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Composition and Substitution: Learning about Language from Algebra Ken Presting University of North Carolina at Chapel Hill Introduction • Intensional contexts are defined by substitution failure – Johnny heard that Venus is the Morning Star – Johnny heard that Venus is Venus • Composition accounts for indefinite application of finite knowledge – ‘p and q’ is a sentence – ‘p and q and r’ is a sentence –… Role of Recursion • Syntax – Atomic symbols – Combination rules – Closure principle • Finiteness – Limited symbols, rules – Infinitely many expressions Compositional Semantics • The usual: – Choose assignments to atoms – Forced valuations for molecules The Two-Element Boolean Algebra • The Truth Values • Just two atomic objects: 2BA = {0, 1} – Disjunction = max(a, b) – Conjunction = min(a, b) – Negation = 1 – a It’s almost familiar • Boolean arithmetic –01=1 –01=0 • Boolean algebra –AB=C – (A B) ~C = C ~C – (A B) ~C = 0 A Homomorphism to 2BA • Take any old function that labels sentences with 0 or 1. • For example: – f(S) = 0 – f(PQ) = 1 – etc. A Homomorphism to 2BA • Ask: Does this function have the ‘distributive’ – a(b + c) = ab + ac – f(S P) = f(S) f(P) • and ‘commutative’ properties? – ac = ca – f(~S) = ~f(S) A Homomorphism to 2BA …is a compositional semantics for propositional calculus Sentence Diagrams • Tree diagrams – Binary – Associativity allows n-ary nodes • (advanced topic: add leaves for empty expression) Repetition • Identical Subtrees – In many sentences, certain letters appear twice or more • P&QP – Sometimes whole expressions recur • (P & R) (P & R) Reducing the diagram • • • • Identify like-labeled leaves Identify like-labeled nodes Form equivalence classes Redraw tree as lattice – (advanced topics: empty expression as zero; quotient) Set Membership Model • Mapping sentences to sets – Set of letters = conjunction – Singleton set = negation – Associativity • And vs. Nand – Naturalness of negation – Failure of associativity Comparing lattices • Embeddings • Homomorphism Substitution for a Letter • Single-letter expressions – Every sentence is a substitution-instance of ‘P’ – Substitution for single letters is easy • Multiple occurrences of a letter Substitution for Expressions • What do these sentences have in common? (P & Q) v ~(P & Q) (T & S) v ~(T & S) Subalgebras • A subalgebra is a subset which follows the same rules as its container • In our case, that means ‘is also a sentence’ Quotients • Ignore specfied details • In our case, treat a subsentence as a letter Sentences as Functions In Algebra, formulas map numbers to each other – F(x) = mx + b • Sentences map the language to itself – (P v ~P)(Q) = Q v ~Q Sentences as Functions • Mapping the language to itself – Atomic Sentence letters map L to itself – No other sentence does • Complex sentences map the language to a subset of itself Image of a Sentence • Image = all the substitution-instances Image of ‘P v ~P’ is: Q v ~Q R v ~R (Q & R) v ~(Q & R) (P & Q) v ~(P & Q) … Composition of mappings • Substitute into a substitution-instance • Start with – P v ~P • Substitute for P – (Q v R) v ~(Q v R) • Substitute for R – (Q v (S & T)) v ~(Q v (S & T)) Sentence Fractions • Here’s a fraction R (P & Q) • The numerator is R • The denominator is (P & Q) Fractions and Substitution • ‘Multiply’ (P & Q) v ~(P & Q) • by the fraction R (P & Q) • This will be a substitution! Sentence Arithmetic Start with – (P & Q) v ~(P & Q) Dividing by (P & Q), gives a lattice with a missing label: – ‘x’ v ~ ‘x’ But R replaces ‘x’ (this step is by fiat) – R v ~R