[Sequence] Analysis

Topology posts

Calculus posts

  • An Introduction to Derivatives
  • An Introduction to Integrals

An Introduction to Topology

Part Of: Analysis sequence
Content Summary: 1000 words, 10 min read

Motivating Example

Can you draw three lines connecting A to A, B to B, and C to C?  The catch: the lines must stay on the disc, and they cannot intersect.

Topology- Motivating Problem (2)

Here are two attempts at a solution:

Topology- Potential Solutions (1)

Both attempts fail. In the first, there is no way for the Bs and Cs to cross the A line. In the second, we have made more progress… but connecting C is impossible.

Does any solution exist? It is hard to see how…

Consider a simplified puzzle. Let’s swap the inner points B and C.

Topology- Original vs Easy Puzzle (2)

In the new puzzle, the solution is easy: just draw straight lines between the pairs!

To understand where this solution breaks down, let’s use continuous deformation (i.e., homeomorphism) to transform this easier puzzle back to the original. In other words, let’s swap point B towards C, while not dropping the “strings” of our solution lines:


Deformation has led us to the solution! Note what just happened: we solved an easy problem, and than “pulled” that solution to give us insight into a harder problem.

As we will see, the power of continuous deformation extends far beyond puzzle-solving. It resides at the heart of topology, one of mathematics’ most important disciplines.

Manifolds: Balls vs Surfaces

The subject of arithmetic is the number. Analogously, in topology, manifolds are our objects. We can distinguish two kinds of primitive manifold: balls and surfaces.

Topology- Balls and Surfaces (1)

These categories generalize ideas from elementary school:

  • A 1-ball B^1 is a line segment
  • A 2-ball B^2 is a disc
  • S^1 is a circle
  • S^2 is a sphere

Note the difference between volumes and their surfaces. Do not confuse e.g., a disc with a circle. The boundary operation \partial makes the volume-surface relationship explicit. For example, we say that \partial B^2 = S^1.

Note that surfaces are one dimension below their corresponding volume. For example, a disc resides on a plane, but a circle can be unrolled to fit within a line.

Importantly, an m-ball and an m-cube are considered equivalent! After all, they can be deformed into one another. This is the reason for the old joke:

A topologist cannot tell the difference between a coffee cup and a donut. Why? Because both objects are equivalent under homeomorphism:


If numbers are the objects of arithmetic, operations like multiplication act on these numbers. Topological operations include product, division, and connected sum. Let us address each in turn.

On Product

The product (x) operation takes two manifolds of dimension m and n, and returns a manifold of dimension m+n. A couple examples to whet your appetite:

Topology- Examples of Product (1)

These formulae only show manifolds of small dimension. But the product operation can just as easily construct e.g. a 39-ball as follows:

B^{39} = \prod_{i=1}^{39} I^1

How does product relate to our boundary operator? By the following formula:

\partial (M x N) = ( \partial M x N) \cup (M x \partial N )

This equation, deeply analogous to the product rule in calculus, becomes much more clear by inspection of an example:
Topology- Product vs Boundary (1)

On Division

Division ( / ) glues together the boundaries of a single manifold. For example, a torus can be created from the rectangle I^{2}:


We will use arrows to specify which edges are to be identified. Arrows with the same color and shape must be glued together (in whatever order you see fit).

Topology- Division Simple Examples (2)

Alternatively, we can specify division algebraically. In the following equation, x=0 means “left side of cylinder” and x=1 means right side:

S^1 x I^1 = Cylinder = \frac{I^2}{(0,y) \sim (1, y) \forall y}

The Möbius strip is rather famous for being non-orientable: it neither has an inside nor an outside. As M.C. Escher once observed, an ant walking on its surface would have to travel two revolutions before returning to its original orientation.

More manifolds that can be created by division on I^{2}. To construct a Klein bottle by division, you take a cylinder, twist it, and fold it back on itself:

Topology- Klein Bottle Construction (5)

In our illustration, there is a circle boundary denoting the location of self-intersection. Topologically, however, the Klein bottle need not intersect itself. It is only immersion in 3-space that causes this paradox.

