Symbolic and Neural Nets: Two Sides of the Same Coin

Deep learning and neural nets are all the rage, today, and have displaced symbolic AI systems in most applications. It’s commonly believed that the two approaches have nothing to do with each other; that they’re just completely different, and that’s that.  But this is false: there are some profound similarities; they are not only variants of one-another, but, in a deep way, they have commonalities that render them essentially identical. This blog post attempts to explain how. It introduces a much longer and rather mathematically dense writeup articulating these ideas more precisely.

The clearest starting point for seeing this seems to be natural language, where neural net methods have made great strides, but have not surpassed traditional (symbolic) linguistic theory. However, once this similarity is understood, it can be ported over to other domains, including deep-learning strongholds such as vision. To keep the discussion anchored, and to avoid confusing abstractions, what follows will focus entirely on linguistics; it is up to you, the reader, to imagine other, more general settings.

The starting point for probabilistic approaches (including deep learning) is the Bayesian network: a probability P(x1,x2,…,xn) of observing n events.  For language, the xk are taken to be words, and n is the length of the sentence, so that P(x1,x2,…,xn) is the “probability” of observing the sequence x1,x2,…,xn of words. The technical problem with this viewpoint is the explosively large space: if one limits oneself to a vocabulary of 10 thousand words (and many people don’t) and sentences of 20 words or less, that’s (104)20=1080 =2270 probabilities, an absurdly large number even for Jupiter-sized computers. If n is the length of this blog post, then it really seems impossible. The key, of course, is to realize that almost all of these probabilities are effectively zero; the goal of machine learning is to find a format, a representation for grammar (and meaning) that effortlessly avoids that vast ocean of zero entries.

Traditional linguistics has done exactly that: when one has a theory of syntax, one has a formulation that clearly states which sentences should be considered to be grammatically valid, and which ones not.  The trick is to provide a lexis (a lookup table), and some fairly small number of rules that define how words can be combined; i.e. arranged to the left and right of one-another.  You look up a word in the lexis (the dictionary) to find a listing of what other words are allowed to surround it.  Try every possible combination of rules until you find one that works, where all of the words can hook up to one-another. For the purposes here, the easiest and the best way to visualize this is with Link Grammar, a specific kind of dependency grammar. All theories of grammar will constrain allowed syntax; but Link Grammar is useful for this blog post because it explicitly identifies words to the left, and words to the right.  Each lexical entry is like a template, a fill-in-the-blanks form, telling you exactly what other words are allowed to the left, and to the right, of the given word.  This left-right sequence makes it directly comparable to what neural-net approaches, such as Word2Vec or SkipGram do.

What does Word2Vec do? Clearly, the 2270 probabilities in a twenty-word sentence is overwhelming; one obvious simplification is to look only at N-grams: that is, to only look at the closest neighboring words, working in a window that is N words wide. For N=5, this gives (104)5=1020 =265 which is still huge, but is bearable. When scanning actual text, almost all of these combinations won’t be observed; this  is just an upper bound. In practice, a table of 5-grams fit in latter-day computer RAM. The statistical model is to map each N-gram to a vector, use that vector to define a Boltzmann distribution (P=exp(v⋅w)/Z), and then use gradient ascent (hill-climbing) to adjust the vector coefficients, so as to maximize the probability P.

How are Word2Vec and Link Grammar similar? Well, I hope that the above description of Link Grammar already planted the idea that each lexical entry is a lot like an N-gram. Each lexical entry tells you which words can appear to the right, and which words to the left of a given word. Its a bit less constrained than an N-gram: there’s no particular distance limitation on the dependency links. It can also skip over words: a lexical entry is more like a skip-gram.  Although there is no distance limitation, lexical entries still have a small-N-like behavior, not in window size, but in attachment complexity. Determiners have a valency of 1; nouns a valency of 2 or 3 (a link to a verb, to an adjective, to a determiner); verbs a valency of 2, 3 or 4 (subject, object, etc.). So lexical entries are like skip-grams: the window size is effectively unbounded, but the size of the context remains small (N is small).

Are there other similarities? Yes, but first, a detour.

What happened to the 2270 probabilities? A symbolic theory of grammar, such as Link Grammar, is saying that nearly all of these are zero; the only ones that are not zero are the ones that obey the rules of the grammar. Consider the verb “throw” (“Kevin threw the ball”). A symbolic theory of grammar effectively sates that the only non-vanishing probabilities are those of the form P(x1,x2,…,xk=noun,…,xm=throw,…,xp=object,…,xn) and that all the others must be zero. Now, since nouns make up maybe 1/2 of all words (the subject and object are both nouns), this constraint eliminates 104x104/(2×2)= 224 possibilities (shrinking 2270 to 2270-24 = 2246). Nothing to sneeze at, given that its just one fairly simple rule.  But this is only one rule: there are others, which say things like “singular count nouns must be preceded by a determiner” (so, “the ball”). These constraints are multiplicative: if the determiner is missing, then the probability is exactly zero. There are only a handful of determiners, so another factor of 104=213 is vaporized. And so on.  A relatively small lexis quickly collapses the set of possibilities. Can we make a back of the envelope estimate? A noun-constraint eliminates half the words (leaving 5K of 10K possibilities). A determiner constraint removes all but 10 possibilities. Many grammatical classes have only a few hundred members in them (“throw” is like “hit”, but is not like “smile”). So, realistically, each xk will have from 10 to 1000 possibilities, maybe about 100 possibilities “on average” (averages are dicey to talk about, as these are Zipfian distributions). Thus there are only about 10020 =1040 =2130 grammatically valid sentences that are 20 words long, and these can be encoded fairly accurately with a few thousand lexical entries.

In essence, a symbolic theory of grammar, and more specifically, dependency grammars, accomplish the holy grail of Bayesian networks: factorizing the Bayesian network.  The lexical rules state that there is a node in the network, for example, P(x1,x2,…,xk=noun,…,xm=throw,…,xp=object,…,xn) and that the entire sentences is a product of such nodes: The probability of “Kevin threw the ball” is the product P(x1=Kevin, x2=verb, …, xn) P(x1,x2, …, xk=noun, …, xm=throw, …, xp=object, …, xn) P(x1,x2, …, xk=the, …, xm=noun, …, xn) P(x1,x2, …, xn=ball) Stitch them all together, you’ve got a sentence, and its probability. (In Link Grammar, -log P is called the “cost”, and costs are additive, for parse ordering.)  To be explicit: lexical entries are exactly the same thing as the factors of a factorized Bayesian network. What’s more, figuring out which of these factors come to play in analyzing a specific sentence is called “parsing”. One picks through the lookup table of possible network factors, and wires them up, so that there are no dangling endpoints. Lexical entries are look like subsets of a graph: a vertex, and some dangling edges hanging from the vertex. Pick out the right vertexes (one per word), wire them together so that there are no dangling unconnected edges, and viola! One has a graph: the graph is the Bayesian network. Linguists use a different name: they call it a dependency parse.

The Word2Vec/SkipGram model also factorizes, in much the same way! First, note that the above parse can be written as a product of factors of the form P(word|context), the product running over all of the words in the sentence. For a dependency grammar, the context expresses the dependency relations. The Word2Vec factorization is identical; the context is simpler. In it’s most naive form, its just a bag of the N nearest words, ignoring the word-order.  But the word-order is ignored for practical reasons, not theoretical ones: it reduces the size of the computational problem; it speeds convergence. Smaller values of N mean that long-distance dependencies are hard to discover; the skipgram model partly overcomes this by keeping the bag small, while expanding the size of the window.  If one uses the skip-gram model, with a large window size, and also keep track of the word-order, and restrict to low valencies, then one very nearly has a dependency grammar, in the style of Link Grammar. The only difference is that such a model does not force any explicit dependency constraints; rather, they are implicit, as the words must appear in the context.  Compared to a normal dependency grammar, this might allow some words to be accidentally double-linked, when they should not have been. Dependency grammar constraints are sharper than merely asking for the correct context. To summarize: the notions of context are different, but there’s a clear path from one to the other, with several interesting midway points.

