NonAxiomatic Logic (NAL)

NonAxiomatic Logic (NAL) is a formal framework for reasoning under uncertainty, designed for intelligent systems that must operate with limited time, knowledge, and computational resources. Developed by Dr. Pei Wang and his research group, NAL departs from classical logics by refusing to assume that all relevant premises are available or that inference can be performed exhaustively. Instead, it embraces a budgeted approach in which each inference consumes a portion of a systems resources, and the most useful conclusions are retained.

Core Principles

• Insufficient Knowledge and Resources (IKR): The world is too complex to be fully known, and agents have bounded processing power. NAL explicitly models these limits.
• Opacity and Openness: Knowledge is represented as a set of statements each equipped with a budget (priority, durability, and confidence). Inferences are made only when the budget permits.
• ExperienceBased Learning: Truth values are updated continuously from the evidence the system encounters, rather than being fixed axioms.

Syntax and Semantics

In NAL a statement (called a sentence) has three parts:

1. Term: The subject and predicate, which can be simple words or compound structures (e.g., <bird → animal>).
2. Relation: The logical connective, usually implication (→) or equivalence (↔), but also inheritance, similarity, and more.
3. Truthvalue: A pair (f, c) where f (frequency) is the estimated proportion of positive evidence, and c (confidence) reflects how much evidence supports f.

Example:

<bird → animal> : (0.9, 0.8)
Most birds are animals, and we are 80% confident about that estimate.

Budget Management

Every sentence carries a budget triple (p, d, c):

• Priority (p): How important the sentence is for future inference.
• Durability (d): How long the sentence should persist in memory.
• Confidence (c): Same as the confidence component of the truthvalue, influencing budget updates.

The inference engine withdraws budget from parent sentences and distributes it to derived conclusions. Sentences whose budget drops below a threshold are eventually forgotten, ensuring the system stays within its resource limits.

Inference Rules

NAL defines a small set of generic inference rules that can be applied to any compatible pair of sentences. The most common ones are:

• Deduction: From A → B and B → C infer A → C.
• Induction: From A → B and C → B infer (A | C) → B, where | denotes a union of evidence.
• Abduction: From A → B and A infer B (hypothesizing a cause).
• Revision: When two sentences have the same term and relation, their truthvalues are combined to improve confidence.
• Choice: When multiple conclusions compete for the same budget, the one with higher priority wins.

Each rule includes a precise formula for computing the resulting truthvalue and budget from the premises. Because the formulas are based on probabilitylike operations, they naturally handle uncertainty.

Temporal and Evidential Extensions

Later versions of NAL (NAL5 onward) introduce explicit temporal operators (at, before, after) and evidential markers that allow a system to reason about events over time and to distinguish between different sources of evidence. This makes NAL suitable for dynamic environments such as robot navigation or realtime data streams.

Comparison with Other Logics

Aspect	Classical Logic	Probabilistic Logic	NonAxiomatic Logic
Assumptions	All premises known, inference unlimited	Known probability distributions	Knowledge and resources are bounded
Truth Representation	True/False	Probability values	Frequency + Confidence (twodimensional)
Handling New Evidence	Requires rederivation	Bayesian update	Revision rule merges evidence incrementally
Resource Management	None (theoretically infinite)	Typically ignored	Explicit budget for each sentence

Practical Applications

Because NAL is designed for openended, realworld environments, it has been used in a variety of projects:

• Artificial General Intelligence (AGI) prototypes: The OpenNARS system implements NAL and demonstrates continual learning, selfmodification, and concept formation.
• Robotics: Robots equipped with NAL can integrate sensor data on the fly, make tentative plans, and discard outdated knowledge without crashing.
• Natural Language Understanding: NALs flexible term structures support parsing of ambiguous sentences and incremental meaning revision.
• Datastream mining: Financial or IoT streams can be processed with limited memory while retaining the most relevant patterns.

Getting Started with NAL

If you are interested in experimenting with NAL, the easiest entry point is the OpenNARS repository. It provides:

• A compact Java implementation of the inference engine.
• Commandline tools for loading and querying knowledge bases.
• Sample scripts illustrating deduction, induction, and revision.

Typical workflow:

1. Write a text file with sentences in NAL syntax (e.g., <bird → animal> %0.9;0.8%).
2. Start the NARS interpreter, feed the file, and let the system run for a few cycles.
3. Issue queries such as ?<bird → flying> and observe the returned truthvalue.
4. Add new evidence dynamically and watch how the confidence adapts.

Limitations and Open Questions

While NAL offers a novel perspective on reasoning under scarcity, several challenges remain:

• Scalability: Even with budgeting, very large knowledge bases can strain memory; hybrid approaches that combine NAL with symbolic indexing are under investigation.
• Explainability: The budgetdriven inference path can be hard to trace for humans; research is exploring visualizations of budget flow.
• Integration with Deep Learning: Bridging NALs symbolic handling of uncertainty with neural representations is an active area of AGI research.

Further Reading

• Pei Wang, Non-Axiomatic Reasoning System, 2020 a comprehensive textbook on NAL theory and practice.
• OpenNARS documentation practical guides and API references.
• Budgeted Inference for ResourceBounded Agents, Journal of Autonomous Agents, 2022 a peerreviewed article on the mathematical foundations of budgeting.

NonAxiomatic Logic challenges the traditional view that logical reasoning must be exhaustive and perfectly certain. By embracing uncertainty and resource limits, NAL provides a flexible engine for systems that must learn, adapt, and act in the messy real world.