Open Horn Type Theory

Abstract: We introduce Open Horn Type Theory (OHTT), an extension of dependent type theory with two primitive judgment forms: coherence and gap, subject to a mutual exclusion law. Unlike classical or intuitionistic negation, gap is not defined via implication but is a primitive witness of non-coherence. Judgments may also be open -- neither coherent nor gapped -- yielding a trichotomy that generalizes the binary derivable/underivable distinction. The central construction is the transport horn: a configuration where a term and a path both cohere, but transport along the path is witnessed as gapped. This captures obstructions that Homotopy Type Theory (HoTT) cannot express, since HoTT's Kan condition guarantees all transport succeeds. We develop the semantics via ruptured simplicial sets -- simplicial sets equipped with coherence and gap structure -- and ruptured Kan complexes, which model types where some horns fill, some are gap-witnessed, and some remain open. We show that HoTT embeds as the coherent fragment of OHTT, recovered by imposing totality. Three classes of obstructions are developed in detail: topological (monodromy, holonomy, characteristic classes), semantic (polysemy, meaning fibrations), and logical (resource-sensitive derivability, substructural failure). In each case, the gap witness is positive structure -- not absence of proof, but certified obstruction.

• The paper introduces gap witnesses as a new primitive to rigorously capture structured obstructions in derivability.
• It develops a three-state logic that distinguishes coherence, gap, and open judgments, thereby generalizing MLTT and HoTT.
• The semantic framework leverages ruptured Kan complexes to model complex failure phenomena in topology, computation, and formal semantics.

Open Horn Type Theory: Formal Foundations, Semantic Innovations, and Theoretical Implications

Introduction

Open Horn Type Theory (OHTT) proposes a radical expansion of the proof-relevant constructive type-theoretic tradition by introducing a new primitive: the gap witness, dual to traditional coherence witnesses. This move reconfigures the logical landscape inherited from both Martin-Löf Type Theory (MLTT) and Homotopy Type Theory (HoTT), enabling OHTT to formally structure not just what is derivable, but to make rigorous, proof-relevant statements about reasons for the failure of derivability—what the theory calls “rupture.” The system is designed to admit both positive evidence for coherence (inhabitation, construction) and positive evidence for non-coherence (structured absence), distinguishing OHTT distinctly from classical, intuitionistic, and paraconsistent approaches.

Primitive Judgments and Structural Trichotomy

OHTT builds on two primitive, proof-relevant judgment forms ${\Gamma}{J}$ (coherence witness) and ${\Gamma}{J}$ (gap witness), interpreted respectively as “$J$ is coherently witnessed” and “$J$ is witnessed gapped”, in context $\Gamma$. The system admits neither classical excluded middle nor intuitionistic duality: it does not assume every $J$ is either coherent or gapped, nor does it treat gap as “negation.” Instead, the absence of a witness for either side is a distinct, structurally significant third case—openness—where the status of $J$ is undetermined but not “false.” The sole axiom, the Exclusion Law, asserts the mutual exclusivity of coherence and gap for the same judgment in the same context, but makes no other ontological commitments.

Gap witnesses are strictly proof-relevant: witnesses of gaps may differ both from each other, and from mere absence of coherence, and may carry obstruction-theoretic or semantic content. This three-state logic underpins the entire calculus, giving rise to a structural trichotomy essential to the remaining developments.

Horn Types and Their Compositional Significance

Central to OHTT is the horn structure, defined for compositional relations between judgments. The horn ${J}{K}{L}$ (for some compositional relation $\circ$) is inhabited by the data $(\gamma_1, \gamma_2, \omega)$, where ${\Gamma}{J}$0 and ${\Gamma}{J}$1 are coherence witnesses for ${\Gamma}{J}$2 and ${\Gamma}{J}$3, and ${\Gamma}{J}$4 is a gap witness for ${\Gamma}{J}$5. This generalizes the Kan horn construction of simplicial sets and is essential in formalizing where local compositional success does not guarantee global closure—a situation invisible in classical and standard constructive logics.

This horn formalism enables OHTT to iteratively define “gaps” and “coherences” at all higher homotopical levels, expanding the system’s reach far beyond first-order failures of proof or derivability, into the territory of higher identity types, non-filling higher-dimensional horns, and structured obstructions.

The Relationship to Dependent and Homotopy Type Theory

OHTT’s coherent fragment (the subtheory where only coherence witnesses are used) is observationally equivalent to standard dependent type theory (MLTT) and, with higher structure, to HoTT. However, OHTT admits an enriched hierarchy:

• MLTT: Only coherence; inability to derive is absence.
• HoTT: Coherence with higher structure; all horns fill (Kan).
• OHTT: Coherence, gap, and open forms at all levels; Kan filling is not assumed.

