CLF: A Concurrent Logical Framework

David WalkerPrinceton

(with I. Cervesato, F. Pfenning, K. Watkins)

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Outline What is a logical framework? Logical frameworks & security CLF: a concurrent logical framework

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Logical Frameworks a logical framework is a meta-language for

representing deductive systems sample deductive systems

logics of all kinds first-order logic, higher-order logic, temporal logic,...

programming languages ML, lambda calculus, pi calculus

specification languages set theory, type theory, multi-set rewriting, ...

compilers translations between languages or logics

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Logical Frameworks

meta-logicalframeworks

logicalframeworks

logics

proofs

Twelf, Nuprl, HOL, ...

Automath, Nuprl,HOL, LF, ...

FOL, linear logic,temporal logic

(A and B) => C

represent propertiesof logics

representlogics

represent proofs

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Logical Frameworks

meta-logicalframeworks

logicalframeworks

programminglanguages

programs

Twelf, Nuprl, HOL, ...

Automath, Nuprl,HOL, LF, ...

C, ML, Javalambda calculus

if (n > 17) {...}

represent propertiesof programming langs

representprogramming langs

represent programs

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Tasks for Logical Frameworks Represent syntax

of a logic of a programming language

Represent valid rules of inference inference rules (modus ponens, law of the

excluded middle) of a logic typing rules, operational rules of a language

Represent valid deductions valid proofs well-typed programs, valid executions

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Outline What is a logical framework? Logical frameworks & security CLF: a concurrent logical framework

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Application I: proof-carrying X Proof-carrying code [Necula, Lee]

Attach a proof of some safety property to mobile code

Represent and check the proof in LF Proof-carrying authentication [Appel,

Felten]

Attach a proof of access rights to database query, service request

Represent and check the proof in LF

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Application II: Mechanical Reasoning

Increase our confidence in system properties through the following process

1. Represent algorithm, deductive system2. Prove adequacy of representation3. Prove security properties within the framework

LF normally small, simple and trustworthy eg: LF has two type constructors Appel et al. (100k lines LF, 0 soundness bugs)

Step 2 is least trustworthy

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Application II: Mechanical Reasoning

Authentication protocols Extremely difficult to get correct

Highly concurrent Errors are costly

Many different specification languages informal notation, security logics, MSR, spi

calculus,... Represent protocols, languages & their

properties/relationships in a logical fwrk

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Outline What is a logical framework? Logical frameworks & security CLF: a concurrent logical framework

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Logical framework design Logical framework design resembles

programming language design Simple logical frameworks are like

simple programming languages few features = simple, trustworthy

implementation but limited expressiveness expressive enough in principle but

awkward and unmanageable in practice adequacy may be unclear, difficult to prove

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The LF family of Logical Frameworks Automath [De Bruijn], LF [Harper et al.]

Dependent types, intuitionistic functions Represents pure PL, logics Higher-order abstract syntax

Linear LF [Cervesato, Pfenning]

LF + linear functions, additive pairs Represents state, effectful programming

paradigms Resource-conscious logics

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Concurrent LF (CLF)

Linear LF + a concurrency monad The monad encapsulates a concurrent

computation Represents concurrent programming

paradigms MSR specifications of security

protocols, pi-calculus, CML, Petri nets

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CLF types Linear LF types + concurrency

monad

Represent the state of a concurrent computation

A,B ::= a | x:A.B | A -o B | T | A & B | {S}

S ::= A | 1 | S1 S2 | x:A.S | !A

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Pure CLF terms Pure Linear LF terms + concurrent

computations A monad encapsulates

computations within the pure language

pure terms M,N ::= c | x:A.M | M N | ... | {E}

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Effectful CLF computations Computation E processes the ambient

state A let expression represents a step in a

computation Patterns p introduce new state

computations E ::= let {p} = M in E | F

patterns p ::= x:A | 1 | p1 p2 | x:A.p | !x

states F ::= M | 1 | F1 F2 | [M,F] | !M

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Example

key : type.msg : type.encrypt : key -> msg -> msg.wire : msg -> type.send : M:msg.{wire M}.rec : M:msg.wire M -o {1}.

{

let {m:wire M} = send M in

let {m’:wire M’} = send M’ in

let {1} = rec M’ m’ in

let {1} = rec M m in

...

}

CLF signature: Valid Computation:

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Concurrency equations LF and related dependent type theories

provide a notion of definitional equality When MN, the framework cannot

distinguish M and N We extend the LF equality:

The framework cannot distinguish the order of independent computation steps

let {p} = M in let {p’} = M’ in E let {p’} = M’ in let {p} = M in E

(M,M’ do not containvariables in p,p’)

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Example

{

let {m:wire M} = send M in

let {1} = rec M m in

let {m’:wire M’} = send M’ in

let {1} = rec M’ m’ in

...

}

Valid Computation: Equivalent Computation:

{

let {m:wire M} = send M in

let {m’:wire M’} = send M’ in

let {1} = rec M’ m’ in

let {1} = rec M m in

...

}

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Results Definition and meta-theory of CLF

A new presentation of LF dependent type theory based on canonical forms

Decidability of type checking Examples and applications

Representation of Petri nets, synchronous and asynchronous pi-calculus with proofs of adequacy

Novel, modular representation of CML in CLF Representation of MSR & examples

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Further reading Cervesato, Pfenning, Walker, Watkins.

A concurrent logical framework. January 2002 (submitted).

Forthcoming technical reports: Watkins et al. A concurrent logical

framework I: Judgments and properties. Cervesato et al. A concurrent logical

framework II: Examples and applications.

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Future work Implementation and logic programming

operational semantics for CLF Represent, check and execute concurrent

computations Meta-logical reasoning about

concurrent computations Properties of concurrent computations,

bisimulation Automatic proof search

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Conclusions Study of logical frameworks is an

important aspect of the development of secure and reliable systems

CLF enables concise and adequate representations of concurrent languages