Notes

Outline

CS 371: Introduction to Artificial Intelligence First-Order Logic

Outline When Propositional Logic Isn’t Enough First-Order Logic (FOL) Practice with FOL FOL Proof Method: Resolution [Why Representing Change in Time is Difficult]

Blocks World Simple domain: Blocks stacked on a table Simple problem: Stack blocks in a given configuration Home to much research in a variety of areas: Vision (Huffman, 1971) Constraint Propogation (Waltz, 1975) Machine Learning (Winston, 1970) Natural Language Understanding (Winograd, 1972) Planning (Fahlman, 1974)

Propositional Block World How would you represent the following: Block A is on Block D Block A is not on Block B If A is on D, A is not on B No block is on A If A is on D and D is on B, A is above B.

When Propositional Logic Isn’t Enough Define Above using transitivity of On “There’s a/there’s no block on A.” “All blocks have red letters.” Need for properties, relationships, and ways to generalize information compactly Practical applications: mathematical theorem proving, common-sense reasoning (CYC image database), circuit/program design/verification

Propositional Logic Propositions - T/F atomic sentences about the world Boolean functions (Ø,Ù,Ú,Þ,Û) - negate and logically combine sentences Parentheses - disambiguates grammar

First-Order Logic Atomic sentences - more complex than propositions Boolean functions (Ø,Ù,Ú,Þ,Û) - make compound sentences Quantifiers (",$) - tell us first-order facts which correspond to sets of sentences Parentheses

Building Atomic Sentences: Symbols Domain of Discourse (DOD) - relevant things in the world that we’re reasoning about Symbols: variables - assigned things in the domain of discourse (DOD) constants - things and relations relations between things in the DOD An interpretation I maps variables to things in the DOD. constants to things and relationships between things in the DOD.

Blocks World Example Domain of Discourse: blocks (for simplicity, exclude table) Constant Symbols (A, B, C, D, On, Above, …) Variables (x, y, z, …) An interpretation maps variables and constant symbols to things in the world. (e.g. A means this block with an “A”.)

Building Atomic Sentences: Constant Symbols object constants - corresponding to things in the DOD relation constants - relationships between things in the DOD relations also called predicates n-ary relation - a set of n-tuples from the DOD 1-ary relation - a property constant

Blocks World Example Object Constants: A, B, C, D correspond to blocks with the same letters on them. Relations Constants: OnTable: set of blocks resting on table On: set of block pairs where the first rests on the second Above: transitive closure of On Clear: set of blocks with no blocks above

Building Atomic Sentences: Constant Symbols (cont.) function constants - mappings between tuples and things in the DOD 0-ary function - an object constant n-ary function (n>0) - special case of n+1-ary relation where n arguments uniquely determine n+1st argument

Atomic Sentences: Putting Symbols Together terms - variables, objects, functions of terms atomic sentence - relationship between terms Under a given interpretation, an atomic sentence is true or false compound (or complex) sentences are built of atomic sentences, Boolean functions (negation, connectives), and...

Blocks World Example Terms: x, A, nextBlock(A) Atomic sentences: Above(A,D), On(x,B), Clear(y) These sentence are true under an interpretation where constants and relations are as we described and x, y are interpreted as D, A or D, C. These sentences are also true under an interpretation where A,B,C,D are the numbers 16, 2, 17, 4, On is “the square of”,  Above is “a power of”, Clear is “greater than 15”.

Universal Quantifier universal (") - “For all” "x f - “For all x, f is true.” "Every block is a red block". "If x>y and y>z then x>z."

Existential Quantifier existential ($) - “There exists” $x f - “There exists x such that f is true.” same as Ø "x Ø f "There is a block on the table." Challenging: "If a block is above another block, then the block is on the other block or on a block that is above it."

Nested Quantifiers nested quantifiers - "x $y is different than $y "x "Everybody likes someone." "Someone likes everyone." Relations: Person, Time, FoolPersonAtTime "You can fool all the people some of the time, …" "… and some of the people all the time, …" "… but you cannot fool all the people all the time."

Compound (Complex) Sentences variable scope - the parts of a sentence for which a variable quantifier is relevant free variables - unquantified variables formula - another word for sentence well-formed formula (wff) - a sentence with no free variables

First-Order Logic Semantics

In-Class Exercise Express the following in FOL using relations On, Above, Clear, OnTable, and objects A, B, C, D: If block A is on block B, then A is above B, B is not clear, and A is not on the table. If one block is on another block, it is also above that block. A or D or C is on the table. If A is on the table, then D and C are not.

Validity, Satisfiability, Models A sentence is valid if it is true for all possible interpretations. A sentence is satisfiable if there exists a “satisfying” interpretation (i.e. an interpretation for which the sentence is true). An interpretation which satisfies a sentence is a model of that sentence.

Propositional KB and Inference entailment - whether a sentences follows from other sentences inference rules - deduce new sentences from old sentences soundness - what can be inferred is entailed completeness - what is entailed can be inferred

Knowledge Representation and Reasoning A logic consists of  a formal language consisting of syntax semantics a proof theory terms: sentences, KB, valid, satisfiable, model, entail, inference rule, sound, complete

Inference with FOL Good News: There exists a complete inference procedure for FOL. (Gödel 1930-31) Robinson’s resolution algorithm Bad News: Can’t tell if a sentence isn’t entailed Þ FOL is semidecidable.

FOL Proof Method: Resolution conversion to normal form generalized resolution unification

Why Representing Change in Time is Difficult situation calculus - situations, actions, result function frame problem - how to say when things don’t change (e.g. cartoon frames) qualification problem - how to define circumstances under which an action will work (e.g. banana in tailpipe) ramification problem - how to infer all of the implicit consequences of actions (e.g. moving luggage)