Continuations and derivational ambiguity Michael Moortgat LACL 2014, Toulouse Abstract In categorial grammar, compositional interpretation is implemented along the lines of Montague's `Universal Grammar' program, taking the form of a structure-preserving mapping (a homomorphism) relating a source logic to a target logic. The homomorphism requirement implies that for an expression to be associated with multiple `readings', there must be derivational ambiguity at the source end of the compositional mapping. In this talk, I study the kind of derivational ambiguity that arises in some recent continuation-based categorial approaches. In Bastenhof (2013) and Moortgat and Moot (2013), the source derivations for compo- sitional interpretation are focused proofs of a polarized sequent calculus or its graphical alternative; different notions of derivational ambiguity result from the choices one can make in fixing the polarity of atomic formulas. I compare this approach to the handling of continuations in terms of the combinatory and multimodal sequent calculi one finds in Barker and Shan (2014). Continuation semantics Turn of the century shift from direct interpretation to continuation-based semantic analysis. I importing ideas from computer science / functional programming . de Groote 2001, Barker 2002; Barker and Shan (. , 2014) . dynamic semantics: de Groote 2006, Asher&Pogodalla 2011, Lebe- deva PhD 2012 I importing ideas from classical linear logic . Lambek-Grishin: equal treatment of hypotheses / conclusions . 2007{. : MM, Bernardi, Kurtonina, Moot, Bransen, Melissen, Bas- tenhof PhD 2013 Some key ideas I direct interpretation: a function A ! B, presented with an argument of type A, returns a value of type B I continuation semantics: makes explicit the context where that value will be used, creating opportunities for exerting control I continuation: the future of the computation, with respect to overall result I target for interpretation: distinguished response type ? (maybe identified with t for NL semantics); for every source type A, distinguish .V A: values .K A: continuations, VA !? .C A: computations, KA !?, i.e. (VA !?) !? A ! B interpretation becomes VA ! (VB !?) !? Outline of the talk I Compositionality, derivational ambiguity I Type shifting and evaluation order (Barker & Shan 2014) I Type shifting and focusing Compositional interpretation Source Target h h (As)s2S;F i h (Bt)t2T ;G i I source algebra, sorts S, operations F I target algebra, sorts T , operations G I interpretation h: homomorphism, mapping respecting sorts/operations Antecedents cf Janssen, Foundations and applications of MG, 1983 I Montague, Universal Grammar, 1970 I ADJ, Initial algebra semantics and continuous algebras, 1977 (Goguen, Thatcher, Wagner, Wright) Compositionality, categorially Source and target: type logics with similar structure. Simplest example: Source Target h np;s;n AB LPe;t =;n ( I h on types: atoms h(np) = e; h(s) = t; h(n) = e ( t, and h(AnB) = h(B=A) = h(A) ( h(B) I h on proofs/terms: h(x) = xe, h(N.M) = h(M/N) = ( h(M) h(N)) source: directional left/right application (Buszkowski, Wansing); target: regular LP/MILL application. Derivational vs lexical interpretation The above abstracts from the contribution of lexical items: one interprets source derivations x1 : A1; : : : ; xn : An ` M : B FV (M): parameters, where lexical constants can be substituted. Adding the lexicon: Source Targetder Targetlex hder hlex np;s;n AB LPe;t ILe;t =;n ( ! I Source signature: type assignments to lexical constants c0; c1;::: I Targetlex signature I hlex ◦ hder translations for ci: terms of Targetlex , possibly non-linear Claim non-linearity encapsuled in interpretation of lexical constants. Compositional variations Source Pick your favourite typelogical framework: I Lambek (N)L, multimodal/hybrid TLG, Displacement Calculus, . I LP/MILL: ACG, Lambda Grammars; MILL1 (Moot 2014) Target Alternatives for Montagovian modeltheoretic interpretation: I PTS, prooftheoretic semantics, Francez c.s. I distributional models (Coecke, Sadrzadeh c.s.): h: `quantisation functor' from monoidal bi-closed category (L) to FVect Derivational ambiguity Input for compositional interpretation: source derivations / proof terms. ; h can `forget' derivational structure (e.g. linear order) but not create struc- ture at the target end. Genuine ambiguity distinct source derivations correspond to distinct target terms. I e.g. s=(npns); (s=(npns))ns `NL s (`something is missing') I two source derivations, su > vp versus vp > su scope construal Spurious ambiguity distinct source derivations mapped to same target term I typical example: multiple sequent derivations for one N.D. proof I proof nets, normalized sequent proofs: eliminate spurious ambiguity Prehistoric remains