Focusing on Binding and Computation Robert Harper Carnegie Mellon University (Joint work with Dan Licata and Noam Zeilberger) June 18, 2008 1 / 41
The Payload • Main Results and Ideas • Main Results and Ideas Motivation Focusing Generalized Datatypes Conclusion The Payload 2 / 41
Main Results and Ideas The Payload Integrate Logical Frameworks and Functional Programming. • Main Results and Ideas • LF level provides a generalized datatype mechanism adequate • Main Results and Ideas for syntax, judgements, rules, proofs. Motivation • FP level provides the means to compute over these datatypes. Focusing In this talk we restrict attention to simple (non-indexed) types (to Generalized Datatypes Conclusion appear, LICS 2008). Current work on extending to dependent types and indexed types (not to appear, ICFP 2008). 3 / 41
Main Results and Ideas The Payload Polarized type systems. • Main Results and Ideas • Positive types are inductively defined by intro/focusing rules, • Main Results and Ideas manipulated by elim/inversion rules. Motivation • Negative types are inductively defined by elim/inversion rules, Focusing manipulated by intro/focusing rules. Generalized Datatypes Conclusion Contextual modal type systems. � Ψ � A has as elements “open terms” with parameters specified • by context Ψ . • Treats binding and scope without reliance on effects/state. 4 / 41
The Payload Motivation • Representation and Computation • Example: Domain-Specific Logics • Example: Domain-Specific Logics • Example: Domain-Specific Logics • Representation and Computation Motivation • Derivability and Admissibility • Representation and Computation Focusing Generalized Datatypes Conclusion 5 / 41
Representation and Computation The Payload Goal: integrate representation and computation in a functional Motivation language. • Representation and Computation 1. Representation: types for syntax including binding and scope. • Example: Domain-Specific Logics 2. Computation: type of higher-order computations over these • Example: Domain-Specific Logics types. • Example: Domain-Specific Logics • Representation and Computation • Derivability and Admissibility • Representation and Computation Focusing Generalized Datatypes Conclusion 6 / 41
Representation and Computation The Payload Goal: integrate representation and computation in a functional Motivation language. • Representation and Computation 1. Representation: types for syntax including binding and scope. • Example: Domain-Specific Logics 2. Computation: type of higher-order computations over these • Example: Domain-Specific Logics types. • Example: Domain-Specific Logics • Representation and Requirements: Computation • Derivability and 1. Sufficiently powerful to represent syntax, judgements, rules, Admissibility • Representation and proofs. Computation 2. Sufficiently flexible to permit computation by structural induction Focusing modulo α -equivalence. Generalized Datatypes Conclusion 3. Purely functional, so that we may index types by syntax. 6 / 41
Example: Domain-Specific Logics The Payload Access control logic (excerpts): Motivation • Representation and sort : type. Computation • Example: princ : sort. Domain-Specific Logics • Example: res : sort. Domain-Specific Logics • Example: Domain-Specific Logics term : sort => type. • Representation and Computation dan : term princ. • Derivability and bob : term princ. Admissibility • Representation and /home/dan/pub : term res. Computation Focusing prop : type. Generalized Datatypes owns : term princ => term res => prop. Conclusion mayrd : term princ => term res => prop. 7 / 41
Example: Domain-Specific Logics The Payload Access control logic (excerpts): Motivation • Representation and true : prop => type. Computation • Example: affirms : term princ => prop => type. Domain-Specific Logics • Example: Domain-Specific Logics impi : (imp A B) true <= (A true => B true). • Example: Domain-Specific Logics impe : B true <= A true <= (imp A B) true. • Representation and Computation • Derivability and aff : K affirms A <= A true. Admissibility • Representation and Computation saysi : (K says A) true <= K affirms A. Focusing sayse : (K affirms C) <= (says K A) <= Generalized Datatypes (K affirms A => K affirms C). Conclusion 8 / 41
