Interacting Conceptual Spaces Martha Lewis (with: Yaared Al-Mehairi, Joe Bolt, Bob Coecke, Fabrizio Genovese, Dan Marsden, Robin Piedeleu) Quantum Group, Department of Computer Science University of Oxford Dagstuhl Seminar 17192, May 7 - 12: Human-like Neural-Symbolic Computing
How can we compose concepts? How can distributed/neural/conceptual representations be combined with symbolic calculations? General strategy Representational choices Symbolic choices
A general programme for compositional models a Choose a compositional structure, such as a pregroup or 1 combinatory categorial grammar. b Interpret this structure as a category, the grammar category . a Choose or craft appropriate meaning or concept spaces, such as 2 vector spaces, density matrices, or conceptual spaces. b Organize these spaces into a category, the semantics category , with the same abstract structure as the grammar category. 3 Interpret the compositional structure of the grammar category in the semantics category via a functor preserving the necessary structure. 4 Bingo! This functor maps type reductions in the grammar category onto algorithms for composing meanings in the semantics category.
Compact closed categories Diagrammatic notation Clowns N Clowns ∈ N := , f : N → M := f N M N tell f M g ◦ f := g , tell ∈ N ⊗ S ⊗ N := N S N N Clowns tell jokes jokes Clowns tell − 1 n n − 1 n s n �→ N N S N N
... originally a calculus for quantum theory! Aleks Bob Aleks Bob ψ U i “ U i ψ Coecke, Bob, and Aleks Kissinger. Picturing quantum processes: A first course in quantum theory and diagrammatic reasoning. Cambridge University Press, 2017.
Pregroup grammar (Lambek, 1999) A pregroup P is a partially ordered monoid where each element p ∈ P has left and right ‘inverses’ p − 1 and − 1 p . p − 1 · p ≤ 1 ≤ p · p − 1 and p · − 1 p ≤ 1 ≤ − 1 p · p The pregroup grammar uses atomic types n , s . Other parts of speech are formed from the concatenation of atomic types e.g. transitive verb = − 1 nsn − 1 adjective = nn − 1
Pregroup grammar If a string reduces to the type s , the sentence is judged grammatical. ‘Clowns tell jokes’ �→ n ( − 1 nsn − 1 ) n : n ( − 1 nsn − 1 ) n ≤ 1 · sn − 1 n ≤ 1 · s · 1 ≤ s This reduction can also be expressed via a graphical calculus: jokes Clowns tell − 1 n n − 1 n s n (1)
Semantic choices Vector spaces Density matrices Conceptual spaces
Word meaning is determined by context U.S. Senate, because they are [?] , like to eat as high on the It made him [?] . sympathy for the problems of [?] beings caught up in the peace and the sanctity of [?] life are not only religious without the accompaniment of [?] sacrifice. a monstrous crime against the [?] race. this mystic bond between the [?] and natural world that the suggests a current nostalgia for [?] values in art. Harbor” in 1915), the [?] element was the compelling an earthy and very [?] modern dance work, To be [?] , he believes, is to seek one’s Ordinarily, the [?] liver synthesizes only enough nothing in the whole range of [?] experience more widely It is said that fear in [?] beings produces an odor that megatons: the damage to [?] germ plasm would be such
Word meaning is determined by context U.S. Senate, because they are human , like to eat as high on the It made him human . sympathy for the problems of human beings caught up in the peace and the sanctity of human life are not only religious without the accompaniment of human sacrifice. a monstrous crime against the human race. this mystic bond between the human and natural world that the suggests a current nostalgia for human values in art. Harbor” in 1915), the human element was the compelling an earthy and very human modern dance work, To be human , he believes, is to seek one’s Ordinarily, the human liver synthesizes only enough nothing in the whole range of human experience more widely It is said that fear in human beings produces an odor that megatons: the damage to human germ plasm would be such
Word meaning is determined by context ... but that doesn’t work for whole sentences. We want the sentence meaning to be a function of the word meanings. s = f ( w 1 , w 2 , ... w n )
Vector space models of meaning cuddly Wilbur iguana cuddly 1 smelly 10 smelly scaly 15 teeth 7 scaly cute iguana 2 Similarity is given by cosine distance: � v , w � sim ( v , w ) = cos( θ v , w ) = || v |||| w ||
Vector space models of meaning Atomic types and adjoints n , s map to vector spaces N , S . Concatenation of types maps to the tensor product ⊗ Reductions are mapped to tensor contraction. Clowns tell jokes jokes Clowns tell − 1 n n − 1 n s n �→ N N S N N Sentences can be directly compared within sentence space S
Sentence meanings are mapped into one space Clowns tell jokes N N S N N Sad clowns tell jokes Clowns tell funny jokes
Effective! Kartsaklis, D., & Sadrzadeh, M. (2013, October). Prior Disambiguation of Word Tensors for Constructing Sentence Vectors. In EMNLP (pp. 1590-1601). Grefenstette, E., & Sadrzadeh, M. (2011, July). Experimenting with transitive verbs in a discocat. In Proceedings of the GEMS 2011 Workshop on GEometrical Models of Natural Language Semantics (pp. 62-66). Association for Computational Linguistics. Grefenstette, E., & Sadrzadeh, M. (2011, July). Experimental support for a categorical compositional distributional model of meaning. In Proceedings of the Conference on Empirical Methods in Natural Language Processing (pp. 1394-1404). Association for Computational Linguistics.
