Vector spaces as Kripke frames

08/15/2019 ∙ by Giuseppe Greco, et al. ∙ 0

In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the lexical aspects of meaning into Lambek's type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces. In this paper, we build on this observation and extend it to a `vector space semantics' for the general Lambek calculus, based on algebras over a fieldK (or K-algebras), i.e. vector spaces endowed with a bilinear binary product. Such structures are well known in algebraic geometry and algebraic topology, since they are important instances of Lie algebras and Hopf algebras. Applying results and insights from duality and representation theory for the algebraic semantics of nonclassical logics, we regard K-algebras as `Kripke frames' the complex algebras of which are complete residuated lattices. This perspective makes it possible to establish a systematic connection between vector space semantics and the standard Routley-Meyer semantics of (modal) substructural logics.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

The extended versions of the Lambek Calculus [8, 9] currently used in computational syntax and semantics can be considered as multimodal substructural type logics where residuated families of n-ary fusion operations coexist and interact. Examples are Multimodal TLG with modalities for structural control [10], Displacement Calculus [12], combining concatenation and wrapping operations for the intercalation of split strings, Hybrid TLG [7], with the non-directional implication of linear logic on top of Lambek’s directional implications. For semantic interpretation, these formalisms rely on the Curry-Howard correspondence between derivations in a calculus of semantic types and terms of the lambda calculus that can be seen as recipes for compositional meaning assembly. This view of compositionality addresses derivational semantics but remains agnostic as to the choice of semantic spaces for lexical items.

Compositional distributional semantics [1, 3, 2, 13] satisfactorily addresses the lexical aspects of meaning while preserving the compositional view on how word meanings are combined into meanings for larger phrases. In [2], the syntax-semantics interface takes the form of a homomorphism from Lambek’s Syntactic Calculus, or its pregroup variant, to the Compact Closed Category of FVect and linear maps; [11] have the same target interpretation, but obtain it from the non-associative Lambek calculus extended with a pair of adjoint modal operators allowing for controlled forms of associativity and commutativity in the syntax. The role of the control modalities, in this approach, is confined to the syntax; moreover, the standard interpretation of the Lambek fusion as tensor product of vector spaces is retained, with the well known issue that the tensor product has more properties (i.e. commutativity, associativity, unitality) than the general Lambek fusion.

In this paper we start exploring a more general interpretation of the Lambek fusion in vector spaces. Our starting point is the notion of algebra over a field (or -algebra). An algebra over a field is a vector space over endowed with a bilinear product (cf. Definition 2.2). Algebras over a field can be regarded as Kripke (Routley-Meyer) frames in the following way. The vector space structure of a given -algebra gives rise to a closure operator on the powerset algebra of the vector space (i.e. the closure operator which associates any set of vectors with the subspace of their linear combinations). The closed sets of this closure operator form a complete general lattice which interprets the additive operations of the Lambek calculus (whenever they are considered). The bilinear product of the -algebra, seen as the graph of a ternary relation, gives rise to a binary fusion operation on the powerset of the vector space, and moreover its bilinearity guarantees that the closure operator mentioned above is a nucleus. This fact makes it possible to endow the set of subspaces of a -algebra with a residuated lattice structure in a standard way. We start developing some instances of correspondence theory in this environment, by characterizing the first order conditions on any given -algebra corresponding to the validity in its associated residuated lattice of several identities involving the Lambek fusion such as commutativity, associativity and unitality. Moreover, using these characterizations, we show that commutativity and associativity fail on the residuated lattice associated with certain well known -algebras. We also explore how this architecture can be further generalized so as to provide a vector space semantics for the Lambek calculus expanded with structural control modalities.

2 Preliminaries

Definition 2.1.

Let be a field. A vector space over is a tuple 111We overload notation and use the same symbols for sum, product and the constant both in the field and in the vector space , and rely on the context to disambiguate the reading. Notice that the operation below does not need to be primitive as it is introduced here, but it can be defined. such that

  • is commutative, associative and with unit ;

  • is s.t.  for any ;

  • (called the scalar product) is an action, i.e.  for all and every ;

  • the scalar product is bilinear, i.e.  and for all and all ;

  • for every .