Example: Domain-Specific Logics The Payload Signature for proof-carrying access control: Motivation • Representation and type file[r:term res] Computation • Example: val paper.tex : file[/home/dan/pub] Domain-Specific Logics • Example: Domain-Specific Logics type iam[p:term princ] • Example: Domain-Specific Logics val iambob : iam[bob] • Representation and Computation • Derivability and val read : Admissibility • Representation and ∀ r. ∀ p. ∀ pf:atom (p mayrd r) true. Computation file[r] -> iam[p] -> string Focusing Generalized Datatypes Implementation of read structurally analyzes proofs at run-time! Conclusion 9 / 41
Representation and Computation There are two different function spaces in play here! The Payload Motivation Representational: A ⇒ B (aka B ⇐ A ). • Representation and 1. Computation Computational: A → B (aka B ← A ). 2. • Example: Domain-Specific Logics • Example: Representational functions: Domain-Specific Logics • Example: Domain-Specific Logics • Representation and Computation • Derivability and Admissibility • Representation and Computation Focusing Generalized Datatypes Conclusion 10 / 41
Representation and Computation There are two different function spaces in play here! The Payload Motivation Representational: A ⇒ B (aka B ⇐ A ). • Representation and 1. Computation Computational: A → B (aka B ← A ). 2. • Example: Domain-Specific Logics • Example: Representational functions: Domain-Specific Logics • Example: Domain-Specific Logics • Adequate for syntax, rules, proofs. • Representation and Computation • Derivability and Admissibility • Representation and Computation Focusing Generalized Datatypes Conclusion 10 / 41
Representation and Computation There are two different function spaces in play here! The Payload Motivation Representational: A ⇒ B (aka B ⇐ A ). • Representation and 1. Computation Computational: A → B (aka B ← A ). 2. • Example: Domain-Specific Logics • Example: Representational functions: Domain-Specific Logics • Example: Domain-Specific Logics • Adequate for syntax, rules, proofs. • Representation and • Closed-ended: schemas built from parameters by composing Computation • Derivability and rules. Admissibility • Representation and Computation Focusing Generalized Datatypes Conclusion 10 / 41
Representation and Computation There are two different function spaces in play here! The Payload Motivation Representational: A ⇒ B (aka B ⇐ A ). • Representation and 1. Computation Computational: A → B (aka B ← A ). 2. • Example: Domain-Specific Logics • Example: Representational functions: Domain-Specific Logics • Example: Domain-Specific Logics • Adequate for syntax, rules, proofs. • Representation and • Closed-ended: schemas built from parameters by composing Computation • Derivability and rules. Admissibility • Representation and Computation Computational functions: Focusing Generalized Datatypes Conclusion 10 / 41
Representation and Computation There are two different function spaces in play here! The Payload Motivation Representational: A ⇒ B (aka B ⇐ A ). • Representation and 1. Computation Computational: A → B (aka B ← A ). 2. • Example: Domain-Specific Logics • Example: Representational functions: Domain-Specific Logics • Example: Domain-Specific Logics • Adequate for syntax, rules, proofs. • Representation and • Closed-ended: schemas built from parameters by composing Computation • Derivability and rules. Admissibility • Representation and Computation Computational functions: Focusing • Compute by pattern matching. Generalized Datatypes Conclusion 10 / 41
Representation and Computation There are two different function spaces in play here! The Payload Motivation Representational: A ⇒ B (aka B ⇐ A ). • Representation and 1. Computation Computational: A → B (aka B ← A ). 2. • Example: Domain-Specific Logics • Example: Representational functions: Domain-Specific Logics • Example: Domain-Specific Logics • Adequate for syntax, rules, proofs. • Representation and • Closed-ended: schemas built from parameters by composing Computation • Derivability and rules. Admissibility • Representation and Computation Computational functions: Focusing • Compute by pattern matching. Generalized Datatypes • Open-ended: any form of computation allowable. Conclusion 10 / 41
Derivability and Admissibility Representational functions witness derivabilities, J 1 ⊢ J 2 . The Payload Motivation • Representation and Computation • Example: Domain-Specific Logics • Example: Domain-Specific Logics • Example: Domain-Specific Logics • Representation and Computation • Derivability and Admissibility • Representation and Computation Focusing Generalized Datatypes Conclusion 11 / 41
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