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opencog-evidence commits / d620956

Expand the general musings
d620956b2b593d468d97ec3a50f65e2b22377dc7
by Linas Vepštas <linasvepstas@gmail.com> · 2025-04-30T05:18:57.000Z

Changed files (2) +391 -3

README.md +7 -0
……@@ -233,6 +233,13 @@ Detailed Design
233233---------------
234234See the [design notes](Design.md) for further rambling design thoughts.
235235
236See also some texts written in Lyx (LaTeX), viewable as PDF's:
237* [Aggregate Satisfaction](aggregate.pdf) attempts to formulate a
238 definition of satsifiability that is "effective": that satisfies as
239 many constraints as is reasonable, while also resolving contradictions
240 as much as reasonable, and providing an effective strategy for
241 arriving at "good" values of aggregate satisfiability.
242
236243
237244Status
238245------
aggregate.lyx +384 -3
……@@ -96,7 +96,7 @@ Aggregate Satisfaction
9696\end_layout
9797
9898\begin_layout Date
9917 April 2025
99Version of 29 April 2025 (Started on 17 April 2025)
100100\end_layout
101101
102102\begin_layout Author
……@@ -111,6 +111,13 @@ Diary entry, attempting to work out some ideas.
111111 Maybe going in circles.
112112\end_layout
113113
114\begin_layout Abstract
115The goal was to try give a precise conception of how to take a collection
116 of statements or assertions, and to satisfy as many of them as possible,
117 all at the same time, while keeping contradictions under control.
118
119\end_layout
120
114121\begin_layout Section*
115122Introduction
116123\end_layout
……@@ -119,7 +126,7 @@ Introduction
119126What follows will be free–form unstructured writing about aggregate satisfiabili
120127ty.
121128 It is an attempt to reconcile a number of quite common and inter–related
122 ideas, and to develop a mathematical frammework for them.
129 ideas, and to develop a mathematical framework for them.
123130 The key ingredients are:
124131\end_layout
125132
……@@ -181,7 +188,8 @@ Ensembles.
181188\end_layout
182189
183190\begin_layout Standard
184So here we go.
191This text is incomplete and imprecise.
192 So here we go.
185193\end_layout
186194
187195\begin_layout Section*
……@@ -208,6 +216,379 @@ with
208216 Read that article if you don't know what this is.
209217\end_layout
210218
219\begin_layout Subsection*
220Motivation
221\end_layout
222
223\begin_layout Standard
224Why start this essay with the Ising Model? For several reasons:
225\end_layout
226
227\begin_layout Itemize
228Imagine one has a collection of
229\begin_inset Quotes eld
230\end_inset
231
232facts
233\begin_inset Quotes erd
234\end_inset
235
236,
237\begin_inset Quotes eld
238\end_inset
239
240assertions
241\begin_inset Quotes erd
242\end_inset
243
244, or
245\begin_inset Quotes eld
246\end_inset
247
248constraints
249\begin_inset Quotes erd
250\end_inset
251
252, each of which can be True or False.
253 Map
254\begin_inset Quotes eld
255\end_inset
256
257True
258\begin_inset Quotes erd
259\end_inset
260
261 to
262\begin_inset Quotes eld
263\end_inset
264
265spin up
266\begin_inset Quotes erd
267\end_inset
268
269 and
270\begin_inset Quotes eld
271\end_inset
272
273false
274\begin_inset Quotes erd
275\end_inset
276
277 to
278\begin_inset Quotes eld
279\end_inset
280
281spin down
282\begin_inset Quotes erd
283\end_inset
284
285.
286 The if there is a relationship between two assertions
287\begin_inset Formula $\sigma_{i}$
288\end_inset
289
290 and
291\begin_inset Formula $\sigma_{j}$
292\end_inset
293
294 then set
295\begin_inset Formula $J_{ij}\ne0$
296\end_inset
297
298 and set it positive, if both have to be true at the same time, else set
299 it negative.
