A SAT Attack on Tarski's High School Algebra Problem

Abstract: Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the positive integers and yet does not follow from Tarski's axioms: \begin{align*} &\left((1+x)^y + (1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x + (1+x^2+x^4)^x\right)^y = \ &\left((1+x)^x + (1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y + (1+x^2+x^4)^y\right)^x. \end{align*} Gurevič gave an algebra on 59 elements that satisfies Tarski's axioms but not Wilkie's identity, and over the years several authors whittled down the size of such a countermodel, culminating in a countermodel of size 12 due to Burris and Yeats. On the other hand, Zhang proved that there is no countermodel with fewer than 11 elements. Using SAT, we prove that the smallest countermodels are of size 12, as conjectured by Burris and Yeats. Moreover, we show that there are exactly 8,957,952 countermodels on 12 elements up to isomorphism and provide a simple classification of them. Our SAT approach outperforms dedicated tools for finding countermodels in equational theories, namely Mace4 and SEM. Furthermore, using autoformalization, we prove the correctness of our main result in Lean.

• The paper proves that no finite high school identity algebra with at most 11 elements violates Wilkie’s identity, while a 12-element countermodel exists.
• The authors combine a semantically exact SAT encoding, algebraic pruning lemmas, normalization, and symmetry breaking to reduce the 11-element unsatisfiability proof to about 627 seconds.
• The paper classifies exactly 8,957,952 pairwise non-isomorphic 12-element countermodels and validates the key negative result with DRAT/LRAT certificates and a Lean formalization.

The paper resolves a longstanding finite-model question associated with Tarski’s high school algebra problem. Tarski asked whether the equational theory of positive integers with operations $+$, $\cdot$, exponentiation, and constant $1$ is axiomatized by the eleven standard high school identities. Wilkie’s construction established a negative answer by exhibiting an identity valid over $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$ but not derivable from those axioms. The present work determines the exact minimum cardinality of a finite algebra witnessing this failure and classifies all witnesses at that cardinality (2608.08421).

Problem setting and main results

An HSI algebra is a finite algebra interpreting addition, multiplication, exponentiation, and $1$ such that the eleven high school identities hold. The central identity is Wilkie’s identity:

$$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$

It is valid over the positive integers because

$1+x^3=(1+x)(1-x+x^2)$

and

$1+x^2+x^4=(1+x+x^2)(1-x+x^2).$

After applying the high school identities, both sides can be transformed into a common expression involving $(1-x+x^2)^{xy}$. This argument is unavailable in the positive-exponentiation language because subtraction is not among the permitted operations. Wilkie showed syntactically that the identity is independent of the eleven axioms; finite countermodels provide an algebraic form of the same independence result.

The paper proves two principal theorems:

• No countermodel exists with at most eleven elements.
• There are exactly $8,957,952$ countermodels on twelve elements up to isomorphism.

The first result confirms the conjecture of Burris and Yeats that the known twelve-element countermodel was minimal. It also strengthens Zhang’s lower bound of eleven, which had previously been obtained through computational model-searching methods. The second result is substantially finer than a minimum-size result: it gives a complete enumeration and a structural parametrization of every twelve-element countermodel.

The cardinality is factored as

$\cdot$0

The classification decomposes this number into $\cdot$1 possible addition tables and $\cdot$2 possible exponentiation tables for each addition table; the multiplication table is uniquely determined in the chosen normal form. All enumerated models are pairwise non-isomorphic.

SAT encoding of HSI algebras

For a fixed cardinality $\cdot$3, the domain is represented by the labeled set $\cdot$4, with label $\cdot$5 interpreting the constant symbol. The encoding uses Boolean variables

• $\cdot$6 for $\cdot$7,
• $\cdot$8 for $\cdot$9,
• $1$0 for $1$1.

Because addition and multiplication are commutative, variables for these operations are introduced only for $1$2. Exponentiation remains fully ordered.

Exact functionality is enforced using one-hot constraints: for every input pair, exactly one output variable must be true. Unit clauses encode identities such as $1$3, $1$4, and $1$5. Commutativity is built directly into the variable representation.

The technically important part of the encoding concerns identities involving nested terms. A naive encoding of associativity, for example, would branch over all intermediate values of the two sides and require $1$6 clauses for addition alone. The authors introduce auxiliary variables such as $1$7 to represent the common value of both parenthesizations. Forward implications connect the primitive operation variables to these auxiliary variables, while an $1$8 constraint ensures that the two parenthesizations cannot yield distinct values. This reduces the associativity encoding to $1$9 clauses.

Analogous auxiliary-variable constructions encode multiplication associativity, distributivity, and the exponentiation identities

$(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$0

and

$(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$1

The full HSI encoding contains $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$2 clauses. The authors note that some individual encodings can be reduced further asymptotically, but experimentally smaller asymptotic bounds did not necessarily produce better solver performance. This distinction is important: the engineering objective is not merely minimizing clause count, but producing a propagation structure favorable to CDCL SAT solving.

