introduction to Genifer -- Deduction

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Introduction to Genifer– deduction

source code: deduction.lisp

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w h a t i sf i r s t - o r d e r l o g i c ?

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“ J o h n l o v e s M a r y .”

l o v e s ( j o h n , m a r y )

t h i s i s a p r o p o s i t i o na l s o c a l l e d a f o r m u l a , o r s e n t e n c e

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“ J o h n l o v e s M a r y .”

l o v e s ( j o h n , m a r y )

t h i s i s ap r e d i c a t e t h e s e a r e

a r g u m e n t s

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“ X l o v e s Y .”

l o v e s ( X , Y )

a r g u m e n t s c a n b ev a r i a b l e s

In my Lisp code variables are denoted by ?1, ?2, … etc

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First order logic has the following connectives and operators:/\ (AND)\/ (OR)┐ (NOT)→ (IMPLY)↔ (EQUIVALENCE)

For example A → B is equivalent to (┐A \/ B) and its truth table is:A B A→BT T TT F FF T TF F T

( I assume you still remember this stuff... )

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Horn form

If a bunch of formulae can be written as:A ← B /\ C /\ D /\ …B ← E /\ F /\ G /\ ...

and if A is the goal we want to solve:A

then we can cross out A and replace it withA, B, C, D, …

as the new sub-goals and repeat the process:B, C, D, ..., E, F, G, …

This makes solving the goals very efficient (similar to production systems in the old days).

Formulae written in this form is called the Horn formand is the basis of the language Prolog.

Genifer's logic is also based on Horn.

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* Optional:

Some first-order formulae cannot be expressed in Horn form.If we want to prove theorems in full first-order logic, we need touse a more general algorithm such as resolution.

The general procedure is:

1. convert all the formulae into CNF (conjunctive normal form)2. eliminate all existential quantifiers ∃ by a process called

Skolemization (see next slide)3. repeatedly apply the resolution rule to formulae in the KB,

until nothing more changes

If we restrict resolution to Horn formulae we get SLD-resolutionwhich is very fast and is the search procedure in Prolog.

I think Horn is expressive enough for making a first AGI prototype.

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First order logic has 2 quantifiers:

Universal: ∀X liar(X)≡ for all X, X is a liar ≡ “Everyone is a liar”.

Existential: ∃X brother(john, X)≡ exists X such that X is John's brother

The existential quantifier can be eliminated by Skolemization:∃X brother(john, X) “Exists X such that X is John's bro”≡ brother(john, f(john))

where f() is called a Skolem function. Its purpose is to map X to X's brother.

With existential quantifiers eliminated, we can assume all variables are implicitly universally quantified, and omit the ∀'s.

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When we have a goal to prove...

goal G

KB

knowledge base

We try to fetch facts and rulesfrom the KB to satisfy the goal.

eg: bachelor(john) ?

(defined in memory.lisp)

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Deduction can start from the goal to the facts. This is known as backward chaining.

goal

each goal has a bunch of rulesthat may apply to itrule1 rule2 rule3

each rule can be a conjunction(/\) of arguments;

arguments are just other goals

A B

G

. . . .

rules and goals alternative down theproof tree

rules

goals

rules

goals

The red link indicates a conjunction (“AND”) that requires all its arguments; The black links represent “OR” (ie, the goal G can be solved by applying either rule1, rule2, rule3, ... etc.

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rule1 rule2 rule3

A B

G

. . . .

1

5 3 2

4 6

. . . .1 2 3

The simplest search is depth-first search but it may not be flexible enough. I think best-first search may be better, which uses a priority queue to rank the nodes.

Each element in the priority queuepoints to a tree node.

priorityqueue

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rule1 rule2 rule3

A B

G

. . . .

fact1 rule1fact2 fact3

A goal / sub-goal can be satisfied by either facts or rules.

example of a fact:bachelor(john)

example of a rule:bachelor(?1) ← male(?1) /\ single(?1)

this is a variable A B

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rule

Q R

P

. . . .

fact fact

bachelor(?1)

bachelor(john)

To satisfy a goal, some variable substitutions occur.

We need to make these 2 termsidentical; this is done by analgorithm known as unification.

The result of unification isa set of substitutions,for example: {?1/john, ...}

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rule

P

fact fact

bachelor(?1)

bachelor(john) bachelor(pete)

. . . .

A successful matching results in a new set of substitutions. Over time, a goal node can acquire multiple sets of substitutions.

{?1/john}{?1/pete}{?1/paul} … etc

these are 3substitution

sets

stored in the node

Managing these set of substitutions can be a great source of complexity,as the next few slides illustrate...

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Each substitution set is associated with a truth value (TV).

{?1/john}

{?1/pete}

{?1/paul}

… etc

TV1

TV2

TV3

...

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rule

Q R

P

fact fact


{?1/mary}{?1/paul}{?1/john} … etc

male(john),male(pete),male(paul), … etc

single(mary),single(paul),single(john), … etc

male(?1) single(?1)

bachelor(?1) ← male(?1) /\ single(?1)

facts from KBthat matchesthe sub-goals

These 2 blocks ofsubstitution-setsmust be mergedto give the resultto the parent node.

When handling substitution setsof a conjunction /\ ...

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rule

Q R



Each set on the left block has to bematched with each set of the right block.

for example: {?1 / john} {?1 / mary} fail {?1 / john} {?1 / john} match

Because the variable ?1can only take on one valueon each instance.

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The merging of substitution-sets is handled in the function propagate. Instead of trying all possible combinations between 2 blocks, we try to do the matching when a new substitution-set arrives. When we get a new set, say {?1 / john}, we try to merge it with each set in the other existing blocks:

{?1 / mary }{?1 / sam } {?1 / paul } {?1 / john } .... etc

A B

{?1/sam}

new resultfrom unification

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rule

Q R

Substitution-sets propagate upwardsalong the proof tree.

....

....

....

parent's block ofsubstitution sets

This part of the Lispcode may still needdeveloping.

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The Lisp code contains these functions:

backward-chain main function,manages the priority queue

process-subgoalprocess-fact builds up the proof tree during searchprocess-rule

retract deletes a dead proof tree node

propagate propagates TVs up the proof tree

TO-DO:* some comments are excessive

Technology

introduction to Genifer -- Deduction