Density matrices We view words as quantum states , combined to form mixed states . I sat on the river I paid some money into the The diagrammatic calculus is the same! Piedeleu, R., Kartsaklis, D., Coecke, B., & Sadrzadeh, M. (2015). Open system categorical quantum semantics in natural language processing. arXiv preprint arXiv:1502.00831.
Positive operators for hyponymy pet =
Positive operators for hyponymy pet =
Positive operators for hyponymy pet =
Positive operators for hyponymy pet =
Positive operators for hyponymy pet =
Positive operators for hyponymy pet =
Graded hyponymy Positive operators A , B have the L¨ owner ordering A ⊑ B ⇐ ⇒ B − A is positive. We say that A is a hyponym of B if A ⊑ B We say that A is a k-hyponym of B for a given value of k in the range (0 , 1] and write A � k B if: B − kA is positive We are interested in the maximum such k . Theorem For positive self-adjoint matrices A, B such that supp ( A ) ⊆ supp ( B ) , the maximum k such that B − kA ≥ 0 is given by 1 /λ where λ is the maximum eigenvalue of B + A.
k -Hyponymy interacts well with compositionality We would like our notion of hyponymy to work at the sentence level. Since sentences are represented as positive operators, we can compare them directly. If sentences have similar structure, we can also give a lower bound on the hyponymy strength between sentences based on the hyponymy strengths between the words in the sentences. Example Suppose � dog � � k � pet � and � park � � l � field � . Then � My dog runs in the park � � ??? � My pet runs in the field �
Conceptual spaces Conceptual spaces (G¨ ardenfors, 2014) can provide a more cognitively realistic semantics. noun ∈ COLOUR ⊗ SHAPE ⊗ · · · Moulton et al. (2015)
Food and drink - the noun space We define a property p property to be a convex subset of a domain. Some nouns:
Food and drink - the noun space We define a property p property to be a convex subset of a domain. Some nouns: banana = × × 0 0.2 0.5 1 apple = × × 0 0.5 0.8 1 beer = × × 0 00.01 1
Food and drink - the sentence space Simple example. Events are either positive or negative, surprising or unsurprising. Sentence space of pairs. First element states whether sentence is positive (1) or negative (0), and the second element states whether sentence is surprising (1) or unsurprising (0). Sentence meanings are convex subsets of the space, for example singletons, or larger subsets such as negative = { (0 , 1) , (0 , 0) } . Adjectives are relations from nouns to nouns Transitive verbs are relations from two nouns to a sentence
Concepts in interaction We form sentences and apply grammatical reductions Sweet bananas are good: bananas taste sweet = ( ǫ N × 1 S × ǫ N )( bananas × taste × sweet ) = ( ǫ N × 1 S )( banana × ( green banana × { (1 , 1) } ∪ yellow banana × { (1 , 0) } ) = { (1 , 1) , (1 , 0) } = positive Sweet beer is not so good: beer tastes sweet = ( ǫ N × 1 S × ǫ N )( beer × taste × sweet ) = { (0 , 1) } = negative and surprising Bolt, J., Coecke, B., Genovese, F., Lewis, M., Marsden, D., & Piedeleu, R. (2017). Interacting Conceptual Spaces I: Grammatical Composition of Concepts. arXiv preprint arXiv:1703.08314.
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