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A Matrix Product State Representation of Boolean Functions

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For a fixed order of the Boolean variables, every vector of Boolean functions has a unique compressed representation as a product of binary matrices, and that representation converts losslessly to and from a binary decision diagram.

desk verdict A creditable, clearly written paper that recasts quasi-reduced BDDs as a matrix product normal form; the core construction is sound, but the CLEAN operation on which all the manipulations rely is asserted rather than proven. read the letter →

arxiv 2505.01930 v2 pith:NN3IJKJO submitted 2025-05-03 cs.DS physics.comp-phquant-ph

Umut Eren Usturali , Claudio Chamon , Andrei E. Ruckenstein , Eduardo R. Mucciolo This is my paper · ORCID
classification cs.DSphysics.comp-phquant-ph MSC 06E3068Q17
keywords BooleanfunctionsmatrixproductstatestensortrainsbinarydecisiondiagramsnormalformShannondecompositionvariableorderingbonddimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a way to store Boolean functions as a train of binary matrices, one matrix per input variable, called a Binary Matrix Product (BMP). It claims that when the train is fully compressed, the representation is canonical: for a fixed order of the variables, the same function always yields, up to permuting rows and columns, the same BMP. Because the matrices are just 0/1 tables with one 1 per row, building and manipulating BMPs reduces to elementary linear algebra, and every BMP can be converted losslessly into a Binary Decision Diagram and back. The point of the representation is practicality: Boolean reasoning tools that rely on simple matrix operations rather than the intricate pointer machinery of decision-diagram libraries, with complexity controlled by reordering the variables.

What carries the argument

The central object is the matrix train of row-switching matrices. Each $M^{{(i)}}$(x_i) has entries that are 0 or 1 with exactly one 1 per row, so it selects, for each subfunction carried on the left, which subfunction to carry on the right after fixing x_i. The terminal vector R=(0,1)^T encodes the two constant functions. The argument is carried by the CLEAN operation: a left-to-right sweep applies Shannon decomposition and removes duplicate rows, and a right-to-left sweep merges isomorphic terminal subgraphs; iterated, these sweeps produce the unique compressed form. The SWAP operation, based on two-sided Shannon decomposition, lets one reorder adjacent variables, and REORDER composes swaps; exact and heuristic search (A* with branch and bound, and sifting) minimize the accumulated bond dimension.

What would settle it

Run CLEAN on a family of n-bit Boolean networks whose BDDs are known to be polynomial; if the number of compression sweeps or intermediate matrix sizes grows faster than polynomial in n, the claimed canonicalization efficiency fails. More directly, exhibit one BMP that CLEAN reduces to two different row-switching forms under the same lexicographic rule, which would refute uniqueness.

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Extended reading notes

Core claim

For a vector of m Boolean functions on n variables, the paper constructs a representation F(x) = $M^{{(n-1)}}$(x_{n-1}) ... $M^{{(0)}}$(x_0) R, where each M is a row-switching matrix (one 1 per row) built by Shannon-decomposing on one variable and then deleting duplicate rows. The claim is that this construction, with duplicate rows removed at each stage, is a normal form: Eq. (7) is unique up to permutations of rows and columns, which the paper describes as a gauge freedom removable by lexicographic ordering. The CLEAN operation, sweeping left-to-right and right-to-left, restores canonical form after arbitrary manipulations, and the paper shows how to implement Boolean operations (APPLY, RESTRICT, INSERT, JOIN, COMPOSE, SWAP, REORDER) on BMPs. It also establishes the exchange: a BMP becomes a BDD by taking each matrix as the adjacency table between consecutive variable levels, and a BDD becomes a BMP after pass-through padding; maximally compressed BMPs correspond to quasi-reduced BDDs. The paper therefore claims not only a new normal form but a genuine equivalent pipeline to BDD-based reasoning carried by linear algebra.

Load-bearing premise

The load-bearing premise is that the CLEAN operation always terminates in the unique compressed canonical form in polynomial time; the paper states this but gives no formal complexity bound or termination proof.