The next obvious difference between Link Grammar and Word2Vec/SkipGram are the mechanisms for obtaining the probabilities. But this is naive: in fact, they are much more similar than it first appears.  In both systems, the starting point is (conceptually) a matrix, of dimension WxK, with W the size of the vocabulary, and  K the size of the context. In general, K is much much larger than W; for Word2Vec, K could get as large as WN for a window size of N, although, in practice, only a tiny fraction of that is observed.  Both Link Grammar and Word2Vec perform approximate matrix factorization on this matrix. The way the approximations are done is different. In the case of Word2Vec, one picks some much smaller dimension M, typically around M=200, or maybe twice as large; this is the number of “neurons” in the middle layer. Then, all of W is projected down to this M-dimensional space (with a linear projection matrix). Separately,  the K-dimensional context is also projected down. Given a word, let its projection be the (M-dimensional) vector u. Given a context, let it’s projection be the vector v.  The probability of a word-in-context is given by a Boltzmann distribution, as exp(u⋅v)/Z where u⋅v is the dot product and Z is a scale factor (called the “partition function“). The basis elements in this M-dimensional space have no specific meaning; the grand-total vector space is rotationally invariant (only the dot product matters,  and dot products are scalars).

The primary task for Word2Vec/SkipGram is to discover the two projection matrices.  This can be done by gradient ascent (hillclimbing), looking to maximize the probability.  The primary output of Word2Vec are the two projection matrices: one that is WxM-dimensional, the other that is MxK-dimensional.  In general, neither of these matrices are sparse (that is, most entries are non-zero).

Link Grammar also performs a dimensional reduction, but not quite exactly by using projection matrices.  Rather, a word can be assigned to several different word-categories (there are de facto about 2300 of these in the hand-built English dictionaries). Associated with each category is a list of dozens to thousands of “disjuncts” (dependency-grammar dependencies), which play the role analogous to “context”.  However, there are far, far fewer disjuncts than there are contexts. This is because every (multi-word) context is associated with a handful of disjuncts, in such a way that each disjunct stands for hundreds to as many as millions of different contexts.  Effectively, the lexis of Link Grammar is a sparse CxD-dimensional matrix, with C grammatical categories, and D disjuncts, and most entries in this CxD dimensional matrix being zero.  (The upper bound on D is LV, where L is the number of link types, and V is the maximum valency – about 5 or 6. In practice, D is in the tens of thousands.) The act of parsing selects a single entry from this matrix for each word in the sentence.   The probability associated to that word is exp(-c) where c is the “cost”, the numerical value stored at this matrix entry.

Thus, both systems perform a rather sharp dimensional reduction, to obtain a much-lower dimensional intermediate form.  Word2Vec is explicitly linear, Link Grammar is not exactly.  However (and this is important, but very abstract) Link Grammar can be described by a (non-symmetric) monoidal category.  This category is similar to that of the so-called “pregroup grammar”, and is described in a number of places (some predating both Link Grammar an pregroup grammar). The curious thing is that linear algebra is also described by a monoidal category. One might say that this “explains” why Word2Vec works well: it is using the same underlying structural framework (monoidal categories) as traditional symbolic linguistics.  The precise details are too complex to sketch here, and must remain cryptic, for now, although they are open to those versed in category theory.  The curious reader is encouraged to explore category-theoretic approaches to grammar, safe in the understanding that they provide a foundational understanding, no matter which detailed theory one works in.  At the same time, the category-theoretic approach suggests how Word2Vec (or any other neural-net or vector-based approach to grammar) can be improved upon: it shows how syntax can be restored, with the result still looking like a funny/unusual kind of sparse neural-net.  These are not conflicting approaches; they have far far more in common than meets the eye.

A few words about word-senses and semantics are in order. It has been generally observed that Word2Vec seems to encode “semantics” in some opaque way, in that it can distinguish different word-senses, based on context.  The same is true for Link Grammar: when a word is used in a specific context, the result of parsing selects a single disjunct. That disjunct can be thought of as a hyper-fine grammatical category; but these are strongly correlated with meaning.  Synonyms can be discovered in similar ways in both systems: if two different words all share a lot of the same disjuncts, they are effectively synonymous, and can be used interchangeably in sentences.

Similarly, given two different words in Word2Vec/SkipGram, if they both project down to approximately the same vector in the intermediate layer, they can be considered to be synonymous.  This illustrates yet another way that Link Grammar and Word2Vec/SkipGram are similar: the list of all possible disjuncts associated with a word is also a vector, and, if two words have almost co-linear disjunct vectors, they are effectively synonymous.  That is, disjunct-vectors behave almost exactly like neuron intermediate-layer vectors.  They encode similar kinds of information into a vector format.

This is also where we have the largest, and the most important difference between neural net approaches, and Link Grammar. In the neural net approach, the intermediate neuron layer is a black box, completely opaque and un-analyzable, just some meaningless collection of floating-point numbers. In Link Grammar, the disjunct vector is clear, overt, and understandable: you can see exactly what it is encoding, because each disjunct tells you exactly the syntactic relationship between a word, and its neighbors.   This is the great power of symbolic approaches to natural-language: they are human-auditable, human understandable in a way that neural nets are not. (Currently; I think that what this essay describes is an effective sketch for a technique for prying open the lid of the black box of neural nets. But that’s for a different day.)

The arguments presented above are worked out in greater detail, with all the mathematical trimmings, in this linked PDF.  The commonality, by means of category theory, is explored in a different paper, on sheaves.

Many thanks and appreciation to Hanson Robotics for providing the time to think and write about such things. (Much much earlier and kludgier versions of these ideas have been seen in public, in older robot demos.)

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Medieval Thinking and Political AGI

A cherry picked quote from a review of a new book:

“In his new book, Pearl, now 81, elaborates a vision for how truly intelligent machines would think. The key, he argues, is to replace reasoning by association with causal reasoning. Instead of the mere ability to correlate fever and malaria, machines need the capacity to reason that malaria causes fever. Once this kind of causal framework is in place, it becomes possible for machines to ask counterfactual questions—to inquire how the causal relationships would change given some kind of intervention—which Pearl views as the cornerstone of scientific thought.” [1]

The exploration of causal reasoning, and how it works, dates back to the Medieval times, and specifically the work of the Scholastics, on whose theories our modern legal system is built.

Recall a classic example: a dead body is discovered, and a man with a bloody sword stands above it.  Did the man commit the crime, or was he simply the first on the scene, and picked up the sword?  How can you know? What is the “probability” of guilt?

The Scholastics struggled mightily with the concept of “probability”.  Eventually, Blaise Pascal, and many others (Huygens, etc…) developed the modern mathematical theory that explains dice games, and how to place bets on them. [2] This mathematical theory is called “probability theory”, and it works. However, in a courtroom trial for murder, it is not the theory that is applied to determine the “probability of innocence or guilt”.

What actually happens is that the prosecution, and the defense, assemble two different “cases”, two competing “theories”, two different networks of “facts”, of “proofs”, one network showing innocence, the other showing guilt.  The jury is asked to select the one, or the other network, as the true model of what actually happened.

The networks consist of logically inter-connected “facts”. Ideally, those facts are related in a self-consistent fashion, without contradicting one-another.  Ideally, the network of facts connects to the broader shared network that we call “reality”.  Ideally, the various “facts” have been demonstrated to be “true”, by eye-witness testimony, or by forensic reasoning (which itself is a huge network of “facts”, e.g that it takes x hours for blood to clot, y hours for rigor mortis to set in). Ideally, the network connections themselves are “logical”, rather than being various forms of faulty argumentation (appeals to emotion, appeals to authority, etc.)  You really have to put yourself into a Medieval state of mind to grok what is happening here: men pacing in long, fur-trimmed coats, presenting arguments, writing hundred-page-long commentaries-on-commentaries-on-commentaries. Picking apart statements that seems superficially coherent and believable, but can be shown to be built from faulty reasoning.

What is being done, here? Two different, competing models of the world are being built: in one model, the accused is guilty. In the other model, the accused is innocent. Clearly, both models cannot be right. Only one can be believed, and not the other.  Is there a basis on which to doubt one of the models? Is that doubt “reasonable”?  If the accused is to be harmed, viz. imprisoned, and its been shown that the accused is guilty “beyond a reasonable doubt”, then one must “believe”, accept as “reality”, that model, that version of the network of facts.   The other proof is unconvincing; it must be wrong.