Our last example of I^{2} division is the real projective plane RP^{2}. This is even more difficult to visualize in 3-space, but there is a trick: cut I^{2} again. As long as we glue both pieces together along the blue line, we haven’t changed the object. 

Topology- Deriving Real Projective Plane First Part (1)

The top portion becomes a Möbius strip; the bottom becomes a disc. We can deform a disc into a sphere with a hole in it. Normally, we would want to fill in this hole with another disc. However, we only have a Möbius strip available.

But Möbius strips are similar to discs, in that its boundary is a single loop. Because we can’t visualize this “Möbius disc” directly, I will represent it with a wheel-like symbol.  Let us call this special disc by a new name: the cross cap.

The real projective plane, then, is a cross cap glued into the hole of a sphere.  It is like a torus; except instead of a handle, it has an “anomaly” on its surface.

Topology- Deriving Real Projective Plane Second Part

These then, are our five “fundamental examples” of division:

Topology- Division Overview (3)

On Connected Sum

Division involves gluing together parts of a single manifold. Connected sum (#), also called surgery, involves gluing two m-dimensional manifolds together. To accomplish this, take both manifolds, remove an m-ball from each, and identify (glue together) the boundaries of the holes. In other words:

\frac{ ( M_1 / B_1 ) \cup ( M_2 / B_2 ) }{ \partial ( M_1 / B_1 ) \sim \partial ( M_2 / B_2 )} = M_1 \# M_2

Let’s now see a couple examples. If we glue tori together, we can increase the number of holes in our manifold. If we attach a torus with a real projective plane, we acquire a manifold with holes and cross-cuts.
Topology- Connected Sum examples (3)


  • In topology, manifolds represent objects in n-dimensional space.
  • Manifolds either represent volumes (e.g., disc) and boundaries (e.g., circles)
  • Manifolds are considered equivalent if a homeomorphism connects them.
  • There are three basic topological operations:
    • Product (x) is a dimension-raising operation (e.g., square can become a cube).
    • Division (/) is a gluing operation, binding together parts of a single manifold.
    • Connected sum (#) i.e., surgery describes how to glue two manifolds together.

The X-Bar Theory of Phrase Structure

Part Of: Language sequence
Followup To: An Introduction to Generative Syntax
Content Summary: 800 words, 8 min read

Explaining Substitution

Consider the sentence “I bought this big book of poems with the red cover”.

XBar- Flat Noun Phrase (1)

In everyday language, we often replace words and phrases with indexing words like “one”. Call this indexing replacement.The meaning of these words can be obtained from the context.

At first glance, indexing replacement seems to target a branch in the syntax tree. For example:

  • I bought that big one of poems with the red cover (“one” replaces the noun)
  • I bought one (“one” replaces the entire noun phrase)

But there are several other substitutions don’t follow from branch replacement:

  • I bought that big one.
  • I bought that small one
  • I bought that big one of poems with the blue cover

Perhaps our notion of noun phrases is too flat. Perhaps we need additional nodes to describe structure within the noun phrase. We will call these intermediate nodes N’, (where N → N’ → N’’ = NP):


This new tree successfully predicts all substitution phenomena, by modeling “one” as replacing various “N-bar” nodes:


We can similarly introduce depth to our verb phrases (VPs), by using intermediate V’ (“V-bar”) nodes:

XBar- Verb Substitution (2)

The X-Bar syntax tree provides a simple explanation of the “do so” substitution effects:

  • I will do so in the office before the party.
  • I will do so before the party.
  • I will do so.

A General Theory of Phrases

We can revise our original NP and VP rules to reflect our intermediate N’ and V’ nodes:

Xbar Theory- Towards XBar Rules

What if noun and verb phrases are instantiations of a more general phrase structure? Just as group theory identifies overlap in the axioms of addition and subtraction, X-bar theory explores the similarity between NP and VP rules.  

