By J. A. Robinson.

Primary source: web.stanford.edu/class/linguist289/robinson65.pdf

Sections 5.3–5.9 and unification theorem.

By David Deutsch.

Primary source: s2.smu.edu/~mitch/class/8381/papers/Deutsch-85-quantum_theory.pdf

Sections 2–3, unitary operations, universal computation and measurement.

By Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone.

Primary source: cacr.uwaterloo.ca/hac/about/chap3.pdf

Section 3.5.1, Algorithm 3.34, modular square roots.

Chapter 8: cacr.uwaterloo.ca/hac/about/chap8.pdf

Cited passages: Section 8.2.1, Algorithms 8.1 and 8.3 and the correctness proof following Algorithm 8.3, printed pages 286–287, RSA key generation and encryption/decryption. Section 8.3, printed pages 292–294, Rabin public-key encryption.

By Alin Bostan, Ryuhei Mori.

Primary source: mathexp.eu/bostan/publications/BoMo21.pdf

Coefficient-extraction algorithm and parity identity.

By Oded Goldreich, Shafi Goldwasser, Silvio Micali.

Primary source: people.csail.mit.edu/silvio/Selected%20Scientific%20Papers/Pseudo%20Randomness/How%20To%20Construct%20Random%20Functions.pdf

Function ensembles and oracle indistinguishability definition.

By Alexander A. Razborov, Steven Rudich.

Primary source: mit6875.github.io/PAPERS/natural_proofs.pdf

Section 4, quantitative pseudorandomness barrier.

By Manindra Agrawal, Somenath Biswas.

Primary source: cse.iitk.ac.in/users/manindra/algebra/identity.pdf

Section 3, Lemma 3.1, Prime(n), Theorem 3.2.

By Michael O. Rabin.

Primary source: math.nyu.edu/~goodman/teaching/Algebra_2_2021/MillerRabinTest.pdf

Sections 1–2, strong probable-prime testing and one-quarter bound.

By Leslie G. Valiant.

Primary source: sites.math.washington.edu/~billey/colombia/references/valiant.permanent.1979pdf.pdf

Sections 1–3 and Lemma 3.1, including variable/clause tracks and signed junction identities.

By Timothy Gowers.

Primary source: gowers.files.wordpress.com/2009/05/razborov2.pdf

Sections 2–4; Lemma 4.1 gives the stronger minimal-member bound.

By Neil Immerman. SIAM Journal on Computing 17(5), 935–938, 1988.

Author-hosted primary paper: people.cs.umass.edu/~immerman/pub/space.pdf

Theorem 1 and Lemmas 1–2 establish complement closure by counting reachable configurations and verifying complete reachable-set enumeration.

Official Clay Mathematics Institute problem description by Stephen A. Cook.

Primary PDF: www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf

Section 1 defines finite-alphabet languages, machine acceptance, worst-case running time, P and polynomially bounded verification for NP. The PDF upload path is not treated as its publication date.

By Alan Turing. The paper's first page records receipt on 28 May 1936 and reading on 12 November 1936.

Primary paper: www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

Sections 1–2 define the finite control, scanned square, tape, elementary moves and complete configurations; §3 gives explicit computing-machine examples.

Official Computer Vision Toolbox explanation of correspondence, calibration and disparity-to-depth geometry.

By Ian Goodfellow, Yoshua Bengio and Aaron Courville. MIT Press, 2016.

Official author-hosted online text: www.deeplearningbook.org/

Chapter 5, Machine Learning Basics, distinguishes learning task, experience and performance measure (§5.1), generalization and overfitting (§5.2), and supervised versus unsupervised algorithms (§§5.7–5.8).

Official font-file specification. The introductory description and font-table inventory distinguish layout, character mapping, outline data and bitmap glyph representations.

Official specification overview. The characters-versus-glyphs section and contextual-form and ligature examples explain why encoded text and visual forms are distinct.

Leslie G. Valiant proved in 1979 that computing the permanent of a zero-one matrix is complete for the counting complexity class $\#\mathbf{P}$ under the paper's arithmetic reductions.^[ref]

For a bipartite graph's zero-one adjacency matrix, the permanent counts perfect matchings. Deciding whether one exists admits a polynomial-time algorithm, while counting all of them has the paper's much stronger completeness property. This makes the difference between existence and counting concrete. The permanent resembles the determinant but lacks its alternating signs, so an efficient determinant algorithm does not automatically compute it. Valiant's permanent reduction encodes logical constraints into weighted combinatorial choices.

Michael O. Rabin analyzed a randomized algorithm for primality testing in his 1980 paper. The Miller-Rabin primality test checks whether a candidate integer satisfies conditions obeyed by primes under modular exponentiation. A failed check certifies compositeness; passing a check gives a probable prime, not a proof of primality.^[ref]

For each fixed odd composite candidate, at most one quarter of the admissible bases are strong liars. Independent uniformly chosen bases therefore reduce the probability of accepting that composite to at most $4^{-k}$ after $k$ rounds. This probability is over the test's choices for a fixed input; it is not a posterior probability that an arbitrary accepted integer is prime.

Manindra Agrawal coauthored Primality and Identity Testing via Chinese Remaindering by Manindra Agrawal and Somenath Biswas with Somenath Biswas in 2003. The paper connects primality testing to polynomial identity testing, using randomized algorithms for circuit-represented polynomials over finite fields, the rationals and, for one procedure, integers modulo an arbitrary candidate.^[ref]

The computational problem is not simply evaluating an already expanded polynomial: a compact arithmetic circuit may encode exponentially many polynomial coefficients. A successful test must exploit that representation without materializing the expansion. This distinction connects an algebraic identity to an implementable randomized algorithm.