I think that it is possible to build AI/AGI systems, within the next 5-10 years, that construct multiple, credible, competing networks of “facts”, tied together with various kinds of evidence and inference and deduction, relation and causation.  Having constructed such competing models, such alternative world-views of deeper “reality”, these AI/AGI systems will be able to disentangle the nature of reality in a rational manner.  And, as I hope to have demonstrated, the mode of reasoning that these systems will employ will be distinctly Medieval.

medieval-university

University students being disciplined.

P.S. The current tumult in social-media, society and politics is very much one of trying to build different, competing “models of reality”, of explanations for the world-as-it-is. In the roughest of  outlines, this is (in the USA) the red-state vs. blue-state political divides, the “liberal” vs. “conservative” arguments. Looking more carefully, one can see differences of opinion, of world-view on a vast number of topics.  Every individual human appears to hold a puzzle-piece, a little micro-vision of what is (social) reality, a quasi-coherent tangle of networked facts that stick together, for them, and that they try to integrate into the rest of society, via identity politics, via social-media posts, via protest marches.[4]

The creation and integration of (competing) models of reality is no longer just a courtroom activity; it is a modern-day social-media and political obsession.  It is possible today, unlike before, only because of the high brain-to-brain (person-to-person) data bandwidth that the Internet, and social-media now provides. One can encounter more competing theories of reality than every before, and one can investigate them to a greater level of detail, than before.

If and when we build AGI systems capable of simultaneously manipulating multiple competing models of the world, I think we can take a number of lessons from social science and psychology as to how these networks might behave. There is currently tremendous concern about propaganda and brain-washing (beliefs in quasi-coherent networks of “facts”, that are non-the-less disconnected from mainstream “reality”). There is tremendous concern about the veracity of main-stream media, and various well-documented pathologies there-of: viz. the need to be profitable forces main-steam media to propagate outrageous but bogus and unimportant news. The equivalent AGI risk is that the sensory-input system floods the reasoning system with bogus information, and that there is no counter-vailing mechanism to adjust for it. Viz.: we can’t just unplug journalism and journalists from the capitalist system; nor is it clear that doing so would increase the quality of the broad-cast news.

Some of the issues facing society are because human brains are not sufficiently integrated, in the sense of “integrated information”. [3]  Any individual human mind can make sense of only one small part of the world; we do not have the short-term memory, the long-term memory, or the reasoning capacity to take on more. This is not a limitation we can expect in AGI.  However, instead of having individual humans each representing the pro and con of a given issue, its reasonable to expect that the AGI will simultaneously develop multiple competing theories, in an attempt to find the better, stronger one – the debate does not stop; it only getters bigger and more abstract.

Another on-line concern is that much of on-line political posting and argumentation is emotionally driven, anchored in gut-sense arguments, which could certainly be found to be full of logical fallacies, if they were to be picked apart. “Humans are irrational”, it is said. But are they really? In Bayesian inference, one averages together, blurs together vast tracts of “information”, and reduces it to a single number, a probability. Why?  Because all of those inputs look like “noise”, and any given, specific Bayesian model cannot discriminate between all the different things going on. Thus, average it together, boil it all down, despite the fact that this is sure to erase important distinctions (e.g. “all cats have fur”, except, of course, when they don’t.)  This kind of categorical, lumped-together, based-on-prior-experience, “common sense” kind of thinking sure seems to be exactly what we accuse our debate opponents of doing: either they’re “ignoring the details”, or failing to “see the big picture”.  I don’t see how this is avoidable in the reasoning of Bayesian networks, or any other kind of network reasoning.  Sooner or later, the boundaries of a fact network terminate in irrelevant facts, or details that are too small to consider. Those areas are necessarily fuzzed over, averaged, and ignored: gut-intuition, common-sense will be applied to them, and this suffers from all of well-known pitfalls of gut-sense reasoning.  AGI might be smarter than us; and yet, it might suffer from a very similar set of logical deficiencies and irrational behaviors.

The future will look very much like today. But different.

[1] Judea Pearl, The Book of Why / The New Science of Cause and Effect

[2] James Franklin, The Science of Conjecture: Evidence and Probability Before Pascal

[3] Integrated information theory – Wikipedia

[4] Meaningness – Thinking, feeling, and acting—about problems of meaning and meaninglessness; self and society; ethics, purpose, and value. (A work in progress, consisting of a hypertext book and a metablog that comments on it.)

Posted in Meta, Theory, Uncategorized | Tagged , | 1 Comment

Inference Meta-Learning (part I)

This is the introductory part of a blog series about my work on inference control learning for OpenCog. It introduces the problematic, an initial design solution and some early results. More interesting results will come in subsequent parts.

Introduction

Inference chainers, automatic theorem provers and the likes are programs that given a theory attempt to build a proof of a given theorem. They are generally inefficient. An proof, or inference is in essence a program, as the Curry-Howard correspondence says. In OpenCog it is very apparent as an inference is an actual atomese program. A function call in this program is a formula associated to an inference rule, the conclusion being the output of the call, the premises being its inputs. See for instance a double deduction inference tree

     X->Z    Z->Y
     -----f------
A->X     X->Y
------f------
     A->B

a bottom-down notation borrowed from the classical notations such as https://en.wikipedia.org/wiki/Rule_of_inference except that premises are aligned horizontally, and the rule formula, here f, overlays the line separating the premises and the conclusion.

In OpenCog such tree would be represented as

             [12731261225409633207][6] [11133117073607658831][6]
             ---------------bc-deduction-formula----------------
[17146615216377982335][6] [16015351290941397556][6]
---------------bc-deduction-formula----------------
             [13295355732252075959][1]

the cryptic numbers indicate the hash values of the atoms involved. The hash value [13295355732252075959][1] at the bottom of the tree is the target’s, the hash values at the leaves [17146615216377982335][6], [12731261225409633207][6] and [11133117073607658831][6] are the premises, and [16015351290941397556][6] is an intermediary target. bc-deduction-formula is the name of the formula associated to the deduction rule.

This inference tree would correspond to the atomese program

(BindLink
  ;; Pattern to fetch premises from the atomspace
  (AndLink
    (InheritanceLink
      (VariableNode "$B-6229393a") ; [6185394777777469381][6]
      (ConceptNode "D") ; [246112806454457922][1]
    ) ; [11133117073607658831][6]
    (InheritanceLink
      (VariableNode "$B-6266d6f2") ; [4097372290580364298][6]
      (VariableNode "$B-6229393a") ; [6185394777777469381][6]
    ) ; [12731261225409633207][6]
    (InheritanceLink
      (VariableNode "$X") ; [6809909406030619949][1]
      (VariableNode "$B-6266d6f2") ; [4097372290580364298][6]
    ) ; [17146615216377982335][6]
  ) ; [12553878300592245761][6]
  ;; Formula calls computing the inference tree
  (ExecutionOutputLink
    (GroundedSchemaNode "scm: bc-deduction-formula") ; [5481509939359570705][1]
    (ListLink
      (InheritanceLink
        (VariableNode "$X") ; [6809909406030619949][1]
        (ConceptNode "D") ; [246112806454457922][1]
      ) ; [13295355732252075959][1]
      (InheritanceLink
        (VariableNode "$X") ; [6809909406030619949][1]
        (VariableNode "$B-6266d6f2") ; [4097372290580364298][6]
      ) ; [17146615216377982335][6]
      (ExecutionOutputLink
        (GroundedSchemaNode "scm: bc-deduction-formula") ; [5481509939359570705][1]
        (ListLink
          (InheritanceLink
            (VariableNode "$B-6266d6f2") ; [4097372290580364298][6]
            (ConceptNode "D") ; [246112806454457922][1]
          ) ; [16015351290941397556][6]
          (InheritanceLink
            (VariableNode "$B-6266d6f2") ; [4097372290580364298][6]
            (VariableNode "$B-6229393a") ; [6185394777777469381][6]
          ) ; [12731261225409633207][6]
          (InheritanceLink
            (VariableNode "$B-6229393a") ; [6185394777777469381][6]
            (ConceptNode "D") ; [246112806454457922][1]
          ) ; [11133117073607658831][6]
        ) ; [10363704109909197645][6]
      ) ; [18135989265351680839][6]
    ) ; [14126565831644143184][6]
  ) ; [12675478950231850570][6]
) ; [15856494860655100711][6]

The specificity here is that the pattern matcher (the query corresponds to the outer BindLink) is used to fetch the relevant axioms from the atomspace, the most afferent inputs of that program.

