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2010), Mathematical foundations for a compositional distributional model of meaning
 Linguistic Analysis (Lambek Festschrift
"... We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the ..."
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Cited by 23 (5 self)
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We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a welltyped sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are ‘lifted ’ to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (welltyped) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the innerproduct can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our ‘categorical model ’ which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montaguestyle Booleanvalued semantics. 1
Breaking Paths in Atomic Flows for Classic Logic
, 2010
"... This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make cruci ..."
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This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make crucial use of the path breaker, an atomicflow construction that avoids some nasty termination problems, and that can be used in any proof system with sufficient symmetry. This paper contains an original 2dimensionaldiagram exposition of atomic flows, which helps us to connect atomic flows with other known formalisms.
Knot polynomial identities and quantum group coincidences
"... Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between smal ..."
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Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D2n planar algebras. We discuss the origins of these coincidences, explaining the role of SO levelrank duality, KirbyMelvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G2 and does not appear to be related to levelrank duality. AMS Classification 18D10; 57M27 17B10 81R05 57R56
Finite dimensional Hilbert spaces are complete for dagger compact closed categories
 In the proceedings of QPL 5
, 2008
"... We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness. ..."
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We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness.
Involutive monoidal categories
, 2010
"... Abstract. In this paper, we consider a nonposetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets ..."
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Abstract. In this paper, we consider a nonposetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets to categories; we also link our subject with other notions already in the literature, such as balanced monoidal categories [JS91] and dagger pivotal categories [Sel09]. 1.
Ribbon Proofs for Separation Logic
"... Abstract—We present a diagrammatic system for constructing and presenting readable program proofs in separation logic. A program proof should not merely certify that a program is correct; it should explain why it is correct. By examining a proof, one should gain understanding of both the program bei ..."
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Abstract—We present a diagrammatic system for constructing and presenting readable program proofs in separation logic. A program proof should not merely certify that a program is correct; it should explain why it is correct. By examining a proof, one should gain understanding of both the program being considered and the proof technique being used. To
Integrity constraints for Linked Data
 In Proceedings of 24th International Workshop on Description Logics
"... Abstract. Linked Data makes one central addition to the Semantic Web principles: all entity URIs should be dereferenceable to provide an authoritative RDF representation. URIs in a linked dataset can be partitioned into the exported URIs for which the dataset is authoritative versus the imported URI ..."
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Abstract. Linked Data makes one central addition to the Semantic Web principles: all entity URIs should be dereferenceable to provide an authoritative RDF representation. URIs in a linked dataset can be partitioned into the exported URIs for which the dataset is authoritative versus the imported URIs the dataset is linking against. This partitioning has an impact on integrity constraints, as a Closed World Assumption applies to the exported URIs, while a Open World Assumption applies to the imported URIs. We provide a definition of integrity constraint satisfaction in the presence of partitioning, and show that it leads to a formal interpretation of dependency graphs which describe the hyperlinking relations between datasets. We prove that datasets with integrity constraints form a symmetric monoidal category, from which the soundness of acyclic dependency graphs follows. 1
Open Graphs and Computational Reasoning
"... We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of halfedges (edges which are drawn with an unconnected end) and enjo ..."
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We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of halfedges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these halfedges. In particular, this allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information. 1
Isomorphic Interpreters from Logically Reversible Abstract Machines
"... Abstract. In our previous work, we developed a reversible programming language and established that every computation in it is a (partial) isomorphism that is reversible and that preserves information. The language is founded on type isomorphisms that have a clear categorical semantics but that are ..."
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Abstract. In our previous work, we developed a reversible programming language and established that every computation in it is a (partial) isomorphism that is reversible and that preserves information. The language is founded on type isomorphisms that have a clear categorical semantics but that are awkward as a notation for writing actual programs, especially recursive ones. This paper remedies this aspect by presenting a systematic technique for developing a large and expressive class of reversible recursive programs, that of logically reversible smallstep abstract machines. In other words, this paper shows that once we have a logically reversible machine in a notation of our choice, expressing the machine as an isomorphic interpreter can be done systematically and does not present any significant conceptual difficulties. Concretely, we develop several simple interpreters over numbers and addition, move on to tree traversals, and finish with a metacircular interpreter for our reversible language. This gives us a means of developing large reversible programs with the ease of reasoning at the level of a conventional smallstep semantics. 1