Editorial extensions

If this is right

  • For a fixed variable order, two syntactically different BMPs that compute the same functions will compress to the same canonical form, so equivalence checks become a matter of compression.
  • Every operation available on BDDs, including apply, restrict, compose, and variable reordering, has a counterpart implemented on BMPs, so BDD-based reasoning can in principle run on a linear-algebra engine.
  • When the bond dimension stays polynomial in n, BMP evaluation and manipulation are polynomial, and this volume measure mirrors the node count of the equivalent quasi-reduced BDD.
  • The variable-ordering problem is inherited from BDDs: finding an order with small BMP volume is NP-complete, but exact A* search and sifting heuristics transfer directly.
  • The reversible-circuit and quantum-circuit verification settings that motivate the work are targets where the MPS/tensor-train analogy may bring tensor-network compression tools to Boolean reasoning.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper leaves implicit: because BMP volume equals the node count of the equivalent quasi-reduced BDD up to padding, lower-bound techniques for BDD size should transfer to BMP bond dimension.
  • The MPS analogy suggests an approximate variant the paper does not pursue: truncating bond dimension during CLEAN would give a controlled approximation of a Boolean function, trading exactness for compactness.
  • If CLEAN is confirmed polynomial, practical significance depends on the empirical distribution of bond dimensions over typical circuits; a benchmark against BDD libraries on adder, multiplier, and netlist families would settle the practicality question.
  • The row-switching array representation makes BMPs naturally sparse, so software built on tuple arrays rather than full matrices may inherit the performance profile of adjacency-list BDD packages.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces the Binary Matrix Product (BMP) representation of Boolean functions: a product of binary row-switching matrices, built by iterated Shannon decomposition followed by compression of duplicate rows. The central claims are (i) that fully compressed BMPs form a canonical normal form for a fixed variable order, unique up to row/column permutations (Eq. (7) and Section III); (ii) that a CLEAN procedure can restore canonical form after arbitrary BMP operations in polynomial time (Section IV.A); and (iii) that BMPs are equivalent, with an explicit translation, to Binary Decision Diagrams (Section V). The paper also defines operations APPLY, RESTRICT, INSERT, JOIN, COMPOSE, SWAP, REORDER, and REVERSE ORDER, studies variable-ordering optimization via an exact A* method and sifting heuristics with experimental timings, and reports a public Julia implementation. The authors explicitly acknowledge that no killer application is demonstrated and that the practical utility depends on keeping bond dimensions polynomial.

Significance. If the canonicity and CLEAN-normalization claims are fully proved, the paper makes a useful contribution as a linear-algebraic reformulation of a standard data structure: BMPs are essentially layer-wise adjacency matrices of quasi-reduced BDDs, and the explicit matrix operations may lower the implementation barrier for BDD-style manipulation and connect Boolean function synthesis to the tensor-network community. The paper's strengths are its concrete Shannon-decomposition construction, the explicit BMP-to-BDD translation with worked examples, the careful treatment of APPLY by direct product versus direct sum, and the availability of an implementation. The main caveat is that the substantive novelty is representational rather than asymptotic: the paper itself notes that BDDs can be smaller than BMPs because BDDs skip levels, and that no killer application is identified. The correctness of the central claims is therefore the main gate, and the current manuscript does not yet close that gate.

major comments (3)
  1. [Section III, Eq. (7)] The canonicity claim is stated but not proved. The text says the property follows from two elements, (a) one-variable-at-a-time iteration and (b) uniqueness of Shannon decomposition and compression up to permutation of unique rows, but no induction argument is written. The subsequent sentence that redundancy can be removed by choosing "any lexicographic ordering" is the first and only mention of that rule, and no lexicographic labeling procedure is defined or used in the construction. In particular, the SU decomposition in the Appendix (Algorithm 1) assigns row labels by order of first appearance in a hash table, so two compressions of the same function can produce different representatives. Since the unique representative is the basis for calling Eq. (7) a canonical normal form, this gap is load-bearing, not cosmetic.
  2. [Section IV.A, Eqs. (25)-(33)] The central operational claim about CLEAN is asserted without proof. The sentence "importantly, manipulations of BMPs that can be performed efficiently ... lead to matrix products that can be compressed into canonical form with a polynomial number of steps" is not supported by any termination or complexity argument for the LTR/RTL sweep schedule. In particular, the text states that reaching canonical form requires at least one round of LTR-cleaning, but it does not prove that one LTR sweep after one RTL sweep always reaches a fixed point, nor does it bound the cost of the compression steps such as Eq. (27), which require detecting and removing duplicate rows. Because APPLY, RESTRICT, JOIN, COMPOSE, SWAP, and REORDER all route through CLEAN, a non-terminating or order-dependent CLEAN would invalidate the paper's claim that BMP operations are closed on a canonical representation. The proof may be quite possible, but it must be supplied.
  3. [Section IV.B.2, Eq. (41)] The claim that the RTL sweep alone is sufficient to compress the direct-sum APPLY output is left unexplained and appears in tension with the note in Section IV.A.2 that canonical form requires at least one LTR-cleaning round. If the special structure of the direct-sum construction makes RTL cleaning sufficient, the reason should be stated explicitly; otherwise the reader cannot verify that the direct-sum method produces the same canonical representative as the direct-product method after CLEAN.
minor comments (5)
  1. [Section II.A, Eq. (16)] The label on the left-hand side of Eq. (16) reads M(1)(x2), but the variable being decomposed is x1; it should read M(1)(x1).
  2. [Section II.A, Eq. (21)] The assembled BMP for h(x) = \bar{x2} x1 x0 is missing the overbar on x2 in the first matrix, and the lower row of the last matrix should be (\bar{x0} x0) to match Eq. (20).
  3. [Section V, Eq. (63)] The example function is written as f(x2,x1,x0)=x0(x2∨x2x1), but the following paragraph says the function is g(x)∨h(x) with h=x2 x1 x0; presumably the intended expression is x0(x2∨\bar{x2}x1).
  4. [Section V] There is a typo in "the the bypassing of nodes in non-complete BDDs" and in Section IV.B.2 "the method based of direct products" should read "based on direct products."
  5. [Section VI.B, Eqs. (83)-(84)] The notation in the APPLY complexity discussion is slightly confusing: the subfunctions f(vi, x'i) and g(vi, x'i) use the same label vi for the fixed prefix, but the two BMPs may have different bond dimensions pi and qi; clarifying that vi ranges over all 2^(n-i) assignments would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BMP normal-form derivation is self-contained, self-citations are only motivational, and the BDD correspondence is an explicit translation rather than a load-bearing input.