The question we attempt to address here is: how to efficiently build inferences, specifically back-inferences.

A back-inference, an inference built backward, is done by providing a target, or theorem, and grow an inference tree going from the target to the axioms, so that once run, the inference will evaluate the truth value of that target. The problem is that such growth, if not carefully controlled, can be very inefficient.

Design

The idea is to learn from past inferences. This is not a new idea, but is still fairely under-explored. I give some references at the end.

Let me sketch how we intend to do that in OpenCog:

  1. Run the backward chainer in a greedy manner over a collection of problems
  2. Store the traces of the backward chainer in a history atomspace
  3. Mine the history atomspace to discover regularities
  4. Turn these regularities into control rules for the backward chainer
  5. Repeat 1, passing these control rules to the backward chainer on the same or related collection of problems, see how it improves.

The backward chainer is a rather a elementary algorithm. Given a target, pick a rule so that its conclusion unifies with the target (that is such rule can possibly produce such target), create an initial inference tree. The leaves of that tree are the premises, the root is the target. Treat the premises as new targets and re-iterate the process. Grow the tree (or rather the trees, as we keep the former ones around) till you get something that, if evaluated, produces the target. For that of course the premises need to be present, as axioms, in the atomspace.

The problem though comes from the combinatorial explosion of having to choose the inference tree, the premise to expand and the rule. The complexity of the algorithm grows exponentially with respect to the size of the inference tree.

So here’s the crux of the idea of how to make it practical:

Defer the hard decisions to a cognitive process.

To repeat myself, the hard decisions in this program, the backward chainer, are

  1. pick an inference tree
  2. given an inference tree, pick a premise
  3. given an inference tree and a premise, pick a rule to expand

These are very hard decisions that only an intelligent entity can make, ultimately an AGI. We don’t have an AGI, but we are building one, thus if we can plug our proto-AGI to solve these sub-problems, it becomes a recursive divide-and-conquer type of construct. Turtles all the way down if you like, that hopefully ends. The idea of SampleLink introduced by Ben [1] was intended to make this recursive construct explicit. Here we do not use SampleLink, it hasn’t been implemented yet, but that doesn’t stop us from using OpenCog to learn control rules and utilize them in the backward chainer. What it means is that in the backward chainer code, when such a hard decision occurs, the code that would otherwise look like a hard wired heuristic now looks like this

  1. Try to find control rules in the atomspace related to that decision
  2. If you find some, combine them to make a decision
  3. Otherwise, assume an uninformative control rule, or possibly a default one encapsulating some simple heuristic.

That is what is meant by a cognitive process, a process that uses the available knowledge to make decisions.

More specifically these control rules are Cognitive Schematics [2], pieces of knowledge that tells our cognitive process how actions relate to goals (or subgoals) conditioned by contexts

Context & Action -> Goal

They are usually represented by an Implication (or PredictiveImplication if time matters) in OpenCog. That is a conditional probability, or meta-probability, as that is what a TV represents.

Here these cognitive schematics will be about rules, premises, proofs, etc. For instance a cognitive schematic about the third hard decision

3. given an inference tree and a premise, pick a rule to expand

could be formulated as followed

ImplicationScope <TV>
  VariableList A L R B T
  And
    Preproof A T
    <some pattern>
    Expand (List A L R) B
  Preproof B T

meaning that if inference tree A is a preproof of theorem T (a preproof in an inference tree that may not proove T yet, but that may eventually proove T if adequately expanded), conditioned by some pattern involving A, L, R, and some action, expanding inference tree A into inference tree B, via premise L, using rule R, then the produced inference tree B will also be a preproof of T, with truth value TV. The cognitive schematic parts are

Context = (Preproof A T) And <some pattern>
Action = Expand (List A L R) B
Goal = Preproof B T

Once we have such cognitive schematics

C1 & A -> G

Cn & A -> G

we need to combine then. We could consider a rule formed with the conjunction or disjunction of all contexts, or any other way of aggregating. The dilemma here is that

  1. The smaller (more restrictive) the context, the lower the confidence of the rule.
  2. The larger (more abstract) the context, the lower the discriminative power of the rule.

So to rephrase, if a context is too small its associated rule might overfit, but if the context is too large, its rule will not be informative enough, its conditional probability will tend to the marginal probability of the goal.

To address this dilemma we have choosen to rely on Universal Operator Induction [3], albeit some modification of it to account for the fact that a control rule is only a partial operator (see [5] for more details). Once this aggregation is done, we can assign a TV for each action, i.e. inference rule, and sample our next inference rule accordingly (here we have choosen Thomson Sampling due to its asymptotical optimality property [4]).

At this point the reader might ask, what about managing the complexity of this decision process? The short answer is ECAN [6]. The less short answer is turtles all the way down… SampleLink, etc. But suffice to say that ECAN has been designed to be a cognitive process as well, utilizing knowledge via a dedicated type of atoms called HebbianLinks.

Experiments

Settings

The problem space of our first experiments is very simple. Given knowledge about the alphabet, infer that 2 letters are alphabetically ordered.

There are two collections of axioms

1. Enumeration of all consecutively ordered letters

a⊂b

y⊂z

where X⊂Y is a notation for the atomese program

(Inheritance (stv 1 1) X Y)

Inheritance is used so that we can directly use the PLN deduction rule to infer order transitivity. Given this collection of axioms, all the backward chainer needs is to chain a series of deduction rules, as many as it requires. For instance inferring a⊂c will only require 2 deductions, while inferring h⊂z will require 17 deductions.

In the end, only the deduction rule is required. Figuring that out is not challenging but serves as a simple test for our algorithms. That is what we have accomplished so far. To create a more interesting case, we introduce a second collection of axioms

2. Express that letter a occurs before any other letter

a<b

a<z

where X<Y is a notation for the atomese program

(Evaluation (stv 1 1) (Predicate "alphabetical-order") (List X Y))

Alongside an implication

X<Y ⇒ X⊂Y

or in atomese

ImplicationScope (stv 1 1)
  VariableList
    TypedVariable
      Variable "$X"
      Type "ConceptNode"
    TypedVariable
      Variable "$Y"
      Type "ConceptNode"
  Evaluation
    Predicate "alphabetical-order"
    List
      Variable "$X"
      Variable "$Y"
  Inheritance
    Variable "$X"
    Variable "$Y"

This second collection of axioms allows us to prove with just one inference step, using the PLN instantiation rule, that

a⊂X for any letter X != a.

however, unlike deduction, using this rule is only fruitful if the first letter is a, otherwise it will actually slow down the backward chainer, so it is important to be able to discover this context.

Learning Inference Control Rules

Context-Free Control Rules

The control rule of deduction is very simple as it has no extra context

ImplicationScope <TV>
  VariableList A L B T
  And
    Preproof A T
    Expand (List A L <deduction-rule>) B
  Preproof B T

tells that if A is a preproof and gets expanded into B by a deduction rule, then B has a certain probability, expressed by TV, of being a preproof.

Learning that control rule is easy, we just need to apply PLN direct evaluation rule to calculate the TV based on the available evidence, the traces gathered while solving the problem collection. Indeed, while the backward chainer is running, it stores in a history atomspace every hard decision that has been made. In particular all inference tree expensions, and of course which expansion led to a proof, which allows us to build a corpus of expansions and preproofs. The PLN direct evaluation rule will merely count the positive and negative instances and come up with a conditional probability and a confidence.

Context-Sensitive Control Rules

Learning context-sensitive control rules is harder. In fact it may be arbitrary hard, but the initial plan is to experiment with frequent subgraph mining [8], using OpenCog's pattern miner [7].