Crucially, HoTT emerges as the subtheory of OHTT in which all gap witnesses are absent and every horn is coherently filled (the full Kan condition). In contrast, OHTT generalizes this by permitting both inhabited coherence and gap witnesses and allows open horns, thus admitting a broader array of mathematical phenomena—specifically, structured obstructions and transport failures.

Semantic and Topological Models: Ruptured Kan Complexes

OHTT’s semantics are constructed via ruptured simplicial sets and ruptured Kan complexes, which generalize standard (Kan) simplicial sets by distinguishing, at every horn, between coherently filled, gap-witnessed, and open horns. This provides a new semantic perspective where the geometry of type theory internalizes both local and global obstruction data:

• In topology, this captures monodromy and holonomy phenomena, where a path in the base fails to lift to a path in the fiber not through mere absence, but with a positive obstruction—e.g., the deck transformation in the Möbius strip example.
• In semantics, particularly linguistic or computational semantics, the framework models polysemy and metaphor: a term and a path of lexical identity may exist, but the transported meaning may be structurally gapped, precisely capturing nuanced failures of compositional semantic transfer.
• In logic, especially resource-sensitive or substructural logics, the failure of derivability under resource-violating substitutions becomes a gap with a witness: a formal certificate of underivability, not mere non-derivation.

The semantics extend to mapping spaces, higher structure, and model-theoretic interpretations in categories of ruptured ${\Gamma}{J}$6-groupoids.

Strong Results, Expressivity, and Obstruction Theory

Summary of Claims and Results

• Structured Obstructions: OHTT can formalize obstruction invariants (characteristic classes, cohomological obstructions, non-fillability data) as first-class gap witnesses, not external to the type-theoretic syntax.
• Failure of Functoriality: Functorial transport may strictly fail in OHTT, even when stepwise transport is witnessed as coherent—semantic drift and cumulative obstruction are rendered explicit.
• Ruptured Higher Inductive Types: OHTT allows construction of types with both path and gap constructors (ruptured HITs), extending expressivity beyond that of conventional HoTT.
• Decidability and Open Judgments: In decidable theories, OHTT partitions all judgments into coherent or gapped; in undecidable settings, the open class is inevitable and formally meaningful.

These features are unattainable in MLTT/HoTT or standard logics, due to lack of proof-relevant failure structure.

Contrasts with Negation and Non-Classical Logics

OHTT’s gap operator is neither classical nor intuitionistic negation, nor does it have the explosion properties of paraconsistent logics. Gap is a primitive, proof-relevant, positive structure. The logic thus avoids both excluded middle and the paradoxical properties seen in bilattice or paraconsistent logics, instead occupying a unique position emphasizing constructive witnessing—both for the existence and for the structured non-existence of constructions.

Theoretical and Practical Implications

The ability to formally structure, witness, and reason about failure is a substantial extension to both the theoretical analysis and practical application of type systems, semantics, and higher-categorical models:

• In mathematics, OHTT brings the internal perspective of homotopy type theory to obstruction theory, cohomological invariants, and characteristic classes, potentially streamlining reasoning about existence and non-existence of mathematical structures.
• In programming language theory, it enables encoding of resource failures, failed transports, and constraint violations as structured, checkable artifacts, with implications for certified compilation, proof-carrying code, and effect systems.
• In formal semantics and computational linguistics, OHTT opens avenues for precise modeling and computation over semantic gaps, ambiguities, and systematic partial transport, as appear in polysemy.
• In higher category theory, it introduces a trichotomous path-filling landscape, offering new categorical and semantic invariants.

Perspectives and Directions for Future Research

OHTT raises challenging foundational, semantic, and practical questions. Key future developments include:

• Computational interpretation: Understanding how gap witnesses interact with computation, effects, exceptions, and certificate-carrying processes.
• Model structures and higher categorical semantics: Formalizing the appropriate ambient model category or ${\Gamma}{J}$7-category for OHTT, including notions of weak equivalence, fibrancy, and universes closed under gap structure.
• Empirical applications: Implementing OHTT-style obstructions in natural language processing, using topological data analysis to compute semantic gaps.
• Proof assistant implementation: Designing systems that can explicitly construct and check both coherence and gap witnesses, extending existing proof assistants.

Conclusion

Open Horn Type Theory marks a profound conceptual and technical extension of proof-relevant constructive logic and type theory. By treating both coherence and rupture as primitive, proof-relevant phenomena, OHTT enables new forms of structural and compositional analysis hitherto unavailable in formal systems. The introduction of gap witnesses and the systematic trichotomy of coherent/gapped/open fundamentally reframe the logical analysis of existence, non-existence, and potentiality, with wide-ranging implications across mathematics, computer science, and formal semantics. The further development and application of OHTT, including computational content, categorical models, and proof-theoretic systems, remains a significant and promising direction for foundational research.