A subspace of a vector space as above is uniquely identified by a subset which is closed under .

Definition 2.2.

An algebra over (or -algebra) is a pair where is a vector space over and is bilinear, i.e. left- and right-distributive with respect to vector sum, and compatible with scalar product:

  • and for all ;

  • for all and all .

Definition 2.3.

A -algebra is:

  1. associative if is associative;

  2. commutative if is commutative;

  3. unital if has a unit ;

  4. idempotent if for every ;

  5. monoidal if is associative and unital.

Example 2.4.

Quaternions [4] form a 4-dimensional vector space over the field of real numbers, with as the fixed basis. A quaternion is an expression of the form , where are real numbers and are the fundamental units.222Notice that we adopt the usual notational convention that omits and omits the coordinate whenever the scalar is .

Quaternions form an -algebra with the Hamilton product which is defined on the basis elements as indicated in the following table and then on arbitrary pairs of quaternions by bilinearity as usual.

The Hamilton product is monoidal (cf. Definition 2.3)333Given our convention, in this case is an abbreviation for . and, notably, not commutative.

Example 2.5.

Octonions [4] form a 8-dimensional vector space over the field of real numbers, with as fixed basis. An octonion is an expression of the form , where are real numbers. Octonions form an -algebra with the product which is defined on the basis elements as indicated in the following table and then on arbitrary pairs of quaternions by bilinearity as usual.

3 A Kripke-style analysis of algebras over a field

For any -algebra , the set of subspaces of is closed under arbitrary intersections, and hence it is a complete sub -semilattice of . Therefore, by basic order-theoretic facts (cf. [5]), gives rise to a closure operator s.t.  for any . As is well known, the elements of can be characterized as linear combinations of elements in , i.e. for any ,

If is a -algebra, let be defined as follows:

Lemma 3.1.

If is a -algebra, is a nucleus on , i.e. for all ,

Proof.

By definition, . Let and , and let us show that . Since for , we can rewrite as follows: ; likewise, since for , we can rewrite each as . Therefore:

which is a linear combination of elements of , as required. ∎

Hence, by the general representation theory of residuated lattices [6, Lemma 3.33], Lemma 5.2 implies that the following construction is well defined:

Definition 3.2.

If is a -algebra, let be the complete residuated lattice generated by , i.e. for all ,

(1)

where

  1. ;

  2. ;

  3. .

Definition 3.3.

If is a -algebra, and is the complete lattice of the subspaces of , let be defined as follows:

Lemma 3.4.

If is a -algebra, then the operation on is completely -preserving in each coordinate.

Proof.

Let . To show that , notice that by definition, . Hence, , which shows that . Conversely, to show that , it is enough to show that for every . This straightforwardly follows from the definitions involved, since for every and every , clearly . The proof of the distributivity in the first coordinate is analogous and omitted. ∎

Since is a complete lattice, Lemma 3.4 implies that the residuals exist such that for all ,

(2)
Lemma 3.5.

If is a -algebra, then for all ,

  1. ;

  2. .

Proof.

1. By (2), , so to show that we need to show that if and then . Let , let and such that ; then implies that , as required. Conversely, to show that , it is enough to show that if is such that then . By definition, . By assumption, all the generators of are in , which proves the statement. The proof of item 2 is similar and omitted. ∎

Definition 3.6.

If is a -algebra, let be the complete residuated lattice generated by .

4 Sahlqvist correspondence for algebras over a field

Definition 4.1.

If is a -algebra, is:

  1. associative if is associative;

  2. commutative if is commutative;

  3. unital if there exists a 1-dimensional subspace such that ;

  4. contractive if ;

  5. expansive if ;

  6. monoidal if is associative and unital.