Xbar Theory- XBar Parameterization (1)

There are only four kinds of phrase constituents:

  1. The head carries the central meaning of the phrase. Consider the sentence “The tall student who is wearing the red shirt asked questions of her professor, after the lecture.” The central meaning is retained if we remove all non-head words: “student asked questions”.
  2. The specifier points to the head. For nouns, specifiers include determiners (“the”) and possessives (“her”). For verbs, adverbs occasionally fill this role (“quickly”).
  3. The complement tends to feel intimately related to the head of a phrase (e.g., “of poems” in “a book of poems”).
  4. Adjuncts, on the other hand, tend to feel more optional (e.g., “big” in “big book”).

Xbar Theory- Phrase Structure

Adjuncts vs Complement

Given that adjuncts and complements both often inhabit prepositional phrases, it is perhaps surprising that they should behave differently. The distinction between adjuncts and complements explains why this should be the case. Let us look at four behavioral differences:

Difference #1. Adjuncts can be reordered freely. 

Consider our example verb phrase:

Xbar Theory- VBar and NBar Example

This rule means that our two adjuncts can be shuffled, but the complement NP must retain its original position

  • I will read the letter in the office before the party (Original order: valid)
  • I will read the letter before the party in the office (Adj reorder: valid)
  • *I will read in the office before the party the letter (Compl reorder: invalid)

Difference #2. Indexing replacement cannot strand the complement.

For example,

  • I will do so in the office before the party (Adj is stranded: valid)
  • *I will do so the letter before the party (Compl is stranded: invalid)

Consider another part of speech we have not yet considered: conjunction words like “and” and “or”.

Difference #3. Conjunction words bind adjuncts together, and complements together. But adjunct-complement bindings are non-grammatical.

Consider our example noun phrase:

Xbar Theory- NBar Example (1)

Three examples to illustrate how conjunction works:

  • I bought the book of poems and of short stories. (Compl-compl conjunction: valid)
  • The book with the red cover and the black spine. (Adj-adj conjunction: valid)
  • *The book of poems and with the red cover. (Compl-adj conjunction: invalid)

What X-Bar Theory Tells Us About Memory

Earlier, I introduced the distinction between episodic and semantic memory:

  • Semantic: ability to remember facts and concepts (e.g., hands have five fingers)
  • Episodic: ability to remember events or episodes (e.g., dinner last Tuesday night)

Concepts are learned by extracting commonalities from episodic memories. If you see enough metallic blocks moving around on four cylinders, you’ll eventually consolidate these objects into the CAR concept:

XBar Theory- Semantic vs Episodic Memory

In philosophy, I suspect the concepts of necessity and contingency relate to semantic and episodic memory, respectively.

In linguistics, I suspect complements help locate concepts in semantic memory, whereas adjuncts assist episodic localization. In the sentence “I bought the book of poems with the red cover”, the complement helps us activate the concept POEM-BOOK, whereas the adjunct creates sense-predictions that locate it within our episodic memory.


  • With flat syntax trees, it is difficult to explain indexing substitution (e.g., “bought a book” → “bought one”)
  • If we make syntax trees binary, by introducing intermediate  X’ (“X-Bar”) nodes, substitution becomes more straightforward.
  • Noun and verb phrases thus parameterize a more general phrase structure.
  • Phrases have four kinds of constituents: head, specifier, complement, and adjuncts.
  • The differences between complements and adjuncts are instructive:
    • Only adjuncts can be reordered.
    • Indexing replacement cannot strand the complement.
    • Conjunction cannot bind across categories
  • In human cognition, complements and adjuncts may correspond to semantic and episodic memory, respectively.

An Introduction to Generative Syntax

Part Of: Language sequence
Content Summary: 900 words, 9 min read

Syntax vs Semantics

In language, we distinguish between syntax (structure) and semantics (meaning).

Compare the following:

  • “Colorless green ideas sleep furiously”
  • “Sleep ideas colorless green furiously”

Both sentences are nonsensical (a semantic transgression). But the first is grammatically correct, whereas the second is malformed.