The paper establishes a bidirectional correctness result. Every satisfying assignment determines a unique finite algebra satisfying the high school identities, and every HSI algebra determines a unique satisfying assignment. Consequently, satisfiability of the encoding is equivalent to existence of an HSI algebra of the specified size, rather than merely being a heuristic representation of the algebraic constraints (2608.08421).

Encoding failure of Wilkie’s identity

The countermodel condition requires that some pair $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$3 violate Wilkie’s identity. Several structural results from earlier work show that, in any countermodel, the distinguished elements

$(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$4

are all distinct. The authors therefore normalize the labels by setting

$(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$5

This is a symmetry-preserving reduction, not an assumption that restricts the class of countermodels. The encoding then constructs one-hot variables for the values of all subexpressions in the parse trees of the left- and right-hand sides of Wilkie’s identity evaluated at $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$6. Each composed expression is connected to its operands through implication clauses and is constrained to have exactly one output. Finally, the two root values are required to be distinct.

The use of an explicit parse-tree encoding avoids expanding the identity into an unwieldy collection of nested case distinctions. The resulting clauses are negligible relative to those required for the HSI axioms.

The authors further incorporate mathematically derived necessary conditions for a Wilkie countermodel. These include non-equalities involving $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$7, $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$8, and powers of $(\mathbb{Z}_{>0}, +, \cdot, \uparrow, 1)$9, divisibility constraints among the expressions

$1$0

and restricted instances of Jackson’s polynomial-exclusion lemma. These constraints substantially prune the SAT search space. Their logical status is explicit: they are consequences of being a countermodel, so adding them preserves completeness, provided the cited lemmas are correctly formalized.

Symmetry breaking and computational performance

The dominant computational difficulty is the large permutation symmetry among unlabeled domain elements. After fixing labels $1$1 through $1$2, the remaining elements are interchangeable. Without symmetry breaking, the solver must rediscover equivalent structures under many permutations.

The authors impose lex-leader constraints for every transposition among labels $1$3. The variables for addition, multiplication, and exponentiation are concatenated into a fixed sequence, and the sequence is required to be lexicographically no greater than its image under each transposition. This selects a canonical representative from each orbit under the residual symmetry group.

The performance effect is substantial. Without symmetry breaking, the solver already requires approximately $1$4 seconds to establish unsatisfiability at size ten, compared with approximately $1$5 seconds with symmetry breaking. The experiments therefore support the paper’s central methodological claim that generic SAT techniques become effective on this problem when combined with algebra-specific normalization and symmetry reduction.

The full formulas have the following scale and outcomes:

Domain size	Variables	Clauses	Outcome	Runtime
7	10,581	404,292	UNSAT	1.38 s
8	18,571	819,202	UNSAT	1.79 s
9	30,921	1,540,377	UNSAT	7.45 s
10	49,110	2,726,169	UNSAT	65.84 s
11	74,845	4,589,476	UNSAT	626.52 s
12	110,061	7,409,052	SAT	3,053.55 s

The size-eleven instance is thus refuted in $1$6 seconds on a single core of an Apple M5 system, while a twelve-element countermodel is found in approximately $1$7 minutes. This contrasts sharply with earlier experiments using Mace4 and SEM, in which the eleven-element lower bound was not established despite weeks of computation and the known twelve-element model required months to obtain.

The result also directly contradicts Zhang’s earlier assessment that SAT-based methods were not especially advantageous for this benchmark. The paper attributes the improvement not to SAT in isolation, but to the combination of auxiliary variables, symmetry breaking, normalization lemmas, and a solver capable of exploiting the resulting CNF structure.

Certified negative results and Lean formalization

A satisfying assignment at size twelve can be checked directly by evaluating the three operation tables. The unsatisfiability result at size eleven is more delicate because it excludes every possible algebra in a large search space. The authors address this with independently checkable proof certificates and formal verification.

Kissat produces a DRAT certificate for the unsatisfiability of the eleven-element formula. The certificate is approximately $1$8 MB and can be checked by drat-trim; checking takes approximately $1$9 seconds on the reported hardware. The certificate verifies unsatisfiability of the generated CNF, but it does not by itself establish that the CNF correctly formalizes the mathematical problem.

To address this second issue, the authors formalize the encoding and its semantic correctness in Lean. The formalization contains more than 10,000 lines of code and includes the HSI encoding, the countermodel condition, the auxiliary lemmas used for pruning, and the symmetry-breaking constraints. The Lean development proves that the executable encoder produces a CNF satisfiable exactly when a countermodel of the corresponding size exists. The authors report byte-for-byte agreement between the Python-generated CNF and the formally specified encoder.

The DRAT proof is converted to LRAT and imported into Lean using LRAT-Catcher. The resulting textual LRAT certificate is approximately $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$0 GB, and certificate import takes $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$1 seconds. The formal theorem established is that no general countermodel exists for any $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$2.