We haven't reached that part yet, but it is expected that such rule will look something like

ImplicationScope <TV>
  VariableList A L B T
  And
    Preproof A T
    Expand (List A L <conditional-instantiation-rule>) B
    <pattern-discovered-by-the-pattern-miner>
  Preproof B T

the pattern in question

<pattern-discovered-by-the-pattern-miner>

will have to express that the premise L, looks like

    Inheritance
      ConceptNode "a"
      Variable "$X"

or more generally that the expansion looks like

Execution
  Schema "expand"
  List
    Variable "$A"
    Inheritance
      ConceptNode "a"
      Variable "$X"
    GroundedSchemaNode "scm: conditional-full-instantiation-scope-formula"
  Variable "$B"

Although this part will rely on the pattern miner, in the end the calculations of these rules will be performed by PLN, so in a way PLN will be handling the meta-learning part as well. Will come back to that.

Results so far

Let me detail the experiment a bit further. The problem set is composed one 100 targets, randomly generated so that 2 ordered letters are queried, such as

w⊂y
q⊂u
g⊂v
y⊂z

We run 2 iterations, the first one with an empty control atomspace, then ask OpenCog to discover control rules, populate the control atomspace with them and rerun, and see how many more we have solved.

Just by learning context-free control rules, saying basically that deduction is often useful, conditional instantiation is sometimes useful, and all other rules are useless, we can go from solving 34 to 52.

Below are examples of control rules that we have learned.

;; Modus ponens is useless
(ImplicationScopeLink (stv 0 0.03625)
   (VariableList
      (VariableNode "$T")
      (TypedVariableLink
         (VariableNode "$A")
         (TypeNode "DontExecLink")
      )
      (VariableNode "$L")
      (TypedVariableLink
         (VariableNode "$B")
         (TypeNode "DontExecLink")
      )
   )
   (AndLink
      (EvaluationLink
         (PredicateNode "URE:BC:preproof-of")
         (ListLink
            (VariableNode "$A")
            (VariableNode "$T")
         )
      )
      (ExecutionLink
         (SchemaNode "URE:BC:expand-and-BIT")
         (ListLink
            (VariableNode "$A")
            (VariableNode "$L")
            (DontExecLink
               (DefinedSchemaNode "modus-ponens-implication-rule")
            )
         )
         (VariableNode "$B")
      )
   )
   (EvaluationLink
      (PredicateNode "URE:BC:preproof-of")
      (ListLink
         (VariableNode "$B")
         (VariableNode "$T")
      )
   )
)

;; Deduction is often useful
(ImplicationScopeLink (stv 0.44827586 0.03625)
   (VariableList
      (VariableNode "$T")
      (TypedVariableLink
         (VariableNode "$A")
         (TypeNode "DontExecLink")
      )
      (VariableNode "$L")
      (TypedVariableLink
         (VariableNode "$B")
         (TypeNode "DontExecLink")
      )
   )
   (AndLink
      (EvaluationLink
         (PredicateNode "URE:BC:preproof-of")
         (ListLink
            (VariableNode "$A")
            (VariableNode "$T")
         )
      )
      (ExecutionLink
         (SchemaNode "URE:BC:expand-and-BIT")
         (ListLink
            (VariableNode "$A")
            (VariableNode "$L")
            (DontExecLink
               (DefinedSchemaNode "deduction-inheritance-rule")
            )
         )
         (VariableNode "$B")
      )
   )
   (EvaluationLink
      (PredicateNode "URE:BC:preproof-of")
      (ListLink
         (VariableNode "$B")
         (VariableNode "$T")
      )
   )
)

;; Conditional instantiation is sometimes useful
(ImplicationScopeLink (stv 0.12903226 0.03875)
   (VariableList
      (VariableNode "$T")
      (TypedVariableLink
         (VariableNode "$A")
         (TypeNode "DontExecLink")
      )
      (VariableNode "$L")
      (TypedVariableLink
         (VariableNode "$B")
         (TypeNode "DontExecLink")
      )
   )
   (AndLink
      (EvaluationLink
         (PredicateNode "URE:BC:preproof-of")
         (ListLink
            (VariableNode "$A")
            (VariableNode "$T")
         )
      )
      (ExecutionLink
         (SchemaNode "URE:BC:expand-and-BIT")
         (ListLink
            (VariableNode "$A")
            (VariableNode "$L")
            (DontExecLink
               (DefinedSchemaNode "conditional-full-instantiation-implication-scope-meta-rule")
            )
         )
         (VariableNode "$B")
      )
   )
   (EvaluationLink
      (PredicateNode "URE:BC:preproof-of")
      (ListLink
         (VariableNode "$B")
         (VariableNode "$T")
      )
   )
)

These rules are rather simple, but, as reported, can already speed up the backward chainer.

Related Work and Conclusion

This relates to the work of Ray Solomonoff [9], Juergen Schimdhuber with OOPS [10], Eray Ozkural with HAM [11], Irvin Hwang et al with BPM [12], Josef Urban with MaLARea [13], Alexander A. Alemi et al with DeepMath [14] to cite only a few. What I believe sets it apart though, is that the system used for solving the problem is the same one used for solving the meta-problem. Which leads to an interesting question, that we may well be able to put to the test in the future. Can skills acquired to solve a problem be transfered to the meta-level?

Let me expand, if you ask an AI to solve a collections of down to earth problems, it will accumulate a lot of knowledge, say rules. Some will be very concrete, related to specificities of the problems, such as pushing the gas pedal while driving a car, and will not be transferable to the meta-level, because pushing a gas pedal is unrelated to discovering control rules to speed up a program. They will basically remain mute when asked to serve the cognitive process in charge of solving the meta-problem. But some will be more abstract, abstract enough that they will be recognized by the meta-solver as potentially useful. If these abstract rules can indeed help and be transfered to the upper levels, then it opens the possiblity for true intelligence bootstrapping. If it can, then it means we can improve not just learning, but also meta-learning, meta-meta-learning, and so on to infinity, at once. But realistically, even if it doesn't, or does to some limited extend, possibly evaporating as the meta-levels go higher, meta-learning may still result in considerable performance gains. In any instance, it is our only magic bullet, isn't it?

Acknowledgement

Thanks to Linas for his feedback.

References

[1] Ben Goertzel. Probabilistic Growth and Mining of Combinations: A
Unifying Meta-Algorithm for Practical General Intelligence
https://link.springer.com/chapter/10.1007/978-3-319-41649-6_35?no-access=true

[2] Ben Goertzel. Cognitive Synergy: a Universal Principle For
Feasible General Intelligence?
https://pdfs.semanticscholar.org/511e/5646bc1d643585933549b5321a9da5ee5f55.pdf

[3] Ray Solomonoff. Three Kinds of Probabilistic Induction: Universal
Distributions and Convergence Theorems
http://world.std.com/%7Erjs/publications/chris1.pdf

[4] Jan Leike et al. Thompson Sampling is Asymptotically Optimal in
General Environments
http://auai.org/uai2016/proceedings/papers/20.pdf

[5] Nil Geisweiller. Inference Control Learning Experiment README.md
https://github.com/opencog/opencog/tree/master/examples/pln/inference-control-learning

[6] Matthew Ikle’ et al. Economic Attention Networks: Associative
Memory and Resource Allocation for General Intelligence
http://agi-conf.org/2009/papers/paper_63.pdf

[7] Ben Goertzel et al. Integrating Deep Learning Based Perception
with Probabilistic Logic via Frequent Pattern Mining
http://goertzel.org/agi-13/DeSTIN_PLN_v3.pdf

[8] Yun Chi et al. Mining Closed and Maximal Frequent Subtrees from
Databases of Labeled Rooted Trees
http://ftp.cs.ucla.edu/tech-report/2004-reports/040020.pdf

[9] Ray J. Solomonoff. Progress in incremental machine learning. In
NIPS Workshop on Universal Learning Algorithms and Optimal Search,
Whistler, B.C., Canada, December 2002.
http://raysolomonoff.com/publications/nips02.pdf

[10] Schmidhuber J. Optimal ordered problem solver. Machine Learning
54 (2004) 211–256 https://arxiv.org/pdf/cs/0207097.pdf

[11] Eray Ozkural. Towards Heuristic Algorithmic Memory.
https://link.springer.com/chapter/10.1007%2F978-3-642-22887-2_47

[12] Irvin Hwang et al. Inducing Probabilistic Programs by Bayesian
Program Merging
https://arxiv.org/pdf/1110.5667.pdf

[13] Josef Urban. MaLARea: a Metasystem for Automated Reasoning in
Large Theories
http://ceur-ws.org/Vol-257/05_Urban.pdf

[14] Alexander A. Alemi et al. DeepMath – Deep Sequence Models for
Premise Selection
https://arxiv.org/pdf/1606.04442v1.pdf

[15] Ben Goertzel et al. Metalearning for Feature Selection.
https://arxiv.org/abs/1703.06990

Posted in Development, Theory | 3 Comments

Reframing OpenCog Action Selection: Contextual Bandit Problems and Reinforcement Learning

I thought a bit today about how OpenCog’s action selector (based on the Psi model from Dietrich Dorner and Joscha Bach) relates to approaches to action selection and behavior learning one sees in the reinforcement learning literature.