The brain responds differently to errors of syntax and semantics, as measured by an EEG machine. Semantic errors produce a negative voltage after 400 milliseconds (“N400”); syntactic errors produce a positive voltage after 600 milliseconds (“P600”):

Syntax- Linguistic ERPs (1)

Parts of Speech

To understand syntax more precisely, we must differentiate parts of speech. Consider the following categories:

  • Noun (N).  cat, book, computer, peace, …
  • Verb (V). jump, chase, eat, sleep, …
  • Adjective (A). long, purple, young, old, …
  • Determiner (D) the, this, many, all, …
  • Preposition (P) in, on, to, for, with…

Nouns and verbs correspond to perception- and action- representations, respectively. They are an expression of the perception-action cycle. But to study syntax, it helps to put aside semantic context, and explore how parts of speech relate to one another.

Phrases as Color Patterns

To understand syntax intuitively, start by adding color to sentences.  Then try to find patterns of color unique to well-formed sentences.

Let’s get started!

Syntax- Noun Phrase Abstraction (3)

“Noun-like” groups of words appear on either side of the verb. Let noun phrase (NP) denote such a group. Optional parts of speech are indicated by the parentheses. Thus, our grammar contains the following rules:

  1. S → NP V NP
  2. NP → (D) (A) N

These rules explain why the following sentences feel malformed:

  • “Chase dogs cats” (violates rule 1)
  • “Old some dogs chase cats” (violates rule 2)

But these rules don’t capture regularities in how verbs are expressed. Consider the following sentences:

Syntax- Verb Phrase Abstraction (1)

A verb phrase contains a verb, optionally followed by a noun, and/or a preposition.

  1. S → NP VP
  2. NP → (D) (A) N
  3. VP → V (NP) (P NP)

This is better. Did you notice how we improved our sentence (S) rule? 🙂 Subject-only sentences (e.g. “She ran”) are now recognized as legal.

Prepositions are not limited to verb phrases, though. They also occur in noun phrases. Consider the following:

Syntax- Prepositional Phrase Abstraction

Prepositions are sometimes “attached to” a noun phrase. We express these as a prepositional phrase, which includes a preposition (e.g. “on”) and an optional noun phrase (e.g. “the table”).

  1. S → NP VP
  2. NP → (D) (A) N (PP)
  3. VP → V (NP) (PP)
  4. PP → P (NP)

Notice how we cleaned up the VP rule, and improved the NP rule.

Congratulations! You have discovered the rules of English. Of course, a perfectly complete grammar must include determiners (e.g., “yours”), conjunction (e.g., “and”), interjection (e.g., “wow!”). But these are fairly straightforward extensions to the above system.

These grammatical rules need not only interest English speakers. As we will see later, a variant of these rules appear in all known human languages. This remarkable finding is known as universal grammar. Language acquisition is not about reconstructing syntax rules from scratch. Rather, it is about learning the parameters by which your particular natural language (English, Chinese, Egyptian) varies from the universal script.

From Rules to Trees

Our four rules are polymorphic: they permit more than one kind of structure. Unique rule sets are easier to analyze, so let’s translate our rules into this format:

Syntax- Compressed vs Unique Ruleset (1)


Importantly, we can conceive of these unique rules as directions to construct a tree. We can conceive of the sentence “Dogs chase cats” as:

Syntax- Simple Tree (1)

Sentences are trees. These trees are not merely used to verify whether grammatical correctness. They play a role in speech production: which transforms the language of thought (Mentalese) to natural language (e.g., English). For more on this, see my discussion of the Tripartite Mind.

How can (massively parallel) conscious thought be made into (painfully serial) speech utterances? With syntax! Simply take the concepts you desire to communicate, and construct a tree based on (a common set of) syntactical rules.


Tree construction provides much more clarity on the phenomena of wordplay (linguistic ambiguity). Consider the sentence “I shot a wolf in my pajamas”. Was the gun fired while you were wearing pajamas? Or was the wolf dressed in pajamas?