There remains a stated trust boundary. For a proof of this scale, LRAT-Catcher is used through Lean’s native_decide, so the Lean compiler remains part of the trusted computing base. The paper does not claim complete foundational independence from all implementation infrastructure; instead, it substantially separates the mathematical correctness argument from the correctness of the SAT solver and encoder.

Classification of the twelve-element countermodels

The classification is obtained by projected model enumeration followed by isomorphism analysis. The solver enumerates $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$3 addition tables that extend to full countermodels. For every such addition table, there is exactly one multiplication table and exactly $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$4 exponentiation tables. The multiplication table is fixed across the normal-form classification, whereas addition and exponentiation contain independent parameters.

The addition tables depend on three choices for one group of entries and eight independent binary choices for another group, giving

$$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$5

possibilities. The exponentiation tables depend on three choices for one parameter block, five independent ternary choices, and a constrained block with $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$6 admissible assignments, giving

$$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$7

Their product yields the full enumeration:

$$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$8

The classification is not merely an enumeration of labeled SAT assignments. The authors explicitly verify that all models are non-isomorphic. They use nauty by reducing isomorphism of commutative Cayley tables to colored graph isomorphism. Automorphisms of the unique multiplication table are then used to check that no two classified addition tables lie in the same isomorphism orbit. This establishes the count up to isomorphism rather than only as a count of canonical solver outputs.

The classification also reveals structural regularities not apparent from the original countermodel search. For example, in every twelve-element model, the element representing $$\begin{aligned} &\left((1+x)^y+(1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x+(1+x^2+x^4)^x\right)^y \ &= \left((1+x)^x+(1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y+(1+x^2+x^4)^y\right)^x. \end{aligned}$$9 is the label $1+x^3=(1+x)(1-x+x^2)$0 in the chosen classification. The identity fails uniformly at the pair $1+x^3=(1+x)(1-x+x^2)$1. By selecting one exponentiation parameter configuration, one obtains $1+x^3=(1+x)(1-x+x^2)$2 at $1+x^3=(1+x)(1-x+x^2)$3, which induces a univariate identity with $1+x^3=(1+x)(1-x+x^2)$4 substituted for $1+x^3=(1+x)(1-x+x^2)$5. Exactly

$1+x^3=(1+x)(1-x+x^2)$6

twelve-element countermodels fail this univariate identity.

By contrast, no twelve-element countermodel in the classification satisfies the analogous substitution with $1+x^3=(1+x)(1-x+x^2)$7. The authors nevertheless find a thirteen-element countermodel for this latter identity. This result bears directly on Gurevič’s conjecture that the $1+x^3=(1+x)(1-x+x^2)$8 version may be minimal, with respect to growth rate, among univariate identities valid over the positive integers but independent of the high school identities. The paper does not settle that conjecture; it supplies a small countermodel and establishes that size twelve is insufficient.

Limitations and open questions

The main negative result concerns Wilkie’s identity specifically. It proves that no HSI algebra of size at most eleven violates that identity, but it does not prove that every identity valid over the positive integers is satisfied by all HSI algebras of size at most eleven. The smallest size at which a finite HSI algebra fails some positive-integer-valid identity remains unknown. The paper records $1+x^3=(1+x)(1-x+x^2)$9 as an upper bound, supplied by Wilkie’s identity, while the best published lower bound is $1+x^2+x^4=(1+x+x^2)(1-x+x^2).$0.

The classification also depends on a substantial collection of prior structural lemmas used as pruning constraints. Although these lemmas are formalized in Lean, the computational argument relies on their exact formal statements and on the correspondence between the formal encoder and the Python implementation. The authors make this correspondence explicit and machine-check it, but the compiler trust boundary remains.

Finally, the asymptotic encoding size remains large: the principal formulation uses $1+x^2+x^4=(1+x+x^2)(1-x+x^2).$1 clauses, with further reductions possible at an unfavorable empirical cost. The experiments demonstrate effectiveness through size twelve, not scalability to substantially larger finite algebras. The specific unresolved mathematical question is therefore not whether the present encoding can be generalized in principle, but whether analogous combinations of structural lemmas and symmetry constraints can settle the minimum size for other identities or for the full equational theory.

Conclusion

The paper establishes that twelve is the minimum size of a finite HSI algebra violating Wilkie’s identity and classifies all such algebras up to isomorphism. Its contribution combines a semantically correct SAT encoding, strong symmetry reduction, exhaustive model enumeration, graph-based isomorphism testing, and Lean-checked certification of the central unsatisfiability result. The resulting enumeration of $1+x^2+x^4=(1+x+x^2)(1-x+x^2).$2 non-isomorphic countermodels gives a detailed structural description of the first finite level at which Wilkie’s independence phenomenon appears, while leaving open the minimum size for countermodels to arbitrary identities valid over the positive integers (2608.08421).