Towards completeness for axiomatic extensions, we introduce a variant of the previous notions. These are to be regarded as first-order conditions on the -algebra, seen as ‘Kripke frames’.

Definition 4.2.

A -algebra is:

  1. pseudo-commutative if s.t. ;

  2. pseudo-associative if s.t.  and s.t. ;

  3. pseudo-unital if s.t.  and and and ;

  4. pseudo-contractive if s.t. ;

  5. pseudo-expansive if s.t. ;

  6. pseudo-monoidal if pseudo-associative and pseudo-unital.

In what follows, we sometimes abuse notation and identify a -algebra with its underlying vector space . Making use of definition 4.2 we can show the following:

Proposition 4.3.

For every -algebra ,

  1. is commutative iff is pseudo-commutative;

  2. is associative iff is pseudo-associative;

  3. is unital iff is pseudo-unital;

  4. is contractive iff is pseudo-contractive;

  5. is expansive iff is pseudo-expansive;

  6. monoidal iff is pseudo-monoidal.

Proof.

1. For the left-to-right direction, assume that is commutative and let . Then . Notice that and . Hence, implies that , i.e.  for some , as required.

Conversely, assume that is pseudo-commutative, and let . To show that , it is enough to show that for every and . By the assumption that is pseudo-commutative, there exists some such that , as required. The argument for is similar, and omitted.

2. For the left-to-right direction, assume that is associative and let . Then . Notice that and . Hence, implies that for some and for some , as required.

Conversely, assume that is pseudo-associative, and let . To show that , it is enough to show that for every , and . Since is pseudo-associative, there exists some such that , as required. The argument for is similar, and omitted.

3. For the left-to-right direction, assume that is unital and let such that . Then for any . Hence, and , for some , as required. Analogously, from one shows that and for some .

Conversely, assume that is pseudo-unital, and let . To show that , it is enough to show that for every . By assumption, there exists some such that , as required. The remaining inclusions are proven with similar arguments which are omitted.

4. For the left-to-right direction, assume that is contractive and let . Then . Hence, for some , as required.

Conversely, assume that is pseudo-contractive, and let . To show that , it is enough to show that for every . By assumption, there exists some such that , as required.

5. For the left-to-right direction, assume that is expansive and let . Then, letting denote the subspace generated by and , we have , and since we conclude , i.e.  for some , as required.

Conversely, assume that is pseudo-expansive, and let . To show that , it is enough to show that for every . By assumption, there exist some such that , as required.

6. Immediately follows from 2. and 3. ∎

4.1 Examples

Fact 4.4.

The algebra of quaternions is not pseudo-commutative.

Proof.

Let and , then and . By contradiction, let us assume that is pseudo-commutative, then there exists a real number s.t. . It follows that , and . We observe that holds only for , but then all the other equalities do not hold contradicting the assumption that is pseudo-commutative. ∎

Corollary 4.5.

is not commutative.

Proof.

Immediate by Fact 4.4 and Proposition 4.3. ∎

Fact 4.6.

The algebra of octonions is not pseudo-associative.

Proof.

Let , then . In order to show that is enough to check the first two coordinates: . By contradiction, let us assume that is pseudo-associative, then there exists a real number s.t. . It follows that and . We observe that holds only for , but then does not hold contradicting the assumption that is pseudo-associative. ∎

Corollary 4.7.

is not associative.

Proof.

Immediate by Fact 4.6 and Proposition 4.3. ∎

5 Modal algebras over a field

Definition 5.1.

A modal algebra over (or modal -algebra) is an algebra over equipped with a unary operator which is bilinear, i.e. left- and right-distributive with respect to vector sum, and compatible with scalar product:

  • ;

  • for all and all .

Notice that in the definition above, no further condition is imposed on .

If is a modal -algebra, let be defined as follows:

Lemma 5.2.

If is a -algebra, is a -nucleus on , i.e. for all ,

Proof.

