Sentence complexity of theorems in Mizar

As one of the longest-running computer-assisted formal mathematics projects, large tracts of mathematical knowledge have been formalized with the help of the Mizar system. Because Mizar is based on first-order classical logic and set theory, and because of its emphasis on pure mathematics, the Mizar library offers a cornucopia for the researcher interested in foundations of mathematics. With Mizar, one can adopt an experimental approach and take on problems in foundations, at least those which are amenable to such experimentation. Addressing a question posed by H. Friedman, we use Mizar to take on the question of surveying the sentence complexity (measured by quantifier alternation) of mathematical theorems. We find, as Friedman suggests, that the sentence complexity of most Mizar theorems is universal ($\Pi_{1}$, or $\forall$), and as one goes higher in the sentence complexity hierarchy the number of Mizar theorems having these complexities decreases rapidly. The results support the intuitive idea that mathematical statements, even when carried out an abstract set-theoretical style, are usually quite low in the sentence complexity hierarchy (not more complex than $\forall\exists\forall$ or $\exists\forall$).

💡 Research Summary

The paper investigates the logical sentence complexity of theorems stored in the Mizar Mathematical Library (MML) by measuring quantifier alternation using the familiar Π/Σ hierarchy of prenex normal forms. The authors begin by recalling the long history of computer‑assisted formalization and the specific design of Mizar, which internally represents formulas only with universal quantifiers, conjunction, and negation. Because of this internal representation, existential quantifiers appear as negated universals, and the authors therefore compute complexity on the internal form rather than on the surface syntax written by human formalizers.

Complexity is defined syntactically: Σₙ consists of formulas that can be written as a block of existential quantifiers followed by a Πₙ₋₁ formula, while Πₙ consists of a block of universal quantifiers followed by a Σₙ₋₁ formula. Every first‑order formula is equivalent to some prenex form (Fact 1). Since a formula may have several prenex equivalents, the authors adopt a “minimal prenex” rule, selecting the equivalent with the lowest index in the hierarchy. This choice reflects the intuition that a human author would not deliberately encode a more intricate quantifier pattern than necessary.

The empirical study uses Mizar version 8.1.02 and MML version 5.20.1189. The notion of “theorem” is deliberately broadened to include not only items explicitly labeled theorem but also lemmas, diffuse reasoning blocks, type‑changing statements, properties, rewrite rules, identifications, scheme instances, existence/uniqueness conditions for new function symbols, existence conditions for new types, and definitional theorems. In total, 89 506 items are classified as theorems, and 10 657 as definitional theorems.

Two tables present the distribution of Πₙ and Σₙ complexities. For all theorems, Π₀ (quantifier‑free) and Π₁ (pure universal) dominate: 3 254 Π₀ items and 67 252 Π₁ items. Higher universal levels (Π₂, Π₃, …) appear in rapidly decreasing numbers, and no Πₙ or Σₙ occurs for n > 7. The Σ‑side is dramatically smaller: only 1 374 Σ₁ items and a handful of Σ₂–Σ₅, with Σₙ empty for n > 5. For definitional theorems, Σₙ never appears for n > 0, confirming that definitions in Mizar are inherently non‑existential; the Π‑distribution again drops steeply, with the highest observed level Π₆.

The discussion interprets these findings. First, the prevalence of Π‑formulas aligns with the conventional mathematical practice of stating results as universal claims; genuinely existential theorems are rare because parameters are implicitly universally quantified at the outermost level. Second, the absence of Σₙ definitions is explained by the Mizar definition scheme, which requires a definitional theorem to be an equation without quantifiers. Third, the few highly complex theorems (Π₄–Π₆) fall into identifiable categories: (a) characterizations that combine several mildly complex properties, (b) representation theorems asserting that every object of a class is isomorphic to a canonical form, and (c) existence/uniqueness conditions for functions whose definitions themselves are elaborate. An illustrative example is a theorem characterizing finite commutative groups via direct products, whose prenex form reaches Π₆.

Limitations are acknowledged. The conversion to prenex form may reorder quantifiers in ways that differ from the author’s intended logical structure, and the broad definition of “theorem” blurs the line between propositions and auxiliary statements. Moreover, because Mizar’s internal language lacks explicit existential quantifiers, some formulas acquire a higher apparent complexity than they would in a richer connective set.

Future work is outlined: (i) isolate definitional theorems and analyze them separately to obtain a cleaner Σ/Π separation, (ii) compare the Mizar complexity profile with those of other proof assistants such as Coq or Isabelle to test the generality of the observed distribution, and (iii) explore correlations between sentence complexity, proof length, and the success rate of automated theorem provers, which could inform the design of more efficient proof‑search strategies.

In sum, the paper provides a large‑scale empirical confirmation of Friedman’s conjecture that mathematically meaningful sentences are “logically tight” and tend to occupy low levels of the quantifier‑alternation hierarchy. The data show that the overwhelming majority of formalized mathematics in Mizar is at the Π₁ level, with a rapid decay for higher alternations. This insight has practical implications for formalization efforts and automated reasoning, suggesting that most mathematical knowledge can be captured with relatively simple logical forms, while the few complex statements may benefit from auxiliary definitions or alternative formulations.