English as a Programming Language - Cryptography | MATH 0209A, Papers of Cryptography and System Security

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English as a programming language

Yiannis N. Moschovakis UCLA and University of Athens

Tarski Lecture 2, March 5, 2008

Frege on sense

“[the sense of a sign] may be the common property of many people” Meanings are public (abstract?) objects

“The sense of a proper name is grasped by everyone who is sufficiently familiar with the language... Comprehensive knowledge of the thing denoted... we never attain” Speakers of the language know the meanings of terms

“The same sense has different expressions in different languages or even in the same language”

“The difference between a translation and the original text should properly not overstep the [level of the idea]” Faithful translation should preserve meaning

The methodology of formal Fregean semantics

I (^) An interpreted formal language L is selected I (^) The rendering operation on a fragment of English:

English expression + informal context −^ render−−→ formal expression + state

I (^) Semantic values (denotations, meanings, etc.) are defined rigorously for the formal expressions of L and assigned to English expressions via the rendering operation

I (^) Montague: L should be a higher type language (to interpret co-ordination, co-indexing,... ) I (^) Claim: L should be a programming language (to interpret self-reference and to define meanings properly)

The typed λ-calculus with recursion Lλ r (K ) - types

An extension of the typed λ-calculus, into which Montague’s Language of Intensional Logic LIL can be easily interpreted (Gallin)

Basic types b ≡ e | t | s (entities, truth values, states)

Types: σ :≡ b | (σ 1 → σ 2 )

Abbreviation: σ 1 × σ 2 → τ ≡ (σ 1 → (σ 2 → τ ))

Every non-basic type is uniquely of the form

σ ≡ σ 1 × · · · × σn → b

level(b) = 0 level(σ 1 × · · · × σn → b) = max{level(σ 1 ),... , level(σn)} + 1

Lλ r (K ) - denotational semantics

• We are given basic sets Ts , Te and Tt ⊆ Te for the basic types

Tσ→τ = the set of all functions f : Tσ → Tτ Pb = Tb ∪ {⊥} = the “flat poset” of Tb Pσ→τ = the set of all functions f : Tσ → Pτ

Tσ ⊆ Pσ and Pσ is a complete poset (with the pointwise ordering)

• We are given an object c : Pσ for each constant c : σ I (^) Pure variables of type σ vary over Tσ; recursive ones over Pσ I (^) If A : σ and π is a type-respecting assignment to the variables, then den(A)(π) ∈ Pσ I (^) Recursive terms are interpreted by the taking of least-fixed-points

Rendering natural language in Lλ r (K )

˜t ≡ (s → t) (type of Carnap intensions) ˜e ≡ (s → e) (type of individual concepts)

Abelard loves Eloise −render−−→ loves(Abelard,Eloise) : ˜t Bush is the president −render−−→ eq(Bush,the(president)) : ˜t liar −render−−→ p where {p = ¬p} : t truthteller −render−−→ p where {p = p} : t

Abelard, Eloise, Bush : ˜e president : ˜e → ˜t, eq : ˜e × ˜e → ˜t ¬ : t → t, the : (˜e → ˜t) → ˜e

den(liar) = den(truthteller) = ⊥

Can we say nonsense in Lλ r (K )?

Yes! In particular, we have parameters over states—so we can explicitly refer to the state (even to two states in one term); LIL does not allow this, because we cannot do this in English

Consider the terms

A ≡ rapidly(tall)(John), B ≡ rapidly(sleeping)(John) : ˜t

A and B are terms of LIL, not the renderings of correct English sentences I (^) The target formal language is a tool for defining rigorously the desired semantic values and it needs to be richer than a direct formalization of the relevant fragment of English —to insure compositionality, if for no other reason

Meaning and synonymy in Lλ r (K )

I (^) For a sentence A : ˜t, the Montague sense of A is den(A) : Ts → Tt , so that

there are infinitely many primes is Montague-synonymous with 1 + 1 = 2 I (^) In Lλ r (K ): The meaning of a term A is modeled by an algorithm int(A) which computes den(A)(π) for every π I (^) The referential intension int(A) is compositionally determined from A I (^) int(A) is an abstract (not necessarily implementable) recursive algorithm of Lλ r (K ) I (^) Referential synonymy: A ≈ B ⇐⇒ int(A) ∼ int(A)

Is this notion of meaning Fregean?

