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Outline

CS 371: Introduction to Artificial Intelligence Propositional Logic

Review/Introduction Review: goal-based agent, simple problem-solving agent How would we design a Minesweeper-playing agent? Capacity for simple logical reasoning useful for some problem solving.

Knowledge-based agent

Examples of KB Agents Toy example: Minesweeper Agent Real-world example: Reaction Control System (RCS)

Propositional Logic Syntax propositions - T/F atomic sentences about the world propositional symbols (a.k.a. variables) P, Q, R, …, and constants True and False e.g. P: “There is a mine at (1,2).”; Q: “There is a mine at (2,2).” not (Ø) - negates meaning of a sentence P, ØP, Q, ØQ, … are called literals e.g. ØQ: “There is not a mine at (2,2)

Propositional Logic Syntax (cont.) connectives (Ù,Ú,Þ,Û) - logically combine sentences  PÙQ, PÚQ, PÞQ, PÛQ are complex sentences e.g. PÞØQ: “If there is a mine at (1,2), then there is not a mine at (2,2).” parentheses - disambiguates grammar e.g. PÞØQ ÙQ PÞ(ØQ ÙQ) vs. (PÞØQ) ÙQ

Propositional Logic Semantics commitments to the meanings of propositions (“P” without an interpretation is meaningless.) propositional symbol interpretations commitments to the meanings of connectives, etc. (“Þ” without an interpretation is meaningless.) truth tables

Truth Tables truth table - table of truth assignments to propositions and resulting truth values for complex sentences truth tables for defining connectives

Truth Tables (cont.) tables can easily become large! Exponential growth with increase in propositional symbols

Validity, Satisfiability, Models A sentence is valid if it is true for all possible truth assignments (a.k.a. a tautology) e.g. (P ÚØP) A sentence is satisfiable if there exists a “satisfying” truth assignment (i.e. a truth assignment for which the sentence is true). A truth assignment which satisfies a sentence is a model of that sentence.

Entailment entailment - whether a sentences follows from other sentences KB Ö f means knowledge base KB entails formula f If a truth assignment that satisfies all sentences in KB also satisfies formula f, f is entailed by KB

Inference inference rules - deduce new sentences from old sentences KB | f means knowledge base KB infers or proves formula f If there are a set of inference actions which derive the truth of formula f from the sentence in KB, f is inferred by KB

Semantics: Relating the Representation to the World

Relationships Between Entailment and Inference soundness - all inferences correct? completeness - all conclusions reachable?

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

Knowledge Representation and Reasoning (cont.) All KR&R languages have syntax, semantics, and ontological and epistemological commitments.

Knowledge-Based Agent Review

Advantages KB-agents can achieve competence quickly, by learning or being told facts KB-agents can adapt.  Changes in the world cause updates to KB Flexible; no reprogramming!

Difficulties expressivity - limits of what knowledge a logic can express tractability - limits of what can be reasoned within computational constraints fundamental tradeoff between expressivity and tractability knowledge acquisition - the difficulty of expressing and codifying human knowledge

Example: SSA Circuit Fault Testing Problem Overview Combinational circuits Single-Stuck-At (SSA) faults Automatic Test Pattern Generation Notation translation for Larrabee paper Normal circuit example Conversion to Conjunctive Normal Form (CNF)

Problem Overview Suppose: You’re working in circuit testing You test a great many combinational circuit designs (many quite complex). You wish to find a way to automate testing of circuits for Single Stuck-At (SSA) faults: A single circuit wire “stuck at” a constant value (e.g. 0 = false=grounded) For any given fault, it would be nice to have a circuit input which would give a different result if fault / no fault.

