completeness, projections, Reductions}
}
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