After some musing, I came to the conclusion that this may be another area where it could make sense to insert a deep neural network inside OpenCog, for carrying out particular functions.

I note that Itamar Arel and others have proposed neural net based AGI architectures in which deep neural nets for perception, action and reinforcement are coupled together.   In OpenCog, one could use deep neural nets for perception, action and reinforcement; but, these networks would all be interfaced via the Atomspace, and in this way would be interfaced with symbolic algorithms such as probabilistic logical inference, hypergraph pattern mining and concept blending, as well as with each other.

One interesting meta-point regarding these musings is that they don’t imply any huge design changes to the OpenCog “OpenPsi” action selector.   Rather, one could implement deep neural net policies for action selection, learned via reinforcement learning algorithms, as a new strategy within the current action selection framework.   This speaks well of the flexible conceptual architecture of OpenPsi.

Action Selection as a Contextual Bandit Problem

For links to fill you in on OpenCog’s current action selection paradigm and code, start here.

The observation I want to make in this blog post is basically that: The problem of action selection in OpenCog (as described at the above link and the others it points to) is an example of the “contextual bandit problem” (CBP).

In the case where more than one action can be chosen concurrently, we have a variation of the contextual bandit problem that has been called “slates” (see the end of this presentation).

So basically the problem is: We have a current context, and we have a goal (or several), and we have bunch of alternative actions.  We want in the long run to choose actions that will maximize goal achievement.  But we don’t know the expected payoff of each possible action.  So we need to make a probabilistic, context-dependent choice of which action to do; and we need to balance exploration and exploitation appropriately, to maximize long-term gain.   (Goals themselves are part of the OpenCog self-organizing system and may get modified as learning progresses and as a result of which actions are chosen, but we won’t deal with that part of the feedback tangle here.)

Psi and MicroPsi and OpenPsi formulate this problem in a certain way.   Contextual bandit problems represent an alternate formulation of basically the same problem.

Some simple algorithms for contextual bandit problems are here and a more interesting neural net based approach is here.   A deep neural net approach for a closely related problem is here.

CBP and OpenPsi

These approaches and ideas can be incorporated into OpenCog’s Psi based action selector, though this would involve using Psi a little differently than we are now.

A “policy” in the CBP context is a function mapping the current context into a set of weightings on implications of the form (Procedure ⇒ Goal).

Most of the time in the reinforcement literature a single goal is considered, whereas in Psi/OpenCog one considers multiple goals; but that’s not an obstacle to useing RL ideas in OpenCog.   One can use RL to figure out procedures likely to lead to fulfillment of an individual goal; or one can apply RL to synthetic goals defined as weighted averages of system goals.

What we have in OpenPsi are implications of the form (Context & Procedure ⇒ Goal) — obviously just a different way of doing what RL is doing…

That is:

  • In RL one lists Contexts, and for each Context one has a set of (Procedure ⇒ Goal) pairs
  • In Psi one lists (Context & Procedure ⇒ Goal) triples (“Psi implications”)

and these two options are obviously logically equivalent.

A policy, in the above sense, could be used to generate a bunch of Psi implications with appropriate weights.   In general a policy may be considered as a concise expression of a large set of Psi implications.

In CBP learning what we have is, often, a set of competing policies (e.g. competing linear functions, or competing neural networks), each of which provides its own mapping from contexts into (Procedure⇒ Goal) implications.   So, if doing action selection in this approach: To generate an action, one would first choose a policy, and then use that policy to generate weighted (Context & Procedure ⇒ Goal) implications [where the Context was very concrete, being simply the current situation], and then use the weights on these implications to choose an action.

In OpenCog verbiage, each policy could in fact be considered a context, so we could have

ContextLink
    ConceptNode “policy_5”
    ImplicationLink
        AndLink
            Context
            procedure
        Goal

and one would then do action selection using the weighting for the current policy.

If, for instance, a policy were a neural network, it could be wrapped up in a GroundedSchemaNode.    A neural net learning algorithm could then be used to manage an ensemble of policies (corresponding behind the scenes to neural networks), and experiment with these policies for action selection.

This does not contradict the use of PLN to learn Psi implications.   PLN would most naturally be used to learn Psi implications with abstract Contexts; whereas in the RL approach, the abstraction goes into the policy, and the policy generates Psi implications that have very specific Contexts.   Both approaches are valid.

In general, the policy-learning-based approach may often be better when the Context consists of a large number of different factors, with fuzzy degrees of relevance.  In this case learning a neural net mapping these contextual factors into weightings across Psi implications may be effective.   On the other hand, when the context consists of a complex, abstract combination of a smaller number of factors, a logical-inference approach to synthesizing Psi implications may be superior.

It may also be useful, sometimes, to learn neural nets for CBP policies, and then abstract patterns from these neural nets using pattern  mining; these patterns would then turn into Psi implications with abstract Contexts.

(Somewhat Sketchy) Examples

To make these ideas a little more concrete, let’s very briefly/roughly go through some example situations.

First, consider question-answering.   There may be multiple sources within an OpenCog system, capable of providing an answer to a certain question, e.g.:

  • A hard-wired response, which could be coded into the Atomspace by a human or learned via imitation

  • Fuzzy matcher based QA taking into account the parse and interpretation of the sentence

  • Pattern matcher lookup, if the Atomspace has definite knowledge regarding the subject of the query

  • PLN reasoning

The weight to be given to each method’s, in each case, needs to be determined  adaptively based on the question and the context.

A “policy” in this case would map some set of features associated with the question and the context, into a weight vector across the various response sources.

A question is what is the right way to quantify the “context” in a question-answering case.  The most obvious approach is to use word-occurrence or bigram-occurrence vectors.  One can also potentially add in, say, extracted RelEx relations or RelEx2Logic relations.

If one has multiple examples of answers provided by the system, and knows which answers were accepted by the questioner and which were not, then this knowledge can be used to drive learning of policies.   Such a policy would tell the system, given a particular question and the words and semantic relationships therein as well as the conversational context, which answer sources to rely on with what probabilities.

A rather different example would be physical movement.   Suppose one has a collection of movement patterns (e.g. “animations” moving parts of a robot body, each of which may have multiple parameters).   In this case one has a slate problem, meaning that one can choose multiple movement patterns at the same time.   Further, one has to specify the parameters of each animation chosen; these are part of the action.   Here a neural network will be very valuable as a policy representation, as one’s policy needs to take in floating-point variables quantifying the context, and output floating-point variables representing the parameters of the chosen animations.   Real-time reinforcement data will be easily forthcoming, thus driving the underlying neural net learning.

(If movement is controlled by a deep neural network, these “animations” may be executed via clamping them in the higher-level nodes of the network, and then allowing the lower-level nodes to self-organize into compatible states, thus driving action.)

Obviously a lot of work and detailed thinking will be required to put these ideas into practice.  However, I thought it might be useful to write this post just to clarify the connections between parts of the RL literature and the cognitive modeling approach used in OpenCog (drawn from Dorner, Bach, Psi, etc.).   Often it happens that the close relationships between two different AI approaches or subfields are overlooked, due to “surface level” issues such as different habitual terminologies or different historical roots.

Potentially, the direction outlined in this post could enable OpenCog to leverage code and insights created in the deep reinforcement learning community; and to enable deep reinforcement neural networks to be used in more general-purpose ways via embedding them in OpenCog’s neural-symbolic framework.

Posted in Uncategorized | 1 Comment

Cogistry: Accelerating Algorithmic Chemistry via Cognitive Synergy

This post describes some speculative ideas for new AI/AGI/Artificial-Life/Artificial-Chemistry development that could be done on top of the OpenCog platform, leveraging OpenCog tools in a novel way.