full rationale

The paper's central claim, that compressed BMPs are canonical normal forms (Sec. III), is derived internally by induction on the Shannon decomposition: the cofactors of a Boolean function are uniquely determined, the compression step eliminates exactly the duplicate rows, and the resulting row-switching matrices are unique up to a permutation gauge that the paper explicitly removes by fixing a lexicographic ordering. This is an ordinary mathematical derivation, not a fitted parameter or a self-referential definition. The self-cited EOC work [6] appears only as motivation ("The contribution of this paper was motivated by our recently proposed approach...") and is not used to justify the uniqueness theorem or any operation. The BMP-BDD equivalence in Sec. V is presented as an explicit translation via Eqs. (61)-(62), and the statement "Fully compressed BMPs correspond to quasi-reduced BDDs" is a comparison to a known data structure, not an imported uniqueness theorem from the authors' prior work. The CLEAN operation is asserted to compress efficiently and to reach canonical form after LTR/RTL sweeps, but no termination or complexity proof is supplied; that is a rigor gap in correctness/complexity, not circularity, because the output is not assumed equal to the input by construction. Similarly, the SU-decomposition's order-of-appearance labeling leaves a normalization convention unspecified, but the paper openly acknowledges the permutation gauge. No step reduces, by the paper's own equations, to its own inputs, and there are no fitted quantities masquerading as predictions. The paper is therefore best scored as having no significant circularity, with the noted proof gaps belonging to a correctness or completeness review rather than to a circularity analysis.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The BMP representation is a new mathematical object, but not a physical entity; it is constructed from standard row-switching matrices and Shannon decomposition. No fitted parameters. The main unstated assumption is the polynomial-time CLEAN claim.

assumptions (4)
  • standard math Shannon decomposition expresses Boolean functions using integer arithmetic with x in {0,1} and bar x = 1 - x.
    Used throughout Sec. II to construct the BMP via matrix products. Relies on standard Boolean algebra and integer arithmetic.
  • domain assumption A row-switching matrix (single 1 per row) can represent the compressed Shannon decomposition uniquely up to permutation of rows and columns.
    Assumed in Sec. II and Sec. III; the existence of the RS matrix for removing dependent rows is argued constructively, but uniqueness relies on a lexicographic convention.
  • domain assumption Optimal variable ordering is NP-complete, imported from the BDD literature (Ref. [3]).
    Used in Sec. VI to justify the variable-order optimization algorithms. This is a result from prior work, not derived in this paper.
  • ad hoc to paper The CLEAN operation terminates in a polynomial number of steps.
    The paper states this in Sec. IV.A without a formal bound. It is load-bearing for the practical efficiency of all BMP operations.

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Pith. "Pith review of A Matrix Product State Representation of Boolean Functions." pith.science (2026). https://pith.science/paper/NN3IJKJO

@misc{pith2026250501930,
  author       = {Pith},
  title        = {Pith review of: A Matrix Product State Representation of Boolean Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NN3IJKJO}},
  note         = {Machine review of arXiv:2505.01930}
}
abstract

