Symbolic Event Tensors

• Symbolic event tensors are multi-dimensional algebraic structures that encode event-argument bindings using tensor operations for precise, lossless compositional semantics.
• They integrate tensor-based composition in vector symbolic architectures and enable dynamic dependency encoding for parallel computation and scheduling in compilers.
• Empirical evaluations show that tensor-based models outperform additive baselines by offering exact unbinding and improved accuracy in tasks such as schema induction and semantic similarity.

Symbolic event tensors are multi-dimensional algebraic structures that encode events or event-argument bindings using tensor operations. They serve distinct technical roles in symbolic representation learning, high-dimensional computing, and advanced scheduling abstractions for dynamic parallel computation. Their principal mechanisms span tensor-based composition for semantic event representations, universal binding in vector symbolic architectures, and dynamic dependency encoding in machine learning compilers.

1. Symbolic Event Representations and Tensor-Based Composition

Events in symbolic systems are formally represented as tuples, typically of the form ⟨predicate, Agent, Patient⟩—for example, ⟨threw, she, football⟩, where "threw" is the predicate, "she" occupies the Agent (subject) role, and "football" the Patient (object). In distributed or vector-space semantics, each slot (predicate, subject, object) is associated with a $d$-dimensional embedding ($p$, $s$, $o \in \mathbb{R}^d$ respectively) (Weber et al., 2017).

Event tensors enable multiplicative composition of these embeddings, capturing higher-order interactions beyond additive models. For a general event embedding $e \in \mathbb{R}^{d'}$, a predicate-specific third-order tensor $P \in \mathbb{R}^{d' \times d \times d}$ defines:

$$e_i = \sum_{j=1}^d \sum_{k=1}^d P_{ijk} \cdot s_j \cdot o_k,$$

facilitating nuanced distinctions based on argument variations. This approach is further generalized: predicate-specific parameters $P$ can be generated on demand from the predicate embedding $p$ via shared base tensors $W, U \in \mathbb{R}^{d \times d \times d}$, as in:

$p$0

The Role-Factored Tensor Model further factorizes role interactions via a shared $p$1 and role-specific matrices, computing intermediate vectors for each role-filler pair, then combining them:

$p$2

with

$p$3

Parameter learning proceeds with event-prediction (margin ranking) and word-prediction (cross-entropy) objectives (Weber et al., 2017). Empirically, these tensor models outperform additive or shallow neural baselines on semantic similarity, cloze, and schema induction tasks, particularly when distinguishing semantically distinct events with similar lexical content.

2. Tensor Products in Vector Symbolic Architectures

Symbolic event tensors form the central class of representations in vector symbolic architectures (VSA) and hyperdimensional computing (HDC), where the tensor product realizes the most expressive and compact representation with exact unbinding (Qiu, 2023).

Given a real inner-product space $p$4 of dimension $p$5 with orthonormal basis $p$6, roles ($p$7) and fillers ($p$8) are embedded as vectors in $p$9. Events are bound as tensor products: $s$0. Superposition—crucial for representing sets—is vector addition in $s$1.

Binding and Unbinding:

• Binding: $s$2.
• Superposition of bindings: $s$3 for a set of role-filler pairs.
• Unbinding: Left-unbinding retrieves the filler: $s$4 if $s$5 is orthonormal to other roles.

Orthogonality Guarantees: If $s$6 and $s$7 are orthonormal, then $s$8 forms an orthonormal basis of $s$9, ensuring zero-error unbinding (Qiu, 2023).

Compressed Representations: The high dimension $o \in \mathbb{R}^d$0 of $o \in \mathbb{R}^d$1 can be reduced by random projection (Johnson–Lindenstrauss), maintaining inner product approximations and therefore approximate unbinding for practical system sizes.

3. Symbolic Event Tensors in Compilers: Dependency Encoding for Megakernel Scheduling

The "Event Tensor" abstraction in modern GPU compiler infrastructure encodes fine-grained dependencies between tiled tasks for both static and data-dependent scheduling (Jin et al., 14 Apr 2026).

Formal Definition:

Given a shape $o \in \mathbb{R}^d$2, an Event Tensor is

$o \in \mathbb{R}^d$3

where each $o \in \mathbb{R}^d$4 is a wait-counter for event index $o \in \mathbb{R}^d$5. Producer-to-event and event-to-consumer maps link tasks and the corresponding synchronization events.

• notify: Atomic decrement: $o \in \mathbb{R}^d$6.
• wait: Spin-wait until counter reaches zero: while (E[e] > 0) {}.