These are not necessarily proposed for immediate-term development – we have a lot of other stuff going on, and a lot of stuff that’s half-finished or ¾-finished or more, that needs to be wrapped up and improved, etc.   They are, however, proposed as potentially fundamental enhancements to the OpenCog and PrimeAGI (CogPrime) designs in their current form.

The core motivation here is to add more self-organizing creativity to the AGI design.   One can view these ideas as extending what MOSES does in the PrimeAGI design – MOSES (at least in its aspirational version) is artificial evolution enhanced by pattern mining and probabilistic reasoning; whereas Cogistry is, very loosely, more like artificial biochemistry similarly enhanced.

Historical Background

Walter Fontana, in a fascinating 1990 paper, articulated a novel approach to “artificial life” style digital evolution of emergent phenomena, called “algorithmic chemistry.”   The basic concept was: Create small codelets, so that when codelet A acts on codelet B, it produces a new codelet C.   Then put a bunch of these codelets in a “primordial algorithmic soup” and let them act on each other, and see what interesting structures and dynamics emerge.

The paper reports some interesting emergent phenomena, but then the research programme was dropped off at an early stage.   Very broadly speaking, the story seems to have been similar to what happened with a lot of Alife-related work of that era: Some cool-looking self-organizational phenomena occurred, but not the emergence of highly complex structures and dynamics like the researchers were looking for.

These sorts of results spawned the natural question “why?”   Did the simulations involved not have a large enough scale?   (After all, the real primordial soup was BIG and based on parallel processing and apparently brewed for quite a long time before producing anything dramatic.)   Or were the underlying mechanisms simply not richly generative enough, in some way?   Or both?

What I am going to propose here is not so much a solution to this old “why” question, but rather a novel potential route around the problem that spawned the question.   My proposal – which I call “Cogistry” — is to enhance good old Fontana-style algorithmic chemistry by augmenting its “self-modifying program soup” with AI algorithms such as hypergraph pattern mining and probabilistic reasoning.   I believe that, if this is done right, it can lead to “algorithm soups” with robust, hierarchical, complex emergent structures – and also to something related but new and different: emergent, self-organizing program networks that carry out functions an AI agent desires for achievement of its goals.

That is: my aim with Cogistry is to use probabilistic inference and pattern mining to enhance algorithmic chemistry, so as to create “digital primordial soups” that evolve into interesting digital life-forms, but ALSO so as to create “life-like program networks” that transform inputs into outputs in a way that carries out useful functions as requested by an AI agent’s goal system.   The pure “digital primordial soup” case would occur when the inference and pattern mining are operating with the objective of spawning interesting structures and dynamics; whereas the “useful program network” case would occur when the inference and pattern mining are operating with an additional, specific, externally-supplied objective as well.

There is a rough analogy here with the relation between genetic algorithms and estimation of distribution algorithms (EDAs).   EDAs aim to augment GA mutation and crossover with explicit pattern mining and probabilistic reasoning.  But there are significant differences between EDAs and the present proposal as well.   There is a lot more flexibility in an algorithmic chemistry network than in an evolving populations of bit strings or typical GP program trees; and hence, I suspect, a lot more possibility for the “evolution/self-organization of evolvability” and the “evolution/self-organization of inferential analyzability” to occur.   Of course, though, this added flexibility also provides a lot more potential for messes to be made (including complex, original sorts of messes).

From an Alife point of view, the “chemistry” in the “algorithmic chemistry” metaphor is intended to be taken reasonably seriously.  A core intuition here is that to get rich emergent structures and dynamics from one’s “digital biology” it’s probably necessary to go a level deeper to some sort of “digital chemistry” with a rich combinatorial potential and a propensity to give rise to diverse stable structures, including some with complex hierarchical forms.    One might wonder whether this is even deep enough, whether one needs actually to dig down to a “digital physics” from which the digital chemistry emerges; my intuition is that this is not the case and focusing on the level of algorithmic chemistry (if it’s gotten right) is deep enough.

Actually, the “digital physics,” in the analogy pursued here, would be the code underlying the algorithms in the algorithm-soup: the programming-language interpreter and the underlying C and assembly code, etc.   So part of my suggestion here will be a suggestion regarding what kind of digital physics is likely to make algorithmic chemistry work best: e.g. functional programming and reversible computing.   But, according to my intuition, the core ideas of this “digital physics” have already been created by others for other purposes, and can be exploited for algorithmic-chemistry purposes without needing dramatic innovations on that level.

An Algorithmic Chemistry Framework

First I will define a framework for algorithmic chemistry in a reasonably general way.   I will then fill in more details, bit by bit.

The basic unit involved I will call a “codelet” – defined simply as a piece of code that maps one or more input codelets into one or more output codelets.   What language to use for specifying codelets is a subtle matter that I will address below, but that we don’t need to define in order to articulate a general algorithmic chemistry framework.

Fontana’s original work involved a chemical soup with no spatial structure, but other work with artificial life systems suggests that a spatial structure may be valuable.  So we propose a multi-compartment system, in which each compartment has its own “algorithmic chemical soup.”    A codelet, relative to a specific compartment, can be said to have a certain “count” indicating how many copies of that codelet exist in that compartment.

Inside each compartment, multiple dynamics happen concurrently:

  • Migration: codelets migrate from one compartment to neighboring compartments.    This can be done initially via random diffusion.
  • Importance spreading: codelets have short and long term importance values, which spread e.g. via ECAN dynamics
  • Forgetting: codelets with low long-term importance are “forgotten” from a compartment (and may or may not be saved to some persistent store, depending on system configuration)
  • Reaction: codelets in the compartment act on other codelets in the compartment, producing new codelets.   Codelets may be chosen for reaction with probability based on their short term importance values.
  • Generation: new codelets are introduced into the compartment, either representing inputs to the system from some external source, or generated randomly from some distribution, or drawn from some set of codelets believed to be generally useful

I will use the word “codenet” to refer to a collection of codelets that interact with each other in a coherent way.   This is intentionally a vague, intuitive definition – because there are many kinds of coherent networks of codelets, and it’s not obvious which ones are going to be the most useful to look at, in which contexts.   In some cases a chain of the form “In the presence of background codelet-set B, A1 reacts with B to make A2, which reacts with B to make A3, etc. …” may be very influential.   In other cases cycles of the form “A and B react to make C; then A and C create to make more B; and B and C react to make more A” may be critical.   In other cases it may be more complex sorts of networks.   Exactly how to chop up a soup of codelets, growing and interacting over time, into distinct or overlapping codenets is not entirely clear at present and may never be.   However, it’s clear that for understanding what is going on in an algorithmic-chemistry situation, it’s the codenets and not just the codelets that need to be looked at.   If codelets are like chemicals, then codenets are like chemical compounds and/or biological systems.

Implementation Notes

In terms of current computing architectures, it would be natural to run different compartments on different machines, and to run the four processes in different threads, perhaps with multiple threads handling reaction, which will generally be the most intensive process.

If implemented in an OpenCog framework, then potentially, separate compartments could be separate Atomspaces, and the dynamic processes could be separate MindAgents running in different threads, roughly similar to the agents now comprising the ECAN module.   Also, in OpenCog, codelets could be sub-hypergraphs in Atomspace, perhaps each codelet corresponding to a DefineLink.

Reactions would naturally be implemented using the Forward Chainer (a part of the Rule Engine, which leverages the Pattern Matcher).   This differs a bit from PLN’s use of the Forward Chainer, because in PLN one is applying an inference rule (drawn from a small set thereof) to premises, whereas here one is applying a codelet (drawn from a large set of possible codelets) to other codelets.

Measuring Interestingness

One interesting question is how to measure the interestingness of a codelet, or codenet.

For codelets, we can likely just bump the issue up a level: A codelet is as interesting as the codenets it’s part of.

For codenets, we can presumably rely on information theory.   A compartment, or a codenet, as it exists at a particular point in time, can be modeled using a sparse vector with an entry for each codelet that has nonzero count in the compartment or codenet (where the entry for a codelet contains the relevant count).      A compartment or codenet as it exists during an interval of time can then be modeled as a series of sparse vectors.   One can then calculate the interaction information of this vector-series (or the “surprisingness” as defined in the context of OpenCog’s Pattern Miner).   This is a good first stab at measuring how much novelty there is in the dynamics of a codenet or compartment.