Syntax- Multiple Interpretation Ambiguity

Both interpretations agree on parts of speech (colors). It is the higher-order structure that admits multiple choices. In practice, semantics constrain syntax: we tend to select the interpretation is feels the most intuitive.

The Sociology of Linguistics

The above presentation uses a simple grammar, for pedagogic reasons. I will at some point explain the popular X’ theory (pronounced “X bar”), which explores similarities between different phrase structures (e.g., NP vs PP). Indeed, there is a wide swathe of possible grammars that we will explore.

Syntax- Sociology of Linguistic Research

Generative grammar is part of the Symbolist tribe of machine learning. As such, this field has rich connections with algebra, production systems, and logic. For example, propositional logic was designed as the logic of sentences; predicate logic is the logic of phrases.

Other tribes besides the Symbolists care about language and grammar, of course. Natural Language Processing (NLP) and computational linguistics have been heavily influenced by the Bayesian tribe, and use probabilitic grammars (i.e., PCFGs).

More recently, the Connectionist tribe (and deep learning technologies) are taking a swing at producing language. In fact, I suspect neural network interpretability will only be achieved once a Connectionist account of language production has matured.


  • Language can be understood via syntax (structure) and semantics (meaning).
  • Syntax requires delineating parts of speech (e.g., nouns vs verbs).
  • Parts of speech occur in patterns called phrases. We can express these patterns as the rules of syntax.
  • Sentences are trees. Syntax rules are instructions for tree construction.
  • Sentence-trees provide insight into problems like sentence ambiguity.

For more resources on syntax trees, I recommend this lecture, this website, and this Youtube channel.

Until next time.

Logic Inference: Sequent Calculus

Part Of: Logic sequence
Followup To: Natural Deduction
Content Summary: 600 words, 6 min read

Motivating Sequent Calculus

Last time, we labelled propositions in the language of verification.

  • ↑ represents conjecture: propositions that require verification
  • ↓ represents assumption: propositions that can be used for verification.

Two of our connective rules (⊃I and ∨E) expanded our set of assumptions, which we could use at any later time. Logic acumen is invoking the right assumption at the right time.

In contrast to natural deduction, sequent calculus explicitly tracks the set of assumptions) as they vary across different branches of the proof tree.

We will use the turnstile to distinguish assumptions from conjecture: { assumptions } ⊢ { conjectures }

In natural deduction, progress in bidirectional: we are done when we found a connection between assumptions and conjecture. In sequent calculus, progress is unidirectional. Instead, we start with no assumptions, and finish when we have no conjectures left to demonstrate.

Sequence Calculus- Different Schematics (1)

Both logical systems rely on two sets of five rules. They bear the following relationships:

  • R = I. Right rules are very similar to Introduction rules.
  • L = E-1. Left rules must be turned “upside down”.

Right and Left rules

We here define capital gamma Γ to represent the context, or current set of assumptions.

Right rules simply preface Introduction rules with  “Γ ⊢”. The exception ⊃R is instructive. There, A is added to the context, and our “target” conjecture shrinks to just B.
Sequent Calculus- Right vs Introduction (2)

Left rules are less transparently related to Elimination. They are more easily understood by an English explanation:

Sequent Calculus- Left Rule English Interpretation (2)

The entire structure of sequent calculus, then, looks like this:

Sequent Calculus- Left and Right Rules (1)

Enough theory! Let’s use sequent calculus to prove stuff.

Example 1: Implication

Show that (A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (B ⊃ C)).

Here, ⊃R serves us well:

Sequent Calculus- Implication Step0 (1)

We have parsed the jungle of connectives, and arrived at a clear goal. We need to prove C. How?

Recall what ⊃L means: “if you have assumed A ⊃ B, you may also assume B (right branch) if you can prove A with your current assumptions (left branch).

Let’s apply ⊃L to the A ⊃ B proposition sitting in our context. To save space, let us here define Γ with the following three elements: { A⊃(B⊃C), A⊃B, and A }.

Sequent Calculus- Implication Step1.png

We can solve the left branch immediately. Since A ∈ Γ, we can invoke the hyp rule.