The abstraction supports symbolic tensor shapes (dimensions known only at runtime) and data-dependent index expressions, allowing one Event Tensor to encode highly dynamic computation graphs (e.g., mixture-of-experts or dynamic routing in transformers). Task execution is guarded by these wait-counters, enabling precise, scalable dependency enforcement on hardware.

4. Applications: Language Semantics, Compilation, and Cognitive Modeling

Event Semantics and Schema Learning

Symbolic event tensors are leveraged in script learning, scenario modeling, and narrative schema induction. Continuous event representations allow schema generation via nearest-neighbor search in embedding space, outperforming discrete co-occurrence approaches in human-rated relevance and coherence metrics (mean relevance: 2.45 for role-factored tensor schemas vs. 1.51 for count-based methods) (Weber et al., 2017).

Dynamic Scheduling in Parallel Systems

In compiler contexts, Event Tensors unify static and dynamic scheduling strategies within the same intermediate representation. The abstraction abstracts away millions of individual semaphores into a tensor indexable by (possibly symbolic) task coordinates (Jin et al., 14 Apr 2026):

• Static scheduling: Per-SM queues are computed ahead-of-time; dependencies tracked by Event Tensor indices.
• Dynamic scheduling: On-GPU push/pop queues plus counter-based triggers from Event Tensors enable on-the-fly task dispatch for unpredictable workloads.

Performance evaluations show up to 1.5× speedup on LLM serving, with substantial reductions in system warmup time (e.g., 583s for SGLang (JIT+CUDA Graph) to 35s for ETC AOT compilation).

Cognitive and Symbolic Computing

In the theory of symbolic reasoning, vector symbolic architectures use event tensors to implement noise-free, compositional memory structures. Compared to lower-dimensional alternatives (e.g., circular convolution), the full tensor product uniquely guarantees errorless unbinding for orthogonal symbols; no lower-dimensional bilinear scheme provides this property (Qiu, 2023).

5. Comparative Properties and Evaluation

Expressivity and Recovery

Scheme	Dimension	Supports Errorless Unbinding	Expressivity
Tensor Product ($o \in \mathbb{R}^d$7)	$o \in \mathbb{R}^d$8	Yes (orthonormal)	Universal (most general)
Circular Convolution	$o \in \mathbb{R}^d$9	No (approximate only)	Less (linear quotient)
Hadamard Product	$e \in \mathbb{R}^{d'}$0	No (approximate only)	Lower

Superiority of tensor-based schemes is empirically observed:

• Similarity tasks: Tensor models improve Spearman’s $e \in \mathbb{R}^{d'}$1 by 3–5 points.
• Hard similarity: 10–15% higher accuracy than additive baselines.
• Cloze tasks: Role-factored tensor models achieve 72.1% vs. 68.5% for neural net baselines (Weber et al., 2017).
• Schema induction: Substantially more coherent event scripts from continuous, tensor-based embeddings.

6. Implementation and Runtime Considerations

Symbolic event tensor logic is realized with ordinary integer buffers and atomic arithmetic in system memory, requiring no specialized runtime support beyond hardware atomics and, optionally, lock-free queues (Jin et al., 14 Apr 2026). At the semantic level, training over event compositions is done via standard optimizers (e.g., AdaGrad over GloVe-initialized embeddings) (Weber et al., 2017).

In highly dynamic execution environments (e.g., LLM serving with symbolic and data-dependent batch dimensions), Event Tensors manage all dependency logic and kernel scheduling, enabling both static and dynamic schemes to co-exist and to outperform traditional approaches in both throughput and latency. The abstraction lowers to standard primitives—atomic counters and busy-wait spin-loops—delivering fine-grained pipelining with minimal overhead.

7. Principal Limitations and Theoretical Guarantees

Tensor product-based event representations uniquely support exact unbinding provided role and filler vectors are orthonormal; this property is mathematically necessary and sufficient (Qiu, 2023). While compressed representations (e.g., random projections) retain approximate retrieval with bounded distortion, alternatives such as convolutional or Hadamard products do not scale to large numbers of bindings without retrieval noise, making full tensor products essential for zero-error, scalable symbolic representations.

Trade-offs for scheduling include minor dynamic-queue overhead in dense static workloads and modest gains from dynamic load balancing in highly data-dependent workloads (Jin et al., 14 Apr 2026). The underlying theory guarantees that symbolic event tensors are both maximally expressive and minimally redundant within the defined algebraic constraints.

---

Symbolic event tensors synthesize higher-order compositional semantics, lossless symbolic binding, and scalable parallel dependency management in a unified mathematical and system-theoretic framework, with rigorous empirical and theoretical validation across distributed semantics, cognitive architectures, and dynamic high-performance scheduling (Weber et al., 2017, Qiu, 2023, Jin et al., 14 Apr 2026).