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Book ChapterDOI

A Combinatory Logic

About: This article is published in The Ruffin Series in Business Ethics.The article was published on 1995-04-01. It has received 404 citations till now. The article focuses on the topics: Substructural logic & Dynamic logic (modal logic).
Citations
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Journal ArticleDOI
TL;DR: This paper finds the logic LP of propositions and proofs and shows that Godel's provability calculus is nothing but the forgetful projection of LP, which achievesGodel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s.
Abstract: In 1933 G¨ odel introduced a calculus of provability (also known as modal logic S4 )a nd left open the question of its exact intended semantics. In this paper we give as olution to this problem. We find the logic LP of propositions and proofs and show that G¨ odel's provability calculus is nothing but the forgetful projection of LP .T his also achieves G¨ odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov styleprovability semantics forIntwhich resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and ! -calculus.

448 citations


Additional excerpts

  • ...If ~s : Γ, ∆ ⊢LP F , then there is a proof polynomial t(~x, ~y) such that ~s :Γ, ~y :∆ ⊢LP t(~s, ~y) :F....

    [...]

Journal ArticleDOI
TL;DR: The article proposes a theory of grammar relating syntax, discourse semantics, and intonational prosody and extends an earlier account based on Combinatory Categorial Grammar, which directly pairs phonological and logical forms without intermediary representational levels.
Abstract: The article proposes a theory of grammar relating syntax, discourse semantics, and intonational prosody. The full range of English intonational tunes distinguished by Beckman and Pierrehumbert (1986) and their semantic interpretation in terms of focus and information structure are discussed, including "discontinuous" themes and rhemes. The theory extends an earlier account based on Combinatory Categorial Grammar, which directly pairs phonological and logical forms without intermediary representational levels.

437 citations


Cites background from "A Combinatory Logic"

  • ...Rather than invoking rules of deletion or movement, CCG allows certain further operations on functions related to what Curry called ‘‘combinators’’ (Curry and Feys 1958), including the following rule of functional composition, indexed B in derivations (because Curry called his composition…...

    [...]

Journal ArticleDOI
TL;DR: This article describes a number of log-linear parsing models for an automatically extracted lexicalized grammar and develops a new model and efficient parsing algorithm which exploits all derivations, including CCG's nonstandard derivations.
Abstract: This article describes a number of log-linear parsing models for an automatically extracted lexicalized grammar. The models are “full” parsing models in the sense that probabilities are defined for complete parses, rather than for independent events derived by decomposing the parse tree. Discriminative training is used to estimate the models, which requires incorrect parses for each sentence in the training data as well as the correct parse. The lexicalized grammar formalism used is Combinatory Categorial Grammar (CCG), and the grammar is automatically extracted from CCGbank, a CCG version of the Penn Treebank. The combination of discriminative training and an automatically extracted grammar leads to a significant memory requirement (up to 25 GB), which is satisfied using a parallel implementation of the BFGS optimization algorithm running on a Beowulf cluster. Dynamic programming over a packed chart, in combination with the parallel implementation, allows us to solve one of the largest-scale estimation problems in the statistical parsing literature in under three hours. A key component of the parsing system, for both training and testing, is a Maximum Entropy supertagger which assigns CCG lexical categories to words in a sentence. The supertagger makes the discriminative training feasible, and also leads to a highly efficient parser. Surprisingly, given CCG's “spurious ambiguity,” the parsing speeds are significantly higher than those reported for comparable parsers in the literature. We also extend the existing parsing techniques for CCG by developing a new model and efficient parsing algorithm which exploits all derivations, including CCG's nonstandard derivations. This model and parsing algorithm, when combined with normal-form constraints, give state-of-the-art accuracy for the recovery of predicate-argument dependencies from CCGbank. The parser is also evaluated on DepBank and compared against the RASP parser, outperforming RASP overall and on the majority of relation types. The evaluation on DepBank raises a number of issues regarding parser evaluation. This article provides a comprehensive blueprint for building a wide-coverage CCG parser. We demonstrate that both accurate and highly efficient parsing is possible with CCG.

433 citations


Cites methods from "A Combinatory Logic"

  • ...The parser is able to parse 20 Wall Street Journal (WSJ) sentences per second on standard hardware, using our best-performing model, which compares very favorably with other parsers using linguistically motivated grammars....

    [...]

  • ...The first is forward composition, which Steedman denotes by > B (because B is the symbol used by Curry to denote function composition in combinatory logic; Curry and Feys 1958): X/Y Y/Z ⇒B X/Z (> B) (5) Forward composition is often used in conjunction with type-raising (T), as in Figure 2....

    [...]

Book
04 Apr 2013
TL;DR: The Curry-Howard isomorphism as discussed by the authors is a type-free lambda-calculus, which is a variant of the Lambda-cube, and it can be expressed as follows:
Abstract: Preface Outline Acknowledgements 1. Typefree lambda-calculus 2. Intuitionistic logic 3. Simply typed lambdacalculus 4. The Curry-Howard isomorphism 5. Proofs as combinators 6. Classical logic and control operators 7. Sequent calculus 8. Firstorder logic 9. Firstorder arithmetic 10. Godel's system T 11. Secondorder logic and polymorphism 12. Secondorder arithmetic 13. Dependent types 14. Pure type systems and the lambda-cube 15. Solutions and hints to selected exercises 16. Solutions for chapter 6 Appendix A Mathematical Background Appendix B Solutions to Selected Exercises Bibliography Index