Unfortunately, assuming B is not enough to prove C. We must invoke ⊃L again, this time against our A⊃(B⊃C) assumption.

Sequent Calculus- Implication Step2

And again, on our newfound B⊃C assumption.

Sequent Calculus- Implication Step3.png

Wait! By now our context by now contains A, B, and C. Each leaf of the proof tree is provable by hyp.

Sequent Calculus- Implication Step4

QED. It is instructive to compare this sequent calculus proof with the analogous natural deduction (which we solved together, last time).

Sequent Calculus- Implication Comparing Proofs

Example 2: Distributivity

Show that (A ∨ (B ∧ C)) ⊃ ((A ∨ B)  ∧ (A ∨ C)).

The first two steps here are straightforward. Simplify the conjecture string!

Sequent Calculus- Distributivity Step0 (1)

Note that Γ = { A ∨ (B ∧ C) }. Here, we use ∨L to split this assumption into two components:

Sequent Calculus- Distributivity Step1 (2)

We now have four conjectures to prove. Fortunately, each proof has become trivial:

Sequent Calculus- Distributivity StepF



In this post, we introduced sequent calculus (SC) as an alternative deductive calculus. Sequent calculus makes the notion of context (assumption set) explicit: which tends to make its proofs bulkier but more linear than the natural deduction (ND) style. The two approaches share several symmetries: SC right rules correspond fairly rigidly to ND introduction rules, for example.

If you want to learn sequent calculus for yourself, I recommend solving the converse problems to the two examples above. Specifically,

  • Given (A ⊃ B) ⊃ (B ⊃ C), show that A ⊃ (B ⊃ C).
  • Given (A ∨ B) ∧ (A ∨ C), show  that A ∨ (B ∧ C).

Until next time!


Logic Inference: Natural Deduction

Part Of: Logic sequence
Content Summary: 500 words, 5 min read


Logical systems like IPL have the following ingredients:

  • proposition is an atomic statement that can acquire a truth value.
  • connective takes atomic propositions, and melds them into a composite.

We can label propositions in the language of verification.

  • ↑ represents conjecture: propositions that require verification
  • ↓ represents knowledge: propositions that can be used for verification.

Introduction and elimination rules can be expressed in this language:

Logic Metalanguage- Original Rules (1)

Elimination rules tend to “point down”; introduction rules point up. Roughly, deduction involves applying such rules until the paths meet:

IPL Inference- Schematic

Enough theory! Let’s see how this works in practice.

Exercise One: Implication Exploration

Given A ⊃ (B ⊃ C), show that  (A ⊃ B) ⊃ (B ⊃ C).

We can visualize the challenge as follows. The red line indicates common knowledge.

IPL Inference- Implication Exploration Step0

First, let’s apply elimination on the premises:

IPL Inference- Implication Exploration Step1

Next, let’s apply introduction on the conclusion:

IPL Inference- Implication Exploration Step2.png

Are we done? No: we have not verified A↑ and B↑. If we had, they would have a red line over them.

To finish the proof, we need to invoke our introduction-rule assumptions.

IPL Inference- Implication Exploration Step3

Proving A↑ is trivial. Proving B↑ requires combining assumptions via elimination.

IPL Inference- Implication Exploration Step4 (2)

Done. 🙂 Good work!

Exercise for the Reader

Prove the converse is true. Given (A ⊃ B) ⊃ (B ⊃ C), show that A ⊃ (B ⊃ C).

Example 2: Distributivity

In arithmetic, distributivity refers to how addition and multiplication can interleave with one another. It requires that a + (b * c) = (a*b) + (a*c). For example:

  • 2 * (4+5) = 2 * 9 = 18
  • (2*4) + (2*5) = 8 + 10 = 18

Are logical conjunction and disjunction distributive? Let’s find out!

IPL Inference- Distributivity Exploration Step0 (1)

First, let’s introduce conjunction on the conclusion.

