A Modal Language for the Safety of Mobile Values (work in progress) SNU 4/7/2006 박성우 Sungwoo Park POSTECH

2 Distributed Computation Heterogeneous nodes with different local resources Mobile term (code) abstract datatype local heap

3 Modal Logic Modalities ¤ and } – ¤ A : necessarily A – } A : possibly A Spatially: – ¤ A : everywhere A – } A : somewhere A

4 Modal Type Theory Basic idea: enrich the type system with modal types Staged computation: temporal interpretation –box M : ¤ A M has type A at all subsequent stages Spatial interpretation –box M : ¤ A M has type A at every node, i.e., globally –dia M : } A M has type A at some node

5 Modal Type System for Distributed Computation Borghuis and Fejis '00 Jia and Walker, ESOP '04 –box M : ¤ A, M = mobile term, valid at every node –dia M : } A, M = mobile term, valid at some node –uses hybrid logic Murphy et al, LICS '04 –box M : ¤ A, M = mobile term, valid at every node –dia l : } A, l = reference to local resource Moody, '03 –box M : ¤ A, M = mobile term, valid at every node –dia M : } A, M = mobile term, valid at some node

6 Remote Evaluation box M : ¤ A V : A ??? M : A V : A

7 Remote Evaluation - Okay but not quite good box M : ¤ ¤ A N : A M : ¤ A box N : ¤ A V : A

8 Remote Evaluation in Jia & Murphy N : A M : ¤ A box N : ¤ A V : A

9 Harsh Reality Jia and Walker, ESOP '04 –uses hybrid logic (i.e., indices) Murphy et al, LICS '04 –Every term is mobile! –Then what is the ¤ modality for?

10 Why This Complication? Because they do not take into consideration value mobility! Consider a term M such that: –Term (code) mobility: Is the term M valid at a remote node? –Value mobility: Is the value V valid at a remote node? –These two are independent. M : A V : A

11 ¤ (int ! int), term: mobile, value: immobile let val new_reference = ref 0 val f = fn x => x + !new_reference in f end The term is valid at any node. The result f is local, however.

12 ¤ (int ! int), term: immobile, value: mobile let val v = !some_existing_reference val f = fn x => x + v in f end The term is local. The result f is valid at any node, however.

13 Key Idea box M : ¤ A M is valid at any node. V is valid at the current node, but we know nothing about its mobility. cir M : O A M is valid at the current node, but we know nothing about its mobility. V is valid at any node, however. M : A V : A

14 Outline Introduction V Modal language ¤ O with ¤ and O modalities Modal language with value mobility Logic of direct evidence

15 Plan ¤ O ¤ O

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17 ¤ ¤ O ¤ O

18 ¤ for Term Mobility

19 Type System for ¤

20 O ¤ O ¤ O

21 O for Value Mobility

22 Type System for O

23 Typing Rules in O

24 Reduction Rules in O

25 ¤ O ¤ O ¤ O

26 ¤ O  =  ¤  + O Additional typing rule and substitution Special rules for primitive types –e.g., booleans, integers, …

27 Good Things about ¤ O

28 Bad Things about ¤ O Complexity –when the system is augmented with indices and communication constructs 30 pages of type safety proof even without mutable references –Mechanizing type safety proof seems necessary. POPLMark Challenge Redundancy –'really' serious problem

29 Key Observation: Redundancy Term mobility is a special case of value mobility. Term M is mobile? Value x:_. M is mobile? Value box M is mobile? Value … M … is mobile?, Conclusion: ditch the ¤ modality.

30 Outline Introduction V Modal language ¤ O with ¤ and O modalities V Modal language ¡} with value mobility Logic of direct evidence

31 Key Idea: Value Mobility Only box M : ¡ A V is valid at every node. I.e., ¡ = O ¼ necessity modality dia M : } A V is valid at some node. I.e., } ¼ possibility modality M : A V : A

32 Plan ¡ } ¡}

33 with Call-by-value

34 ¡ ¡ } ¡}

35 ¡

36 Type System for ¡

37 } ¡ } ¡}

38 }

39 Type System for } (1/2)

40 Type System for } (2/2)

41 Soundness of the Type System for }

42 ¡} and Beyond ¡ } ¡} ¡} store ¡} store+communication

43 Application Robotics –communication constructs –does not use code mobility. Grid computing –distributed computation on the network –makes heavy use of code mobility.

44 Outline Introduction V Modal language ¤ O with ¤ and O modalities V Modal language ¡} with value mobility Logic of direct evidence

45 Motivation What is the logic for ¡} under the Curry-Howard isomorphism? –Type-theoretically, we distinguish between values and ordinary terms. –Logically, we distinguish between (weak) normal proofs and ordinary proofs. So we develop a logic of normal proofs, or direct evidence.

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