401 citations

Journal ArticleDOI
TL;DR: This paper explains the motivations and reasoning behind the development of the TPTP (thus implicitly explaining the design decisions made) and describes theTP contents and organization.
Abstract: This paper provides a detailed description of the CNF part of the TPTP Problem Library for automated theorem-proving systems The library is available via the Internet and forms a common basis for development and experimentation with automated theorem provers This paper explains the motivations and reasoning behind the development of the TPTP (thus implicitly explaining the design decisions made) and describes the TPTP contents and organization It also provides guidelines for obtaining and using the library, summary statistics about release v121, and an overview of the tptp2X utility program References for all the sources of TPTP problems are provided

386 citations

References
More filters
Journal ArticleDOI
TL;DR: This paper finds the logic LP of propositions and proofs and shows that Godel's provability calculus is nothing but the forgetful projection of LP, which achievesGodel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s.
Abstract: In 1933 G¨ odel introduced a calculus of provability (also known as modal logic S4 )a nd left open the question of its exact intended semantics. In this paper we give as olution to this problem. We find the logic LP of propositions and proofs and show that G¨ odel's provability calculus is nothing but the forgetful projection of LP .T his also achieves G¨ odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov styleprovability semantics forIntwhich resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and ! -calculus.

448 citations

Journal ArticleDOI
TL;DR: The article proposes a theory of grammar relating syntax, discourse semantics, and intonational prosody and extends an earlier account based on Combinatory Categorial Grammar, which directly pairs phonological and logical forms without intermediary representational levels.
Abstract: The article proposes a theory of grammar relating syntax, discourse semantics, and intonational prosody. The full range of English intonational tunes distinguished by Beckman and Pierrehumbert (1986) and their semantic interpretation in terms of focus and information structure are discussed, including "discontinuous" themes and rhemes. The theory extends an earlier account based on Combinatory Categorial Grammar, which directly pairs phonological and logical forms without intermediary representational levels.

437 citations

Journal ArticleDOI
TL;DR: This article describes a number of log-linear parsing models for an automatically extracted lexicalized grammar and develops a new model and efficient parsing algorithm which exploits all derivations, including CCG's nonstandard derivations.
Abstract: This article describes a number of log-linear parsing models for an automatically extracted lexicalized grammar. The models are “full” parsing models in the sense that probabilities are defined for complete parses, rather than for independent events derived by decomposing the parse tree. Discriminative training is used to estimate the models, which requires incorrect parses for each sentence in the training data as well as the correct parse. The lexicalized grammar formalism used is Combinatory Categorial Grammar (CCG), and the grammar is automatically extracted from CCGbank, a CCG version of the Penn Treebank. The combination of discriminative training and an automatically extracted grammar leads to a significant memory requirement (up to 25 GB), which is satisfied using a parallel implementation of the BFGS optimization algorithm running on a Beowulf cluster. Dynamic programming over a packed chart, in combination with the parallel implementation, allows us to solve one of the largest-scale estimation problems in the statistical parsing literature in under three hours. A key component of the parsing system, for both training and testing, is a Maximum Entropy supertagger which assigns CCG lexical categories to words in a sentence. The supertagger makes the discriminative training feasible, and also leads to a highly efficient parser. Surprisingly, given CCG's “spurious ambiguity,” the parsing speeds are significantly higher than those reported for comparable parsers in the literature. We also extend the existing parsing techniques for CCG by developing a new model and efficient parsing algorithm which exploits all derivations, including CCG's nonstandard derivations. This model and parsing algorithm, when combined with normal-form constraints, give state-of-the-art accuracy for the recovery of predicate-argument dependencies from CCGbank. The parser is also evaluated on DepBank and compared against the RASP parser, outperforming RASP overall and on the majority of relation types. The evaluation on DepBank raises a number of issues regarding parser evaluation. This article provides a comprehensive blueprint for building a wide-coverage CCG parser. We demonstrate that both accurate and highly efficient parsing is possible with CCG.

433 citations

Book
04 Apr 2013
TL;DR: The Curry-Howard isomorphism as discussed by the authors is a type-free lambda-calculus, which is a variant of the Lambda-cube, and it can be expressed as follows:
Abstract: Preface Outline Acknowledgements 1. Typefree lambda-calculus 2. Intuitionistic logic 3. Simply typed lambdacalculus 4. The Curry-Howard isomorphism 5. Proofs as combinators 6. Classical logic and control operators 7. Sequent calculus 8. Firstorder logic 9. Firstorder arithmetic 10. Godel's system T 11. Secondorder logic and polymorphism 12. Secondorder arithmetic 13. Dependent types 14. Pure type systems and the lambda-cube 15. Solutions and hints to selected exercises 16. Solutions for chapter 6 Appendix A Mathematical Background Appendix B Solutions to Selected Exercises Bibliography Index

401 citations

Journal ArticleDOI
TL;DR: This paper explains the motivations and reasoning behind the development of the TPTP (thus implicitly explaining the design decisions made) and describes theTP contents and organization.
Abstract: This paper provides a detailed description of the CNF part of the TPTP Problem Library for automated theorem-proving systems The library is available via the Internet and forms a common basis for development and experimentation with automated theorem provers This paper explains the motivations and reasoning behind the development of the TPTP (thus implicitly explaining the design decisions made) and describes the TPTP contents and organization It also provides guidelines for obtaining and using the library, summary statistics about release v121, and an overview of the tptp2X utility program References for all the sources of TPTP problems are provided

386 citations