H. Hendriks & M. Moortgat, Theory of flexible interpretation. Specification and grammar fragment. DYANA Deliverable R1.2.A, 1990. Hendriks: Flexible MG Objective optimal (=minimal) type assignment in syntax / semantics Source (viewed categorially) a pure application AB syntax Mapping source ; target derivations: weakened to a relation I types: source type is associated with (infinite) set of sem types . basic type map + closure under type-shifting combinators I terms: for primitive and composite terms . basic translation + derived translations for type-shifts ; the FMG architecture moves the origin of derivational ambiguity to the semantic target language. FMG semantic type shifting The following are valid LP/MILL type transitions: ~ VR, value raising from f : A ( B derive ~ λ~xλw:(w (f ~x)) : A ( (B ( D) ( D ~ ~ AL, argument lowering from f : A ( ((B ( D) ( D) ( C ( E derive ~ ~ λ~xλwλ~y:(f ~x (λz:(z w)) ~y): A ( B ( C ( E ~ ~ AR, argument raising from f : A ( B ( C ( D derive ~ ~ λ~xλwλ~y:(w λz:(f ~xz ~y)) : A ( ((B ( D) ( D) ( C ( D (Generalized from Hendriks '90, who imposes the restriction D = E = t.) Back to the source Challenge Can one restore compositionality in its strong form by relocating the origin of derivational ambiguity to the source calculus? Observation VR, AL and a restricted form of AR are obtainable as the image of valid NL type transitions. MM'90, commenting on HH'90. I VR, AL: lifting, recursively generalized under monotonicity I AR: obtain its sem effect via source inference of the form ∆[ y : A] ) N : B Γ[ z : C] ) M : D (qL) Γ[ ∆[ x : q(A; B; C)]] ) M 0 : D 0 where M = M[ z=(x λy:N)], h( q(A; B; C) ) = (h(A)(h(B))(h(C) Logical decomposition: q(A; B; C) = (B " A) # C, Morrill TLG. Type shifting and continuations Barker 2002/2004: rational reconstruction of FMG in terms of continuations. Target recast the ubiquitous lifted types of FMG as computations functions KA (?, i.e. (A (?) (?, wrt response type ? Source to Target extend the interpretation function with continuation level: 0 I types: h lifts h(A) to Ch(A), functions acting on their own continuation: 0 h (A) = (h(A) (?) (? I terms: we need translations for . atomic terms / lexical constants . derivations: left/right application Translation: constants, proof terms Constants at the Targetlex level, the response type is identified with t Default translation: cA ; λk:(k ceh(A)) but `logically interesting' lexical items actually exploit their continuation: (et)t someonenp ; λk:(9 λx.(k x)) Compound terms compositional interpretation of source operations /; . h0(N A .M AnB) = h0(.) h0(M) h0(N) 0 0 0 0 h (.) :: h (AnB) ( h (A) ( h (B) Two possible solutions respecting the interpretation of types: 0 (y) h (/) = λM λN λk:(M λx.(N λy:(k (x y)))) 0 (z) h (/) = λM λN λk:(N λy:(M λx.(k (x y)))) Towers Ambiguity the interpretation of everyone . (tv / someone) depends on the choice between translations (y) and (z)(exercise). But having them both, the Source ; Target correspondence is again a relation. B&S 2014 Source: multimodal type system: I AB logic for regular phrasal composition: operations =; n I separate level for composition of continuations: operations ; C B Tower notation for C (A B) formulas: A Compare binder schema q(A; B; C), infixation/extraction (B " A) # C Interaction slash / fat slash composition The type logic of B&S 2014 (Part I) is a CCG-style combinator system. The combination schema for interaction between the =; n and ; levels realizes default left-to-right surface scope construal: F D D C F C ; ` A AnB B The underlying sequent calculus (Part II) also validates right-to-left construal: D C F D F C ; ` A AnB B CompareLP derivations (thanks to Grail) for the two options. Left to right [y : a]1 [x : anb]2 [nE] (x y): b [k : bnc]3 [nE] (k (x y)): c [P 3] (k (x y)): c [P 2] (k (x y)): c [nI]2 λx:(k (x y)):( anb)nc right : ((anb)nc)nd [nE] (right λx:(k (x y))): d [P 2] (right λx:(k (x y))): d [nI]1 λy:(right λx:(k (x y))): and left :(and)nf [nE] (left λy:(right λx:(k (x y)))): f [P 2] (left λy:(right λx:(k (x y)))): f [nI]3 λk:(left λy:(right λx:(k (x y)))):( bnc)nf [P 3] λk:(left λy:(right λx:(k (x y)))):( bnc)nf λk:(left λy:(right λx:(k (x y)))) Right to left [y : a]1 [x : anb]2 [nE] (x y): b [k : bnc]3 [nE] (k (x y)): c [P 2] (k (x y)): c [nI]1 λy:(k (x y)): anc left :(anc)nd [nE] (left λy:(k (x y))): d [P 2] (left λy:(k (x y))): d [nI]2 λx:(left λy:(k (x y))):( anb)nd right : ((anb)nd)nf [nE] (right λx:(left λy:(k (x y)))): f [P 2] (right λx:(left λy:(k (x y)))): f [nI]3 λk:(right λx:(left λy:(k (x y)))):( bnc)nf λk:(right λx:(left λy:(k (x y)))) Inverse scope To obtain inverse scope, an alternative version of lifting is needed, targeting internal levels (. ?. not generally valid) B (A B) ` B (( C (A C )) B) plus a generalization of the