The document introduces the concepts behind the Genifer logic deduction source code. It discusses key concepts such as Horn clauses, unification, substitution sets, and backward chaining used in logical deduction. The code implements a backward chaining search using a priority queue to efficiently solve goals by applying facts and rules from a knowledge base. Managing substitution sets and propagating truth values during the search process introduces complexity that requires further development.

1. Introduction to Genifer – deduction source code: deduction.lisp

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

6. 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->B T T T T F F F T T F F T ( I assume you still remember this stuff... )

7. 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 with A, 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 form and is the basis of the language Prolog . Genifer's logic is also based on Horn.

8. * Optional: Some first-order formulae cannot be expressed in Horn form. If we want to prove theorems in full first-order logic, we need to use 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-resolution which is very fast and is the search procedure in Prolog. I think Horn is expressive enough for making a first AGI prototype.

9. 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.

10. When we have a goal to prove... goal G KB knowledge base We try to fetch facts and rules from the KB to satisfy the goal. eg: bachelor(john) ? (defined in memory.lisp )

11. Deduction can start from the goal to the facts .

12. This is known as backward chaining . The red link indicates a conjunction (“AND”) that requires all its arguments;

13. The black links represent “OR” (ie, the goal G can be solved by applying

14. either rule1, rule2, rule3, ... etc.

15. The simplest search is depth-first search but it may not be flexible enough.

16. I think best-first search may be better, which uses a priority queue to rank the nodes. Each element in the priority queue points to a tree node. priority queue

17. 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

18. To satisfy a goal, some variable substitutions occur. We need to make these 2 terms identical; this is done by a n algorithm known as unification . The result of unification is a set of substitutions , for example: {?1/john, ...}

19. A successful matching results in a

20. new set of substitutions .

21. Over time, a goal node can acquire