Focusing in Proof-search and

Concurrent Synchronization

Deepak GargCarnegie Mellon University

(Based on joint work with Frank Pfenning)

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Objectives

• Direct translation of asynchronous pi-calculus to intuitionistic linear logic formulas

• Committed Forward chaining in linear logic to simulate pi-calculus reductions

• Combine with focusing to obtain atomicity in synchronization

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Direct translation

• Connectives of pi-calculus map to connectives of linear logic

First attempt at translation

(No focusing, unsound!)

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Pi-calculus, Linear Logic

• Asynchronous pi-calculus (without replication):

• Intuitionistic linear logic fragment:

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Pi-calculus: semantics

• Modified CHAM semantics

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Linear Logic: Forward Chaining

• Judgment:

Term variables Formulas (Linear)

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Linear Logic: Forward Chaining

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Forward Chaining as Rewriting

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Forward Chaining as Rewriting

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Forward Chaining as Rewriting

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Translation

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Simulation example

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Unsound!

• Another possible reduction sequence:

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Unsoundness analysis

• Forward chaining may get stuck incorrectly

• Universal quantifier and implication in inputs must be eliminated simultaneously

• Can be done using focusing– Each communication is exactly one focusing

Focusing in Forward Chaining

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Synchronous and Asynchronous

• Divide formulas into (right) synchronous S and (right) asynchronous A

• Coerce S to A via a monad (CLF, LolliMon)

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Focused Forward Chaining

• Judgments

• Different from CLF focusing– Decomposition of ­, 9 on left is not in focus– No right rules

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Focused Forward Chaining as Rewriting

• We read rules bottom up, ignore synchronous goals and get a conditional rewrite system.

• Judgments:

Unchanged

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Complete Rewrite System

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Translation of the pi-calculus

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Simulation example

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Simulation example

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Correctness of translation

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A Strange External Choice…

• What does A&B correspond to in the pi-calculus?

• It corresponds to external choice between input/output actions

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Extension to the logic

• Extension of asynchronous formulas

• New focusing rules

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Translation of choice

Correctness results remain the same

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Expressiveness

• How expressive is this extension of the asynchronous pi-calculus?

• Conjecture: It is as expressive as the synchronous pi-calculus (with choice, without replication)

- Translation follows from Boudol’s encoding

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3-way synchronization

• Can encode receivers with 2 simultaneous inputs

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Encoding 3-way synchronization

• Works because both implications must be eliminated in one focusing step

• Generalizes to n-way inputs

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Other connectives

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Summary

• Dynamic semantics of the asynchronous pi-calculus can be simulated using focusing and forward chaining

• Focusing – atomicity in synchronization

• Correspondence between connectives of the pi-calculus and intuitionistic linear logic

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Related Work

• Translation idea is not new– Miller92 – pi-calculus as a theory in linear

logic– Cervesato03 – similar idea, no focusing

• Abramsky93 – classical logic and concurrency

• Concurrent Logic Programming