IPL Inference- Distributivity Exploration Step1

Here we reach an impasse. We need to introduce disjunction elimination on the premise. But what should we choose for C?

Let’s set C = A or B.

IPL Inference- Distributivity Exploration Step2.png

Filling in the gaps is straightforward. On the right, we eliminate conjunction and retain B. Then we introduce disjunction on both sides.

IPL Inference- Distributivity Exploration Step2 (1)

Here is where I originally got stuck. How can we use disjunction elimination?

The way forward becomes easier to grasp, when you remember:

  • We can use knowledge as many times as we like.
  • The symbols in the rule schematics are arbitrary.

Let’s set the arbitrary elimination symbol “C”  equal to A or C:

IPL Inference- Distributivity Exploration Step4 (1)

From here, the solution is straightforward.

IPL Inference- Distributivity Exploration Step5 (1)

Exercise for the Reader

Prove the converse is true. Given (A ⊃ B) ⊃ (B ⊃ C), show that A ⊃ (B ⊃ C).


In this post, we saw worked examples of deduction. Specifically:

  • Given A ⊃ (B ⊃ C),  show that  (A ⊃ B) ⊃ (B ⊃ C).
  • Given A ∨ (B ∧ C), show that (A ∨ B) ∧ (A ∨ C).

The best way to learn is practice. For the interested reader, I recommend these exercises:

  • Given (A ⊃ B) ⊃ (B ⊃ C), show that A ⊃ (B ⊃ C).
  • Given (A ∨ B) ∧ (A ∨ C), show  that A ∨ (B ∧ C).

In the latter exercise, you must also “get creative” on how to use disjunction elimination. Instead of choosing an arbitrary C, you must set A^B to a useful value.

… still stuck? Okay, see solution here. 🙂

Until next time.


Five Tribes of Machine Learning

Part Of: Machine Learning sequence
Content Summary: 900 words, 9 min read

ML is tribal, not monolithic

Research in artificial intelligence (AI) and machine learning (ML) has been going on for decades. Indeed, the textbook Artificial Intelligence: A Modern Approach reveals a dizzying variety of learning algorithms and inference schemes. How can we make sense of all the technologies on offer?

As argued in Domingos’ book The Master Algorithm, the discipline is not monolithic. Instead, five tribes have progressed relatively independently. What are these tribes?

  1. Symbolists use formal systems. They are influenced by computer science, linguistics, and analytic philosophy.
  2. Connectionists use neural networks. They are influenced by neuroscience.
  3. Bayesians use probabilistic inference. They are influenced by statistics.
  4. Evolutionaries are interested in evolving structure. They are influenced by biology.
  5. Analogizers are interested in mapping to new situations. They are influenced by psychology.

Expert readers may better recognize these tribes by their signature technologies:

  • Symbolists use decision trees, production rule systems, and inductive logic programming.
  • Connectionists rely on deep learning technologies, including RNN, CNN, and deep reinforcement learning.
  • Bayesians use Hidden Markov Models, graphical models, and causal inference.
  • Evolutionaries use genetic algorithms, evolutionary programming, and evolutionary game theory.
  • Analogizers use k-nearest neighbor, and support vector machines.

Five Tribes- Strengths and Technologies

In fact, my blog can be meaningfully organized under this research landscape.

History of Influence

Here are some historical highlights in the development of artificial intelligence.

Symbolist highlights:

  • 1950: Alan Turing proposes the Turing Test in Computing Machinery & Intelligence.
  • 1974-80: Frame problem & combinatorial explosion caused First AI Winter.
  • 1980: Expert systems & production rules re-animate the field. 
  • 1987-93: Expert systems too fragile & expensive, causing the Second AI Winter.
  • 1997: Deep Blue defeated reigning chess world champion Gary Kasparov.

Connectionist highlights:

  • 1957: Perceptron invented by Frank Rosenblatt.
  • 1968: Minsky and Papert publish the book Perceptrons, criticizing single-layer perceptrons. This puts the entire field to sleep, until..
  • 1986: Backpropagation invented, and connectionist research restarts.
  • 2006: Hinton et al publish A fast learning algorithm for deep belief nets, which rejuvinates interest in Deep Learning.
  • 2017: AlphaGo defeats reigning Go world champion, using DRL.

