sec-symbolic-neural-lincs

0.13 Symbolic and neural reasoning

The relation between symbolic and neural reasoning has been debated since the emergence of connectionist AI. Symbolic accounts emphasize explicit representations, compositional rules, and search over expressions [ Newell and Simon , 1976 ] ; connectionist accounts emphasize distributed representations and learned transformations [ Smolensky , 1988 ] . Neural–symbolic systems often respond by placing a rule layer beside, before, or after a neural model [ d’Avila Garcez et al. , 2002 ] . LINCS suggests a different division of labor. It does not treat symbolic and neural reasoning as rival substances, nor does it identify their reconciliation with a particular two-module architecture.

Three roles must be distinguished.

  1. A structural declaration specifies typed objects, generating arrows, admissible diagrams, equations, covers, and repair operations. This is the role most closely associated with symbolic reasoning.

  2. A computational realization assigns actual representations and maps to that declaration. Neural networks, embeddings, vector fields, and learned stochastic maps can all serve as realizations.

  3. An internal reasoning discipline determines which claims, witnesses, restrictions, and repairs are warranted inside the chosen category. In a smooth or sheaf semantics this discipline may be intuitionistic and context-dependent, even when the external implementation uses ordinary classical arithmetic.

The interface between these roles is a typed obstruction. If \(\mathbb S\) is the declared sketch and \(D_\theta :J\to \mathcal C\) is a neural realization, then

\[ (\mathbb S,D_\theta ) \longmapsto \mathcal O_{\mathbb S}(D_\theta ) \longmapsto \text{localized typed repair}. \]

The obstruction is neither an ordinary symbolic contradiction nor merely a scalar neural loss. It records how the learned realization fails relative to a particular structural promise. An observer may subsequently turn part of that object into a loss or test statistic, but the scalarization does not define the failure.

11. Category theory supplies the grammar of declaration and realization; it does not make a neural representation symbolic by fiat. The typing map, observers, and admission criteria must all be supplied and validated. ↩

This account also makes the symbolic side revisable. A failure can sometimes be repaired by changing parameters inside a fixed sketch. At other times the available declaration is itself inadequate: a new object, arrow, axiom, cover, or observer must be proposed and old evidence transported into the enlarged theory. LINCS therefore distinguishes repair within a symbolic language from repair of the language. The latter is a controlled form of theory change, not the execution of a permanently fixed rule base.

There is consequently no single universal “LINCS logic.” The internal logic depends on the semantic category selected by an application. Deep LINCS uses parametrized differentiable maps to realize declared diagrams; infinitesimal causality uses tangent directions and intervention protocols to reason about local causal variation; CoLT uses constructive witnesses, restrictions, and admission certificates to govern reasoning traces. Their commonality is the declaration–realization–obstruction–repair pattern, not one shared stock of symbols or one neural architecture.

Design principle

LINCS treats symbolic structure as a revisable categorical declaration, neural computation as one family of its realizations, and learning as the diagnosis and repair of failures between the two.

If \(D:J\to \mathcal C\) is a realized diagram, its tangent lift is

\[ TD:J\longrightarrow \mathcal C. \]

The base diagram asks whether declared routes agree. The tangent diagram asks whether their transported first-order behavior agrees along a specified class of probes.

Example 0.20 Adapter order

For two smooth adapter actions \(A\) and \(B\), the base comparison is between \(B\circ A\) and \(A\circ B\). Their tangent lifts compare

\[ T(B\circ A)=TB\circ TA, \qquad T(A\circ B)=TA\circ TB. \]

A base discrepancy measures finite order sensitivity. A tangent discrepancy asks how that sensitivity changes locally around the current representation and parameters.

Vector fields introduce another infinitesimal operation. The Lie bracket \([X,Y]\) is the infinitesimal commutator: it records the leading mixed-order discrepancy between the short flows generated in the two orders \(X\) then \(Y\), and \(Y\) then \(X\). It supports different semantics in different applications:

  • BRIDGE/SKFM uses bracket closure as a falsifiable screen for a candidate intervention distribution;

  • infinitesimal causality studies when intervention fields and their interactions can be given causal meaning;

  • ALLORA uses order interactions among adapter-induced directions; and

  • LASKO uses Lie-algebroid structure to test closure and executability of skill compositions.

22. Automatic differentiation supplies derivatives. It does not supply their semantics. A LINCS tangent site declares which points, directions, transports, covers, and null variations are admissible. ↩