Combinational Circuits

Larrabee Notation In the Larrabee paper, a different logical notation is used:

Conversion to CNF Eliminate Þ and Û Move negations in (form literals) using deMorgan’s Law Distribute Ù and Ú to form conjunction of disjunctions Worked example:  X Û (Y Ù ØZ)

Eliminate Þ and Û Replace XÛY with (XÞY) Ù (YÞX) Replace XÞY with ØXÚY Worked example: X Û (YÙØZ) = (X Þ (YÙØZ)) Ù ((YÙØZ) Þ X) = (ØX Ú (YÙØZ)) Ù (Ø(YÙØZ) Ú X)

Move Negations In deMorgan’s Law: Replace Ø(XÚY) with (ØX ÙØY) Replace Ø(XÙY) with (ØX Ú ØY) Worked example: (ØX Ú (YÙØZ)) Ù (Ø(YÙØZ) Ú X) = (ØX Ú (YÙØZ)) Ù ((ØY Ú ØØZ) Ú X) = (ØX Ú (YÙØZ)) Ù ((ØY Ú Z) Ú X) = (ØX Ú (YÙØZ)) Ù (ØY Ú Z Ú X) At this point, you should only have complex sentences made up of Ù, Ú, and literals

Form Conjunction of Disjunctions (CNF) with Distribution Replace (f0 Ú  (f 1 Ù … Ù f n)) with ((f0 Ú f1) Ù … Ù (f0 Ú fn)) where fi is a formula Continue until you have one conjunction (and) of disjunctions (or’s). (ØX Ú (YÙØZ)) Ù (ØY Ú Z Ú X) = ((ØXÚY) Ù (ØXÚØZ)) Ù (ØY Ú Z Ú X) = (ØXÚY) Ù (ØXÚØZ) Ù (ØY Ú Z Ú X)

Conjunctive Normal Form This is called Conjunctive Normal Form (CNF). Any CNF form can be converted to 3CNF or k-CNF by introducing variables and splitting disjunctions: (PÚQÚRÚSÚT) is equivalent to (PÚQÚU) Ù (ØUÚRÚV) Ù (ØVÚSÚT)

SSA Circuit Fault Testing Example Suppose we wish to test the following circuit for a SSA fault at D with D stuck at true:

Fault Model We model the fault by introducing D’, which is stuck at true and affects the value of X which we rename X’:

Test Pattern Generation We want to find a test pattern (inputs resulting in different outputs for correct/faulty cases) This is accomplished by first creating a circuit with both our correct and fault models, which evaluates to true when the correct and fault circuits have different outputs.

Test Pattern Generation (cont.) Then we fix the output of this combined circuit to true, and treat this as a boolean satisfiability problem, seeking a satisfying truth assignment to the inputs. Such an assignment will be a test pattern suitable for testing for the given fault.

SSA Test Pattern Generation Summary Given: combinational circuit, possible fault Build fault model Build “test circuit” combining normal and fault model, evaluating true iff outputs of each are different Fix output of test circuit to true Convert circuit logic to CNF form Use GSAT to find a satisfying truth assignment to inputs.

Theorem Proving Proof by Contradiction: Given KB, want to show KB Ö f (KB entails f) Let KB’ = KB È {Øf} If we can prove that KB’ Ö false (inconsistent, contradiction), then we’ve proven that KB Ö f If KB’ | false and our inference rules are sound then, KB’ Ö false

Propositional Logic Inference Can use propositional logic inference rules of Figure 6.13, or… Can convert to CNF and do resolution theorem proving Example: KB = {P, PÞQ, QÞR} Does this infer R?

Resolution Theorem Proving Example KB’ = {P, PÞQ, QÞR, ØR} = = (P Ù (ØP Ú Q) Ù (ØQ Ú R) Ù ØR) {P} (KB) {ØP Q} (KB) {ØQ R} (KB) {ØR} (negated goal) {Q} (1,2) {R} (3,5) {} (4,6)  à Contradiction!  Q.E.D.

Summary KB-agent: “Knowledge is power” à “Ability to derive new knowledge is ability to increase power.” Overview of Propositional Logic as a language for a KB-agent Tradeoff: expressivity vs. tractability Practical Example: circuit test pattern generation Automatic theorem proving approaches