300\end_layout
301
302\begin_layout Itemize
303For the conventional Ising model on a 2D grid,
304\begin_inset Formula $J_{ij}\ne0$
305\end_inset
306
307 only when
308\begin_inset Formula $\sigma_{i}$
309\end_inset
310
311 and
312\begin_inset Formula $\sigma_{j}$
313\end_inset
314
315 are nearest neighbors, in which case
316\begin_inset Formula $J_{ij}=J$
317\end_inset
318
319.
320 For zero temperature and zero magnetic field, the Hamiltonian is obviously
321 minimized by having all spins aligned.
322 The model is interesting because of it's thermodynamic behavior, at finite
323 temperature, in which thermal energy is allows some of the spins to be
324 non–aligned.
325\end_layout
326
327\begin_layout Itemize
328The attempt being made here is to draw an analogy to systems of logic, such
329 as the Boolean SAT problem.
330 By convention, Boolean SAT is solved with DPLL.
331 The algo there recasts
332\begin_inset Formula $J_{ij}$
333\end_inset
334
335 as a graph adjacency matrix.
336 The corresponding graph is then some graph with cycles, and a bunch of
337 trees attached to it.
338 The trees are trimmed away, because they can be uniquely determined by
339 the root.
340 This leaves an interconnected kernel.
341 Then, conventionally, one performs a search, possibly an exhaustive search
342 over that kernel, and determine if it is satisfiable.
343 Various techniques can be attempted to speed that search.
344\end_layout
345
346\begin_layout Itemize
347For example, there may be a circular chain of connections,
348\begin_inset Formula $\sigma_{a}\to\sigma_{b}\to\cdots\to\sigma_{p}\to\lnot\sigma_{a}$
349\end_inset
350
351 which cannot be satisfied because it forces a self–contradiction.
352 So strict zero–temperature Boolean SAT indicates this problem is unsatisfiable.
353 However, by using simulated annealing, one might be able to find find truth
354 assignments that satisfy most of the constraints.
355\end_layout
356
357\begin_layout Itemize
358The problem of satisfiability reared its head in Dough Lenat's CYC; the
359 solution there was to propose
360\begin_inset Quotes eld
361\end_inset
362
363microtheories
364\begin_inset Quotes erd
365\end_inset
366
367 which were known to be self–consistent, and then hop between these.
368 The suggestion being made here is that such inconsistencies can be thought
369 of as
370\begin_inset Quotes eld
371\end_inset
372
373lattice frustrations
374\begin_inset Quotes erd
375\end_inset
376
377, and mostly resolved through annealing.
378\end_layout
379
380\begin_layout Itemize
381In the human world of common–sense judgments, it is clear that most humans
382 insist that they are
383\begin_inset Quotes eld
384\end_inset
385
386rational
387\begin_inset Quotes erd
388\end_inset
389
390 and
391\begin_inset Quotes eld
392\end_inset
393
394logical
395\begin_inset Quotes erd
396\end_inset
397
398, even though it is clear that this cannot possibly be true, if one tries
399 to drill down and interrogate the foundations of beliefs.
400 Yet, except for extreme cases involving various forms of mental illness,
401 most humans mostly do have a collection of beliefs that they perceive to
402 be consistent.
403 Those beliefs appear to be malleable via persuasion, propaganda, cult indoctrin
404ation, etc.
405 The proposal here is that perhaps a Hamiltonian approach to a description
406 of belief networks can give some sense of the grounded–ness of those beliefs.
407\end_layout
408
409\begin_layout Itemize
410The relational weights
411\begin_inset Formula $J_{ij}$
412\end_inset
413
414 can be made dynamic.
415 Thus, for example, inconsistent beliefs about what time of day one left
416 the house might be utterly inconsequential, unless this plays a role in
417 a criminal trial, in which case the truth/falsehood coupling
418\begin_inset Formula $J_{ab}$
419\end_inset
420
421 between a statement
422\begin_inset Formula $\sigma_{a}$
423\end_inset
424
425 of when one left the house, and the statement of
426\begin_inset Formula $\sigma_{b}$
427\end_inset
428
429 of the defendant becomes very important.