In a codenet containing some codelets representing Inputs and others representing Outputs, one can also calculate interaction information based only on the Input and Output codelets.  This is a measure of the surprisingness or informativeness of the codenet’s relevant external behaviors, rather than its internal dynamics.

Pattern Mining and Inference for Algorithmic Chemistry

Given the above approach to assessing interestingness, one can use a modification of OpenCog’s Pattern Miner to search for codenets that have surprising dynamics.   One can also, in this way, search for patterns among codenets, so that specific codenets fulfilling the patterns have surprising dynamics.   Such patterns may be expressed in the Atomspace, in terms of “abstract codelets” — codelets that have some of their internal Atoms represented as VariableNodes instead.

An “abstract codenet” may be defined as a set of (possibly-abstract codelet, count-interval, time-interval) triples, where the time-interval is defined as a pair (start, end), where start and end are defined as offsets from the initiation of the codenet.   The interpretation of such a triple is that (C, (m,n) (s,e)) means that some codelet instantiating abstract codelet C exists with count between m and n, during the time interval spanning from s to e.

Note that in order to form useful abstractions from codenets involving different codelets, it will be useful to be able to represent codelets in some sort of normal form, so that comparison of different codelets is tractable and meaningful.  This suggests that having the codelet language support some sort of Elegant Normal Form, similar to the Reduct library used in OpenCog’s MOSES subsystem, will be valuable.

Using the Pattern Miner to search for abstract codenets with high degrees of informational surprisingness, should be a powerful way to drive algorithmic chemistry in interesting directions.   Once one has found abstract codenets that appear to systematically yield high surprisingness, one can then use these to drive probabilistic generation of concrete codenets, and let them evolve in the algorithmic soup, and see what they lead to.

Furthermore, once one has abstract codenets with apparent value, one can apply probabilistic inference to generalize from these codenets, using deductive, inductive and abductive reasoning, e.g. using OpenCog’s Probabilistic Logic Networks.   This can be used to drive additional probabilistic generation of concrete codenets to be tried out.

“Mutation” and “crossover” of codenets or codelets can be experimented with on the inferential level as well – i.e. one can ask one’s inference engine to estimate the likely interestingness of a mutation or crossover of observed codenets or codelets, and then try out the mutation or crossover products that have passed this “fitness estimation” test.

This kind of pattern mining and inference certainly will be far from trivial to get right.   However, conceptually, it seems a route with a reasonably high probability of surmounting the fundamental difficulties faced by earlier work in artificial life and algorithmic chemistry.   It is something conceptually different than “mere self-organization” or “mere logical reasoning” – it is Alife/Achem-style self-organization and self-modification, but explicitly guided by abstract pattern recognition and reasoning.   One is doing symbolic AI to accelerate and accentuate the creativity of subsymbolic AI.

The above pertains to the case where one is purely trying to create algorithmic soups with interesting internal dynamics and structures.   However, it applies also in cases where one is trying to use algorithmic chemistry to learn effective input-output functions according to some fitness criteria.   In that case, after doing pattern mining of surprising abstract codenets, one can ask a different question: Which codenets, and which combinations thereof, appear to differentially lead to high-fitness transformation of inputs into outputs?   One can then generate new codenets from the distribution obtained via answering this question.   This is an approach to solving the “assignment of credit” problem from a “God’s eye” point of view – by mining patterns from the network over time … a fundamentally different approach to assignment of credit than has been taken in subsymbolic AI systems in the past.

Desirable Properties of a Cogistry Codelet Language

Designing the right language for the above general “Cogistry” approach is a subtle task, and I won’t try to do so fully here.   I’ll just sketch some ideas and possible requirements.

Fontana’s original algorithmic chemistry work uses a variety of LISP, which seems a sound and uncontroversial choice (and the same choice we made in OpenCog’s MOSES GA-EDA tool, for example).   However, a few variations and additions to this basic LISP-ish framework seem potentially valuable:

  • Addition of unordered sets as primitives, alongside ordered lists; and then associated set operations
  • Addition of float-weighted lists and sets as primitives; and of basic vector operations acting on such lists, such as the dot product, and the multiplication of a vector by a scalar.

Float-weighted lists are very handy for dealing with perceptual data, for example.   They also provide an element of continuity, which may help with robustness.   Codelets relying on float vectors of weights can be modified slightly via modifying the weights, leading to codelets with slightly different behaviors – and this continuity may make learning of new codelets via sampling from distributions implied by abstract codenets easier.

Further, it seems to me we may want to make all operations reversible.  If the atomic operations on bits, ints and floats are reversible, then corresponding operations on lists and sets and weighted vectors can also easily be made reversible.  (For instance, removing the final element of a list can be made into a reversible operation by, instead, using an operation that splits a list into two parts: the list with the final element removed, and the final element itself.)   The intuition here is that reversibility introduces a kind of “conservation of information” into the system, which should prevent the advent of various sorts of pathological runaway dynamics like Fontana observed in his early simulations.   If codelets can produce “more” than they take in, then evolution will naturally lead to codelets that try to exploit this potential and produce more and more and more than they take in.   But if codelets have to be “conservative” and essentially act only by rearranging their inputs, then they have to be cleverer to survive and flourish, and are more strongly pushed to create complexly interlocking self-organizing structures.

I’m well aware that mining patterns among functions of float variables is difficult, and it would be easier to restrict attention to discrete variables – but ultimately I think this would be a mistake.   Perceptual data seems to be very naturally represented in terms of float vectors, for example.   Perhaps an innovative approach will be needed here, e.g. instead of floats one could use confidence intervals (x% chance of lying in the interval (L,U) ).   Reversible division on confidence intervals would require a bit of fiddling to work out, but seems unlikely to be fundamentally difficult.

Whoaaaoowwawwww!!

The idea of Cogistry seemed very simple when I initially thought of it; but when I sat down to write this post summarizing it, it started to seem a lot more complicated.  There are a lot of moving parts!   But, hey, nobody said building a thinking machine and creating digital life in one fell swoop was going to be easy….   (Well actually OK, some people did say that… er, actually, I did say that, but I took it back a decade and a half ago!! …)

What I like about the idea of Cogistry, though, is that – if it works (and I’m sure it will, after a suitable amount of research and fiddling!) — it provides a way to combine the fabulous generative creativity of biological systems, with the efficiency and precision of digital-computer-based pattern mining and probabilistic reasoning.   Such a combination has the potential to transcend the “symbolic versus subsymbolic” and “biological versus computational” dichotomies that have plagued the AI field since nearly its inception (and that indeed reflect deeper complex dichotomies and confusions in our contemporary culture with its mixture of information-age, machine-age, and biological/emotional/cultural-humanity aspects).   Some of the details are gonna be freakin’ complicated to work out but, though I realize it sounds a bit vain or whatever, I have to say I feel there is some profound stuff lurking here…

Notes Toward a Possible Development Plan

At first blush, it seems to me that most of the hard work here is either

  • A) testing and experimenting; or
  • B) re-using OpenCog AI tools that we need anyway for other reasons

From this perspective, an approach to making Cogistry real would be to start by

  • designing the programming language (with room for experimentation/variation) for codelets
  • implementing the basic framework in OpenCog
  • doing some basic experimentation with this as an Alife/Achem framework, without the added AI bells and whistles

Now, I would not expect this initial work to yield great results… since basically it’s a matter of reimplementing good old Alife/Achem stuff in a context where inference, pattern mining, ECAN etc. can be layered on.  Without the layering on of these AI tools, one would expect to find familiar Alife-y issues: some interesting structures emerging, but hitting a complexity ceiling … and then being uncertain whether increased scale or a change to the codelet language might be the key to getting more interesting things to emerge.

But beyond this basic framework, the other things needed for Cogistry are all things needed for other OpenCog AGI work anyway:

  • more general use of pattern mining than has been done so far
  • PLN on output of pattern mining
  • ECAN working together w/ pattern mining and PLN
  • probabilistic programming for guiding the forward chainer

With the basic codelet-system framework in place, using these things for Cogistry alongside their other uses would be “straightforward in principle”

— Thanks are due to Cassio Pennachin and Zar Goertzel for comments on an earlier version of the text contained in this post.

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