- The difference between propositional and predicate logic
- How to make logic inference fast

## What is the difference between propositional logic and predicate logic?

For example, "rainy", "sunny", and "windy" are**propositions**. Each is a complete sentence (consider, eg "it is sunny today") that describe a truth or fact, and we assume the sentence

__has no internal structure__.

In contrast,

**predicate logic**allows internal structure of propositions in the form of

**predicates**and

**functions**acting on

**objects**. For example: loves(john, mary) where "loves" is a predicate and "john", "mary" are objects. They are called

**first-order**objects because we can use the quantifiers $\forall X, \exists X$ to quantify over them.

**Second-order**logic can quantify over predicates and functions, and so on.

An example of a

*propositional*machine learning algorithm is

**decision tree**learning. This is the famous example:

All the entries are given as data, the goal is to learn to predict whether to play tennis or not, given combinations of weather conditions that are

*unseen*before.

This would be the learned result:

The same data can also be learned by neural networks, and support vector machines. Why? Because we can encode the same data spatially like this:

where the decision boundary can be learned by a neural network or SVM. This is why I say that decision trees, ANNs, SVMs, are all

**classifiers.**

*spatial*The situation is very different in the predicate logic /

**relational**setting. For example, look at this graph where an arrow $X \rightarrow Y$ indicates "$X$ loves $Y$":

That is a sad world. This one is happier:

As a

**digression**, sometimes I wonder if a multi-person cycle can exist in real life (eg, $John \rightarrow Mary \rightarrow Paul \rightarrow Anne \rightarrow John$), interpreting the relation as purely sexual attraction. Such an example would be interesting to know, but I guess it's unlikely to occur. </digression>

So the problem of relational learning is that

*data points cannot be represented in space with fixed labels*. For example we cannot label people with + and -, where "+ loves -", because different persons are loved by different ones. At least, the naive way of representing individuals as points in space does not work.

My "matrix trick" may be able to map propositions to a high-dimensional vector space, but my assumption that concepts are matrices (that are also rotations in space) may be unjustified. I need to find a set of criteria for matrices to represent concepts faithfully. This direction is still hopeful.

## How to make logic inference fast?

### DPLL

The current fastest

__propositional__SAT solvers are variants of

**DPLL**(Davis, Putnam, Logemann, Loveland). Davis and Putnam's 1958-60 algorithm was a very old one (but not the earliest, which dates back to Claude Shannon's 1940 master's thesis).

### Resolution + unification

Robinson then invented

**resolution**in 1963-65, taking ideas from Davis-Putnam and elsewhere. At that time first-order predicate logic was seen as the central focus of research, but not propositional SAT. Resolution works together with

**unification**(also formulated by Robinson, but already in Herbrand's work) which enables it to handle first-order logic.

Resolution + unification provers (such as Vampire) are currently the fastest algorithm for theorem proving in first-order logic. I find this interesting because resolution can also be applied to propositional SAT, but it is not the fastest, which goes to DPLL. It seems that

*unification*is the obstacle that makes first-order (or higher) logic more difficult to solve.

### What is SAT?

For example, you have a proposition $P = (a \vee \neg b) \wedge (\neg a \vee c \vee d) \wedge (b \vee d)$ which is in CNF (conjunctive normal form) with 4 variables and 3 clauses. A

**truth assignment**is a map from the set of variables to {$\top, \bot$}. The

**satisfiability**problem is to find any assignment such that $P$ evaluates to $\top$, or report failure.

Due to the extreme simplicity of the structure of SAT, it has found connections to many other areas, such as integer programming, matrix algebra, and the Ising model of spin-glasses in physics, to name just a few.

The

**DPLL algorithm**is a top-down search that selects a variable $x$ and recursively search for satisfaction of $P$ with $x$ assigned to $\top$ and $\bot$ respectively. Such a step is called a

**branching decision**. The recursion gradually builds up the partial assignment until it is complete. Various heuristics are employed to select variables, cache intermediate results, etc.

Edit: DPLL searches in the space of assignments (or "interpretations", or "models"). Whereas [propositional] resolution searches in the space of

*resolution proofs*. I have explained resolution many times elsewhere, so I'll skip it here. Basically a resolution proof consists of steps of this form:

$$ \frac{P \vee Q, \quad \neg P \vee R} {Q \vee R} $$

As you can see, it is qualitatively different from DPLL's search.

### How is first-order logic different?

It seems that unification is the obstacle that prevents propositional techniques from applying to first-order logic.

*So what is the essence of unification?*For example, if I know

"all girls like to dress pretty" and

"Jane is a girl"

I can conclude that

"Jane likes to dress pretty".

This very simple inference already contains an implicit unification step. Whenever we have a logic statement that is

**general**, and we want to apply it to yield a more

**specific**conclusion, unification is the step that

*matches*the specific statement to the general one. Thus, it is

*unavoidable*in any logic that has

**generalizations**. Propositional logic does not allow quantifications (no $\forall$ and $\exists$), and therefore is not

**compressive**. The compressive power of first-order logic (or above) comes from quantification and thus unification.

### Transform logic inference to an algebraic problem?

I have found a nice algebraic form of logic that has the expressive power of higher-order logic and yet is even simpler than first-order logic in syntax. My hope is to transform the problem of logic inference to an algebraic problem that can be solved efficiently. So the first step is to recast logic inference as algebra. Category theory may help in this.

### Unification in category theory

In category theory, unification can be construed as

**co-equalizer**(see Rydeheard and Burstall's book

*Computational Category Theory*). Let

**T**be the set of terms in our logic. A substitution $\theta$ is said to unify 2 terms $(s,t)$ if $\theta s = \theta t$. If we define 2 functions $f, g$ that points from

**1**to

**T**, so that $f(1) = s$ and $g(1) = t$, then the

**most general unifier**(mgu) is exactly the co-equalizer of $f$ and $g$ in the category

**T**.

The book also describes how to model equational logic in categories, but the problem is that the formulas in an AGI are mainly simple propositions that are not equational. How can deduction with such propositions be recast as solving algebraic equations....?

wow, thanks for the guide

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