430 A large enough value of
431\begin_inset Formula $J_{ab}$
432\end_inset
433
434 can make the prosecutions case fall apart or to firmly establish guilt.
435 A much softer variant of this can occur during attempts to solve quotidian
436 engineering problems, where one wishes to make trade–offs between different
437 design parameters or criteria: the assigned weights
438\begin_inset Formula $J_{ab}$
439\end_inset
440
441 can be raised or lowered to taste.
442\end_layout
443
444\begin_layout Standard
445The above is a motivational sketch of why the Ising Model can be interesting
446 and useful in the pursuit of
447\begin_inset Quotes eld
448\end_inset
449
450common sense
451\begin_inset Quotes erd
452\end_inset
453
454 inference or during evidence gathering.
455 It provides a formalism in which
456\begin_inset Quotes eld
457\end_inset
458
459aggregate satisfiability
460\begin_inset Quotes erd
461\end_inset
462
463 can be discussed.
464\end_layout
465
466\begin_layout Standard
467There are several problems with the direct use of the Ising Model:
468\end_layout
469
470\begin_layout Itemize
471The matrix entries
472\begin_inset Formula $J_{ij}$
473\end_inset
474
475 are (by convention) symmetric:
476\begin_inset Formula $J_{ij}=J_{ji}$
477\end_inset
478
479 whereas conventional logic is causitive or inferential; one writes
480\begin_inset Formula $P\to Q$
481\end_inset
482
483 to say
484\begin_inset Quotes eld
485\end_inset
486
487
488\begin_inset Formula $P$
489\end_inset
490
491 implies
492\begin_inset Formula $Q$
493\end_inset
494
495
496\begin_inset Quotes erd
497\end_inset
498
499 or
500\begin_inset Quotes eld
501\end_inset
502
503
504\begin_inset Formula $P$
505\end_inset
506
507 causes
508\begin_inset Formula $Q$
509\end_inset
510
511 to happen
512\begin_inset Quotes erd
513\end_inset
514
515 or
516\begin_inset Quotes eld
517\end_inset
518
519
520\begin_inset Formula $P$
521\end_inset
522
523 believes that
524\begin_inset Formula $Q$
525\end_inset
526
527 is true
528\begin_inset Quotes erd
529\end_inset
530
531.
532\end_layout
533
534\begin_layout Itemize
535Logical statements typically have multiple antecedents:
536\begin_inset Formula $A,B\to C$
537\end_inset
538
539.
540\end_layout
541
542\begin_layout Itemize
543One may wish to strongly couple inferences: that is, to write
544\begin_inset Formula $A\to B,C$
545\end_inset
546
547.
548 Now, in conventional logic, this can be decomposed into two statements:
549
550\begin_inset Formula $\left(A\to B,C\right)\implies\left(A\to B\right)\wedge\left(A\to C\right) $
551\end_inset
552
553 which is fine for crisp logic, but a problem if one is looking for approximate
554 satisfiability.
555 That is, one may want to strongly avoid the situation of
556\begin_inset Formula $\left(A\to B\right)\wedge\lnot\left(A\to C\right)$
557\end_inset
558
559, choosing instead to negate both
560\begin_inset Formula $B,C$
561\end_inset
562
563 or neither.
564\end_layout
565
566\begin_layout Standard
567A lesser issue, but one that we should get out of the way, is that of hierarchic
568al satisfiability.
569 That is, one may have strongly interacting clusters, with only weak connections
570 between the clusters.
571 That is, the
572\begin_inset Formula $J_{ij}$
573\end_inset
574
575 are large, when
576\begin_inset Formula $i,j$
577\end_inset
578
579 both belong to the same cluster, but are small when
580\begin_inset Formula $i,j$
581\end_inset
582
583 are in different clusters.
584 The Ising Model doesn't prevent this, but perhaps there's some better way
585 of describing natural clustering.
586\end_layout
587
588\begin_layout Standard
589Jigsaws
590\end_layout
591
211592\begin_layout Section*
212593Link Grammar
213594\end_layout