Bayesian highlights:

  • 1953: Monte Carlo Markov Chain (MCMC) invented. Bayesian inference finally becomes tractable on real problems.
  • 1968: Hidden Markov Model (HMM) invented.
  • 1988: Judea Pearl authors Probabilistic Reasoning in Intelligent Systems, and creates the discipline of probabilistic graphical models (PGMs).
  • 2000: Judea Pearl authors Causality: Models, Reasoning, and Inference, and creates the discipline of causal inference on PGMs.

Evolutionary highlights

  • 1975: Holland invents genetic algorithms.

Analogizer highlights

  • 1968: k-nearest neighbor algorithm increases in popularity.
  • 1979: Douglas Hofstadter publishes Godel, Escher, Bach.
  • 1992: support vector machines (SVMs) invented.

We can summarize this information visually, by creating an AI version of the Histomap:

Five Tribes- Historical Size and Competition (2)

These data are my own impression of AI history. It would be interesting to replace it with real funding & paper volume data.

Efforts Towards Unification

Will there be more or fewer tribes, twenty years from now? And which sociological outcome is best for AI research overall?

Theory pluralism and cognitive diversity are underappreciated assets to the sciences. But scientific progress is often heralded by unification. Unification comes in two flavors:

  • Reduction: identifying isomorphisms between competing languages,
  • Generalization: creating a supertheory that yields antecedents as special cases.

Perhaps AI progress will mirror revolutions in physics, like when Maxwell unified theories of electricity and magnetism.

Symbolists, Connectionists, and Bayesians suffer from a lack of stability, generality, and creativity, respectively. But one tribe’s weakness is another tribe’s strength. This is a big reason why unification seem worthwhile.

What’s more, our tribes possesses “killer apps” that other tribes would benefit from. For example, only Bayesians are able to do causal inference. Learning causal relations in logical structure, or in neural networks, are important unsolved problems. Similarly, only Connectionists are able to explain modularity (function localization). Symbolist and Bayesian tribes are more normative than Connectionism, which makes their technologies tend towards (overly?) universal mechanisms.

Symbolic vs Subsymbolic

You’ve heard of the symbolic-subsymbolic debate? It’s about reconciling Symbolist and Connectionist interpretations of neuroscience. But some (e.g., [M01]) claim that both theories might be correct, but at different levels of abstraction. Marr [M82] once outlined a hierarchy of explanation, as follows:

  • Computational: what is the structure of the task, and viable solutions?
  • Algorithmic: what procedure should be carried out, in producing a solution?
  • Implementation: what biological mechanism in the brain performs the work?

One theory, supported by [FP98] is that Symbolist architectures (e.g., ACT-R) may be valid explanations, but somehow “carried out” by Connectionist algorithms & representations.

Five Tribes- Tribes vs Levels (2)

I have put forward my own theory, that Symbolist representations are properties of the Algorithmic Mind; whereas Connectionism is more relevant in the Autonomic Mind.

This distinction may help us make sense for why [D15] proposes Markov Logic Networks (MLN) as a bridge between Symbolist logic and Bayesian graphical models. He is seeking to generalize these technologies into a single construct; in the hopes that he can later find a reduction of MLN in the Connectionist paradigm. Time will tell.


Today we discussed five tribes within ML research: Symbolists, Connectionists, Bayesians, Evolutionaries, and Analogists. Each tribe has different strengths, technologies, and developmental trajectory. These categories help to parse technical disputes, and locate promising research vectors.

The most significant problem facing ML research today is, how do we unify these tribes?


  • [D15] Domingos (2015). The Master Algorithm
  • [M01] Marcus (2001). The Algebraic Mind
  • [M82] Marr (1982). Vision
  • [FP98] Fodor & Pylyshyn (1998). Connectionism and cognitive architecture: A critical analysis