Computational Paths in Type Theory

• Computational paths are formal syntactic objects composed of sequences of definitional rewrites that witness propositional equality in Martin-Löf Intensional Type Theory.
• They employ equality constructors like reflexivity, symmetry, and transitivity, normalized via a terminating term rewriting system ensuring unique normal forms.
• Their combinatorial and categorical properties yield a weak groupoid structure and extend to higher ω-groupoids, supporting mechanized proof verification.

A computational path is a formal, syntactic object representing an explicit sequence of definitional rewrites—such as β, η, reflexivity, symmetry, and transitivity—witnessing propositional equality between two terms of the same type within Martin-Löf Intensional Type Theory. The concept was introduced by de Queiroz and Gabbay (1994) as "sequences or rewrites" and has since been developed into a rigorous framework for the paper of identity types, groupoid semantics, and higher categorical structures. Computational paths provide the internal, syntactic grounds on which propositional equality stands, in contrast to the purely semantic interpretations of equality in homotopy type theory. Their combinatorial and categorical properties make them foundational for a syntactic approach to weak groupoids and higher structures in type theory (Ramos et al., 2015).

1. Formal Definition and Syntax of Computational Paths

A computational path from $a$ to $b$, both of type $A$, is defined as a finite composition of equality axioms and α-renamings, providing explicit witness of $a=b$ at the syntactic level. Formally, for type $A$:

$$a =_{\,s\,} b : A \quad\Longleftrightarrow\quad s\ \text{is a sequence of rewrites taking}\ a\ \text{to}\ b.$$

The core constructors for computational path formation are:

• Reflexivity ($\rho$): $a =_{\rho} a : A$
• Symmetry ($\sigma$): $a =_{s} b : A$ implies $b =_{\sigma(s)} a : A$
• Transitivity ($\tau$): $a =_{s} b : A$ and $b =_{t} c : A$ imply $a =_{\tau(s,t)} c : A$
• Other definitional steps: such as $\beta$-reduction, $\eta$-expansion, and contextual congruence rules

Computational paths constitute terms of the identity type $\mathsf{Id}_A(a, b)$, treating equality proofs as first-class syntactic objects (Ramos et al., 2015).

2. Term Rewriting System and Redundancy Elimination

The composition of basic equality constructors may yield redundant representations of the same equality. To normalize computational paths, a term rewriting system (LND$_{EQ}$–TRS) was established by de Oliveira (1994). It includes 39 local rewrite rules, among them:

• Left and right units: $\tau(\rho_a, s) \rhd s$, $\tau(s, \rho_b) \rhd s$
• Associativity of composition: $\tau(\tau(s, t), u)\rhd \tau(s, \tau(t, u))$
• Inverses: $\tau(s, \sigma(s)) \rhd \rho_a$, $\sigma(\sigma(s)) \rhd s$

The system is confluent and terminating: every computational path admits a unique normal form up to $rw$-equality (Ramos et al., 2015), enabling effective comparison and normalization of equality proofs.

3. Groupoid Structure and Higher Categorical Properties

A key result is that computational paths, together with their rewrite system, induce a weak groupoid structure on any type $A$. The construction is as follows:

• Objects: terms $a, b: A$
• Morphisms: computational paths $s: a =_{s} b$
• Composition: given by $\tau(s,t): a =_{\tau(s,t)} c$
• Identity morphisms: $\rho_a : a =_{\rho_a} a$
• Inverses: $\sigma(s): b =_{\sigma(s)} a$
• Laws: associativity, identity, and inverse laws hold up to $rw$-equality

For example, associativity holds up to rewrite by the $tt$-rule: $\tau(\tau(r, s), t) =_{rw} \tau(r, \tau(s, t))$

This weak groupoid structure extends categorically: when quotienting computational paths by $rw$-equality, the structure becomes a strict groupoid at the quotient level (Ramos et al., 2015).

4. Higher Categorical Extension and ω-Groupoid

The framework naturally extends to higher dimensions:

• 2-cells: rewrites between computational paths (i.e., $rw$-equality proofs between paths $s, t: a =_{(-)} b$), forming the morphisms of a category $A_{2rw}(a, b)$
• 3-cells and beyond: higher-level rewrites among these $n$-cells, with each level governed by an extended term rewriting system $LND_{EQ}-TRS_n$
• Limit structure: this iteration yields a weak ω-groupoid, in which:
  • 0-cells: terms of $A$
  • 1-cells: computational paths
  • 2-cells: rewrites between paths
  • $n$-cells: $(n-1)$-level rewrite equalities

The coherence conditions (e.g., Mac Lane's pentagon and triangle identities) hold only up to further rewrites, whose normal forms and equivalence classes control the strictification at each dimension. The creation of a "tower" of rewrite structures ensures all necessary higher coherences (Ramos et al., 2015).

5. Categorical Semantics and Coherence Laws

Within categorical semantics, the following points are fundamental:

• Composition and identities arise from the syntactic structure of computational paths, with all required coherence conditions (pentagon, triangle for associators and unitors) being witnessed by explicit rewrite transformations at the next dimension.
• Quotienting by suitable higher rewrite relations (e.g., $rw_2$-equality) yields strict $n$-categories at the corresponding dimension.
• Concrete coherence witnesses: for example, associativity in the form

$$\tau(\tau(r, s), t) \xRightarrow[\text{assoc}]{\qquad} \tau(r, \tau(s, t))$$

is mediated by a 2-cell induced by the $tt$-rule; Mac Lane's pentagon expresses the higher associativity, handled by higher rewrite rules (e.g., $tt_2$) (Ramos et al., 2015).

6. Key Results, Proof Sketches, and Significance

The central theoretical propositions are as follows:

• Proposition (Weak Groupoid): In $A_{rw}$, every arrow $s: a \to b$ has a weak inverse $\sigma(s)$, and the triangle and pentagon equations hold up to $rw$ (Ramos et al., 2015).
• Proposition (Weak 2-Category): The quotient by $rw_2$-equality yields a strict 2-category whose 1-cells are computational paths and 2-cells are $rw$-classes, with horizontal/vertical compositions and interchange law structured by $\tau$ and rewrite concatenation.

The explicit rewriting structure removes all ambiguity about higher identity proofs—uniqueness of identity proofs does not hold, and the explicit normal forms of computational paths reveal this combinatorial richness (Ramos et al., 2015).

7. Applications and Implications

• Internalization of Equality: Computational paths supply a syntactic model for the equality structure in type theory, grounding the abstract, semantical groupoid models of identity types (e.g., Hofmann–Streicher 1994) in explicit proof objects.
• Homotopy Type Theory foundations: The computational path perspective aligns with the homotopic viewpoint, where identity types correspond to path spaces, but does so fully syntactically.
• Coherence computation: Algebraic verification of higher coherence laws (e.g., pentagon identity) becomes a normalization problem in the rewrite theory of computational paths.
• Extension to mechanized mathematics: This explicit model is beneficial for formal proof verification, as it offers decidability (via normalization) for equality judgments at all levels and is amenable to mechanization in systems like Lean.

The syntactic, normalization-based framework of computational paths thus underpins a rigorous, mechanizable account of the entire weak ω-groupoid structure implicit in intensional type theory, with ramifications for the foundations of both constructive mathematics and category-theoretic homotopy theory (Ramos et al., 2015).