Evans (in a discussion of Dummett’s similar, computational interpretations of Frege’s sense): “This leads [Dummett] to think generally that the sense of an expression is (not a way of thinking about its [denotation], but) a method or procedure for determining its denotation. So someone who grasps the sense of a sentence will be possessed of some method for determining the sentence’s truth value

... ideal verificationism ... there is scant evidence for attributing it to Frege” Converse question: For a sentence A, if you possess the method determined by A for determining its truth value, do you then “grasp” the sense of A? (Sounds more like Davidson rather than Frege)

The reduction calculus

Bush is the president −render−−→ eq(Bush)(the(president)) ⇒ eq(Bush)(L) where {L = the(president)} ⇒ eq(Bush)(L) where {L = the(p) where {p = president}} ⇒ eq(Bush)(L) where {L = the(p), p = president} ⇒

eq(b) where {b = Bush}

(L) where {L = the(p), p = president} ⇒

eq(b)(L) where {b = Bush}

where {L = the(p), p = president} ⇒cf eq(b)(L) where {b = Bush, L = the(p), p = president}

He is the president −render−−→ eq(He)(the(president)) ⇒cf eq(b)(L) where {b = He, L = the(p), p = president}

Utterances, local meanings, local synonymy

An utterance is a pair (A, u), where A is a sentence, A : ˜t and u is a state; it is expressed in Lλ r (K ) by the term A(¯u)

The local meaning of A at the state u is int(A(¯u))

A ≈u B ⇐⇒ A(¯u) ≈ B(¯u)

Bush is the president(¯u) ⇒cf eq(b)(L)(¯u) where {b = Bush, L = the(p), p = president}

He is the president(¯u) ⇒cf eq(b)(L)(¯u) where {b = He, L = the(p), p = president}

Bush is the president 6 ≈u He is the president even if at the state ¯u, He(¯u) = Bush(¯u)

Three aspects of meaning for a sentence A : ˜t

Referential intension int(A) Referential synonymy ≈ Local meaning at u int(A(¯u)) Local synonymy ≈u Factual content at u FC(A, u) Factual synonymy ≈f ,u

The factual content of a sentence at a state u gives a representation of the world at u (Eleni Kalyvianaki’s Ph.D. Thesis)

Bush is the president 6 ≈u He is the president Bush is the president ≈f ,u He is the president

Claim: The objects of belief are local meanings

The distinction between local meaning and factual content are related to David Kaplan’s distinction between the character and content of a sentence at a state

Is referential synonymy decidable?

Synonymy Theorem. A ≈ B if and only if

A ⇒ cf(A) ≡ A 0 where {p 1 = A 1 ,... , pn = An} B ⇒ cf(B) ≡ B 0 where {p 1 = B 1 ,... , pn = Bn}

so that for i = 0,... , n and all π, den(Ai )(π) = den(Bi )(π). I (^) Synonymy is reduced to denotational equality for explicit, irreducible terms (the truth facts of A) I (^) Denotational equality for arbitrary terms is undecidable (there are constants, with fixed interpretations) I (^) The explicit, irreducible terms are very special — but by no means trivial!

The synonymy problem for Lλ r (K ) (with finite K )

I (^) The decision problem for Lλ r (K )-synonymy is open

Theorem If the set of constants K is finite, then synonymy is decidable for terms of adjusted level ≤ 2 These include terms constructed “simply” from Names of “pure” objects 0 , 1 , 2 , ∅,... : e Names, demonstratives John, I, he, him : ˜e Common nouns man, unicorn, temperature : ˜e → ˜t Adjectives tall, young : (˜e → ˜t) → (˜e → ˜t) Propositions it rains : ˜t Intransitive verbs stand, run, rise : ˜e → ˜t Transitive verbs find, loves, be : ˜e × ˜e → ˜t Adverbs rapidly : (˜e → ˜t) → (˜e → ˜t)

Proof is by reducing this claim to the Main Theorem in the 1994 paper (for a corrected version see www.math.ucla.edu/∼ynm)