By definition, . Let , and let us show that . Since for , we can rewrite as follows:

which is a linear combination of elements of , as required. ∎

Hence, by the general representation theory of residuated lattices [6, Lemma 3.33], Lemma 5.2 implies that the following construction is well defined:

Definition 5.3.

If is a modal -algebra, let be the complete modal residuated lattice generated by , i.e. for all ,

(3)

where

  1. ;

  2. .

5.1 Axiomatic extensions of a modal algebra over

In order to capture controlled forms of associativity/commutativity, we want to consider axiomatic extensions of the modal algebras introduced in the previous section. Below, as an example, we consider bidirectional forms of right-associativity and right-commutativity. Unidirectional forms have been proposed to model phenomena of extraction versus infixation. Such unidirectional forms would require inequalities rather than the equalities of the following definitions. We leave this as a topic for further research.

Definition 5.4.

If is a modal -algebra, is:

  1. right associative if ;

  2. right commutative if .

Definition 5.5.

A modal -algebra is:

  1. right associative if ;

  2. right commutative if .

Proposition 5.6.

If modal -algebra is:

  1. right associative then modal is right associative;

  2. right commutative then modal is right commutative.

Proof.

1. Let us assume that is right associative, let , and let us show that

i.e., that

For , it is enough to show that if for any and , then . The assumptions imply that with , and , and hence , as required. The argument for is similar and omitted.

2. Let us assume that is right commutative, let , and let us show that

i.e., that

For , it is enough to show that if for any and , then . The assumptions imply that with , and , and hence , as required. The argument for is similar and omitted.

References

  • [1] M. Baroni, R. Bernardi, and R. Zamparelli. Frege in space: a program for compositional distributional semantics. Linguistic Issues in Language Technology, 9(241–346), 2014.
  • [2] B. Coecke, E. Grefenstette, and M. Sadrzadeh. Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus. Annals of Pure and Applied Logic, 164(11):1079–1100, 2013.
  • [3] B. Coecke, M. Sadrzadeh, and S. Clark. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394, 2010.
  • [4] J. H. Conway and D. Smith. On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. AK Peters/CRC Press, 2003.
  • [5] B. A. Davey and H. A. Priestley. Introduction to lattices and order. Cambridge university press, 2002.
  • [6] N. Galatos, P. Jipsen, T. Kowalski, and H. Ono. Residuated lattices: an algebraic glimpse at substructural logics, volume 151. Elsevier, 2007.
  • [7] Y. Kubota and R. Levine. Gapping as like-category coordination. In D. Béchet and A. J. Dikovsky, editors, Logical Aspects of Computational Linguistics - 7th International Conference, LACL 2012, Nantes, France, July 2-4, 2012. Proceedings, volume 7351 of Lecture Notes in Computer Science, pages 135–150. Springer, 2012.
  • [8] J. Lambek. The mathematics of sentence structure. The American Mathematical Monthly, 65(3):154–170, 1958.
  • [9] J. Lambek. On the calculus of syntactic types. In R. Jakobson, editor, Structure of Language and its Mathematical Aspects, volume XII of Proceedings of Symposia in Applied Mathematics, pages 166–178. American Mathematical Society, 1961.
  • [10] M. Moortgat. Multimodal linguistic inference. Journal of Logic, Language and Information, 5(3-4):349–385, 1996.
  • [11] M. Moortgat and G. Wijnholds. Lexical and derivational meaning in vector-based models of relativisation. In A. Cremers, T. van Gessel, and F. Roelofsen, editors, Proceedings of the 21st Amsterdam Colloquium, pages 55–64. ILLC, University of Amsterdam, 2017.
  • [12] G. Morrill, O. Valentín, and M. Fadda. The displacement calculus. Journal of Logic, Language and Information, 20(1):1–48, 2011.
  • [13] M. Sadrzadeh, S. Clark, and B. Coecke. The Frobenius anatomy of word meanings I: Subject and object relative pronouns. Journal of Logic and Computation, pages 1293–1317, 2013.