Abstract Data Types

1. Abstract Data Types • Structured (data) objects as terms of a logical language • behavior of operations specified by axioms • symbolic execution • Semantics: Algebraic structures given by equivalence classes of variable free terms • (freely) generated structures • constructors, selectors, recursively defined functions • Proof (tool) support • deduce properties of algebraic structures • induction on the structure of data types

2. Abstract Data Types : Language • Formal language: First Order (Predicate) Logic (FOL) • (user defined) predicates • boolean connectives: AND (Æ), OR (Ç), NOT (:), -> (!) … • equality (=) between terms over user defined (typed) function symbols • universal, existential quantification : ALL (8), EX (9) • types for quantified variables • (outer) universal quantifiers sometimes omitted • + (other) auxiliary symbols • ASCII versus Latex symbols • uppercase / lowercase letters not relevant (here)

3. Abstract Data Types : Natural Numbers • Natural numbers: • Defines theory NATBASIC introducing the type NAT. • Constructor symbols: NULL (nullary), SUCC (unary) • Selector symbol: PRED • Cases enumerated by | : • disjoint union (sum) • possibly also test predicates • see exercise below

4. Abstract Data Types : Example • Axioms (generated from the specification) • 8 x: NAT. NULL  SUCC(x) • 8 x,y: NAT. ( SUCC(x) = SUCC(y) ! x=y ) • 8 x: NAT. PRED(SUCC(x)) = x • Are these axioms independent ?

5. Abstract Data Types : Semantics • Semantics: Set of equivalence classes of (closed) terms { {NULL, PRED(SUCC(NULL)), PRED(PRED(SUCC(SUCC(NULL)))), … }, {SUCC(NULL), PRED(SUCC(SUCC(NULL))), … }, {SUCC(SUCC(NULL))), … }, … } • Freely generated by NULL and SUC: There is a unique representation of each element (equivalence class) by SUC and NULL. • « PRED ¬ ({SUCC(SUCC(NULL))), … }) = {SUCC(NULL), PRED(SUCC(SUCC(NULL))), … }

6. Abstract Data Types : Extending Theories • Extending the theory by additional functions and predicates:

7. Abstract Data Types : Extending Theories

8. Abstract Data Types : Recursion and Induction • Recursive axiomatizations • intended as definitions • like addition above • well definedness: • existence (of a solution) ) consistency • uniqueness ) “termination” according to some well-founded order necessary for inductive proofs (following the recursion structure) • often fixed definition schemes

9. Abstract Data Types : Recursion and Induction • Proof of inductive properties • like 8 x,y: NAT. x+y = y+x • induction schemes critical for consistency • well-founded order • not user defined • for (freely) generated structures: structural induction • size function: reduction (for example) to (complete induction for) natural numbers ) size function itself has to be well defined • complete induction (scheme) on the structure of NAT-terms: ( (NULL) Æ8 y:NAT. ( (y) !(SUCC(y)) ) ) !8 x: NAT. (x) .

10. Abstract Data Types : Recursion and Induction • Prove: 8 x: NAT. ( x = NULL Ç9 y:NAT. x = SUCC(y) ) • What does this mean? • Disjointness?

11. Inductive Protocol Verification • Use these techniques for Protocol Verification ! • Steps: • Define data types for agents, keys, messages, and appropriate auxiliary notions (lists). • Define the attacker model (knowledge and abilities) in terms of parts, analz, and synth. • Define events and execution traces (for a given protocol): • events caused by normal participants • events caused by the attacker (or spy) • Formalize security properties (confidentiality, authentication). • Prove security properties by induction on the (structure) length of traces.

12. Step1: Agents • Secure agents are servers. • NAT parameter allows for arbitrary many agents. • Additional test predicates (is… ): • Give the axiom(s) for the predicate isFriend !

13. Step1: Keys • Formalizes the notions used so far.

14. Step1: Messages • What is the (formal) characteristics of atomic messages? atomic messages

15. Step1: Lists • Use (finite) lists instead of (finite) sets. • Size function used for induction schemes.

16. Step1: Lists • Additional notions for lists (of messages).

17. Step1: Lists • Analogous to lists of messages.

18. Step2: Decomposition of Messages • The function parts computes (the list of) all sub-messages • of messages given by a list (mL).

19. Step2: Decomposition of Messages • What about hashing?

20. Step2: Analysis of Messages(by the spy) • The function analz makes explicit the knowledge • (as a list of messages) the spy is able to obtain (infer) from • a given list of messages.

21. Step2: Analysis of Messages(by the spy) • Uses the auxiliary function hAnalz that takes as arguments two lists of messages. • A new round is started with an extended list.

22. Step2: Analysis of Messages(by the spy) decryption • Why does analz terminate?

23. Step2: Synthesis of Messages (by the spy) • isSynth(msg,mL) : msg is in the set of messages that can be generated (by the spy) from the list mL.

24. Step2: Synthesis of Messages (by the spy) • synth is given by a predicate. Why?

25. Step3: Protocol Events • Says-events have been used so far: a message is sent. • Gets-events explicitly model that a message is actually received. • Notes events model (internal) state transitions • spy learns a key (independently of protocol steps) • setting an internal clock

26. Step3: Protocol Traces • Analogous to lists of messages and keys. • Again: size function for inductive proofs.