Symbolic Logic Foundations: An Overview 1/22/01 Logic (Programming) & AI Selmer Bringsjord selmer@rpi.edu www.rpi.edu/~brings

Logic Programming: Two Perspectives Logic Programming as arising from Herbrand’s Theorem, etc. Logic Programming as using a logical system (in mathematical sense of this phrase) I will take second perspective, which subsumes first E.g., completeness theorem for first-order logic (LI) allows one to affirm Herbrand’s Theorem This theorem fully done in LCU What you need to know to understand second perspective is precisely what you need to know to understand first

Logical Systems (Are Programming Langs in here?) Name Alphabet Grammar Proof Theory Semantics Metatheory LPC Propositional Calculus p, q, r, … and truth-functional connectives Easy Fitch-style and natural deduction Truth tables! Sound, complete, compact, decidable LI First-Order Logic Add variables x, y, … and   Structures and interpretations Sound, complete, compact, undecidable LPML Add “box” and “diamond” for necessity and possibility Wffs created by prefixing new operators to wffs Add necessitation, etc. Possible worlds Same as PC LII New variables for predicates Pretty obvious New adapt quantifier rules Quantification over subsets in domain allowed Sound but not complete

Readings AIMA “Natural Deduction” on Pollock’s web site OTTER manual “Logic and AI: Divorced, Still Married…” http://kryten.mm.rpi.edu/COURSES/ILOGPROG/lai.ed2.pdf LCU http://www.rpi.edu/~faheyj2/SB/LCU/lcu.driver.pdf

LPC (Propositional Calculus) Where we left off: Logic Theorist problems in OTTER… Ad lib in HYPERPROOF… Some problems NYS 1, NYS 2, NYS 3, J-L 1 Semantics of Propositional Calculus: Truth Tables Boole Via HYPERPROOF Full formal view: LCU

“NYS 1” Given the statements a  b b c  a which one of the following statements must also be true? c b c h a none of the above

“NYS 2” Which one of the following statements is logically equivalent to the following statement: “If you are not part of the solution, then you are part of the problem.” If you are part of the solution, then you are not part of the problem. If you are not part of the problem, then you are part of the solution. If you are part of the problem, then you are not part of the solution. If you are not part of the problem, then you are not part of the solution.

“NYS 3” Given the statements c c  a a  b b  d (d  e) which one of the following statements must also be true?  e c e h a all of the above

J-L 1 Suppose that the following premise is true: If there is a king in the hand, then there is an ace in the hand, or else if there isn’t a king in the hand, then there is an ace. What can you infer from this premise? There is an ace in the hand. NO! NO! In fact, what you can infer is that there isn’t an ace in the hand!

Proof Theory of LI (First-order logic) Ad lib in HYPERPROOF Syllogisms in OTTER Dreadsbury Mansion Mystery The Bird Problem

The Dreadsbury Mansion Mystery Someone who lives in Dreadsbury Mansion killed Aunt Agatha. Agatha, the butler, and Charles live in Dreadsbury Mansion, and are the only people who live therein. A killer always hates his victim, and is never richer than his victim. Charles hates no one that Aunt Agatha hates. Agatha hates everyone except the butler. The butler hates everyone not richer than Aunt Agatha. The butler hates everyone Agatha hates. No one hates everyone. Agatha is not the butler. Now, given the above clues, there is a bit of disagreement between three (competent?) Norwegian detectives. Inspector Bjorn is sure that Charles didn’t do it. Is he right? Inspector Reidar is sure that it was a suicide. Is he right? Inspector Olaf is sure that the butler, despite conventional wisdom, is innocent. Is he right? Can you get it, prove it?

Good litmus test for mastery of proof theory in FOL The Bird Problem Is the following assertion true or false? Prove that you are correct. There exists something which is such that if it’s a bird, then everything is a bird. x(B(x) yB(y)) Good litmus test for mastery of proof theory in FOL

Metatheory for PC and FOL Soundness If you start with true propositions in an agent’s knowledge base, deduction from that kb will always yield a true conclusion. Completeness If something intuitively “follows from” a given kb, then the agent can prove it from the kb. Decidability & undecidability If Dec: There is a decision algorithm which can tell you whether a given formula is a theorem. Compactness Not today Herbrand’s Theorem etc. Godel’s Theorem LII not complete Lindstrom’s Theorems Already characterized intuitively at start of lecture