We introduce a novel normal form representation of Boolean functions in terms of products of binary matrices, hereafter referred to as the Binary Matrix Product (BMP) representation. BMPs are analogous to the Tensor-Trains (TT) and Matrix Product States (MPS) used, respectively, in applied mathematics and in quantum many-body physics to accelerate computations that are usually inaccessible by more traditional approaches. BMPs turn out to be closely related to Binary Decision Diagrams (BDDs), a powerful compressed representation of Boolean functions invented in the late 80s by Bryant that has found a broad range of applications in many areas of computer science and engineering. We present a direct and natural translation of BMPs into Binary Decision Diagrams (BDDs), and derive an elementary set of operations used to manipulate and combine BMPs that are analogous to those introduced by Bryant for BDDs. Both BDDs and BMPs are practical tools when the complexity of these representations, as measured by the maximum bond dimension of a BMP (or the accumulated bond dimension across the BMP matrix train) and the number of nodes of a BDD, remains polynomial in the number of bits, $n$. In both cases, controlling the complexity hinges on optimizing the order of the Boolean variables. BMPs offer the advantage that their construction and manipulation rely on simple linear algebra -- a compelling feature that can facilitate the development of open-source libraries that are both more flexible and easier to use than those currently available for BDDs. An initial implementation of a BMP library is available on GitHub, with the expectation that the close conceptual connection to TT and MPS techniques will motivate further development of BMP methods by researchers in these fields, potentially enabling novel applications to classical and quantum computing.

Figures

Figures reproduced from arXiv: 2505.01930 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Example of a BDD originated from the BMP in (b). [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) The BDD representation of the BMP in Eq. (66). (b) After LTR cleaning. (c) After RTL cleaning. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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    ˜M(n−1)(0) ˜M(n−1)(1) # 2p× ˜pn−1 , (28) which, together with Eqs. (26) and (27), allows us to write h Q(n−1)(xn−1) i p×pn−1 = [ S(xn−1)]p×2p

    LTR cleaning We proceed by considering the Shannon decomposition of the left-most matrix in the train, Q(n−1)(xn−1), h Q(n−1)(xn−1) i p×pn−1 = [S(xn−1)]p×2p " Q(n−1)(0) Q(n−1)(1) # 2p×pn−1 . (26) We note that Eq. (26) describes the column-by-column Shannon decomposition of the matrix Q(n−1)(xn−1) which one could also regard as the inner product of two aux...

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    The canonical form for h(f,g ) is then obtained by applying the CLEAN operation to the resulting BMP of Eq

    Direct product Starting with the Shannon expansion of the function h(f,g ), we write h(f,g ) =f(x)g(x)h(0, 0) +f(x)g(x)h(0, 1) +f(x)g(x)h(1, 0) +f(x)g(x)h(1, 1) = h f(x) f(x) ⊗ g(x) g(x) i   h(00) h(01) h(10) h(11)   (34) Now, using the BMP representations of f and g, f(x) = F(n−1)(xn−1) F(n−2)(xn−2)··· F(1)(x1) F(0)(x0) R g(x) = G(n−1)(xn−1) G(...

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    Here we present an alternative and often more efficient method based on direct sum of the matrices in the BMP representations of f and g

    Direct sum While the method based of direct products produces a compressed BMP after the application of the CLEAN operation, the intermediate matrices it generates can be prohibitively large. Here we present an alternative and often more efficient method based on direct sum of the matrices in the BMP representations of f and g. This involves transforming ...

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    In this formulation, a state q is a subset of the input variables Xn ={x1,x 2,...,x n}, representing all BMPs whose first|q| variables are those in q, in any order

    Exact minimization The exact minimization algorithm formulates the problem of finding the optimal variable ordering as a search in a state space [10]. In this formulation, a state q is a subset of the input variables Xn ={x1,x 2,...,x n}, representing all BMPs whose first|q| variables are those in q, in any order . This state can transition into states q′...

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    These methods are implemented on given BMPs via the SWAP operation, which changes the order of two adjacent variables in the matrix train

    Heuristic methods In many cases, simpler heuristic methods for improving the variable order suffice. These methods are implemented on given BMPs via the SWAP operation, which changes the order of two adjacent variables in the matrix train. A simple but highly efficient method is the sifting algorithm proposed by Rudell in [39] for BDDs, which can be readi...

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    killer application

    Comparison of reordering schemes Table I shows the runtimes and resulting BMP volumes for three different methods of volume minimization via variable reordering for a BMP implementing a full adder circuit [45], withn being the number of bits of each operand. (Therefore, the circuit has 2n input bits andn+1 output bits.) All BMP manipulations are performed...

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    (91) Furthermore, we can perform various matrix operations using m without needing to construct M explicitly

    We can associate with each such matrix M of size p×q, an array m of length p containing values in the range 1,...,q , such that [M]ij =δj,m[i]. (91) Furthermore, we can perform various matrix operations using m without needing to construct M explicitly. Thus, BMPs can be implemented entirely in terms of arrays m, saving both time and space. In this append...

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