Probabilistic NetKAT
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1 Probabilistic NetKAT Nate Foster (Cornell) Dexter Kozen (Cornell) Konstantinos Mamouras (Penn) Mark Reitblatt (Facebook) Alexandra Silva (UCL) Steffen Smolka (Cornell)
2 Team ProbNetKAT Nate Foster Dexter Kozen Mark Reitblatt Konstantinos Mamouras [ESOP '16] Alexandra Silva Steffen Smolka
3 Team ProbNetKAT Nate Foster Dexter Kozen Mark Reitblatt Konstantinos Mamouras Alexandra Silva Steffen Smolka
4 Context: Software-Defined Networking
5 Context: Software-Defined Networking High Performance ` Nightmare to Maintain & Control at Scale
6 Networks have become programmable (just now!) high-level program low-level rule tables Firewall Pattern Pattern Pattern ; Route Compiler Actions Actions Actions dstport=22 Drop dstport=22 Drop srcip= dstport=22 Forward Drop srcip= /8 Forward 1 srcip= */8 Forward Forward 1 */8 Forward 2 1 * Forward 2 2 System SDN switches
7 Networks have become programmable (just now!) high-level program low-level rule tables Firewall Pattern Pattern Pattern ; Route Compiler Actions Actions Actions dstport=22 Drop dstport=22 Drop srcip= dstport=22 Forward Drop srcip= /8 Forward 1 srcip= */8 Forward Forward 1 */8 Forward 2 1 * Forward 2 2 System PL Problems: programming abstractions compilation verification synthesis, SDN switches
8 NetKAT: A Success Story programming language, modeling language, programming logic
9 NetKAT: A Success Story programming language, modeling language, programming logic powerful tools Compiler Virtualization Automatic Verification basis of many systems
10 NetKAT: A Success Story programming language, modeling language, programming logic rich theory Enables powerful tools [ p ] denotational semantics p q sound & complete axiomatization Compiler Virtualization Automatic Verification automata theory symbolic representation basis of many systems
11 ProbNetKAT: Not Yet A Success Story programming language, modeling language, programming logic rich theory Enables powerful tools [ p ] denotational semantics
12 ProbNetKAT: Not Yet A Success Story programming language, modeling language, programming logic rich theory Enables powerful tools [ p ] denotational semantics
13 ProbNetKAT: Not Yet A Success Story programming language, modeling language, programming logic rich theory Enables powerful tools [ p ] denotational semantics
14 Language
15 Model Packets are records of values. Programs are functions on packets. { switch = A, port = 3, ethsrc = 8:8:::::8:8, ethdst = 2:2:::::2:2, vlan = 8, ipsrc = , ipdst = ,... }
16 Review: NetKAT Language pol ::= false true field = val field := val pol 1 + pol 2 pol 1 ; pol 2!pol pol * S S'
17 Review: NetKAT Language pol ::= false Boolean Algebra true field = val field := val pol 1 + pol 2 pol 1 ; pol 2!pol pol * S S'
18 Review: NetKAT Language pol ::= false true field = val field := val pol 1 + pol 2 Boolean Algebra + Kleene Algebra "Regular Expressions" pol 1 ; pol 2!pol pol * S S'
19 Review: NetKAT Language pol ::= false true field = val field := val pol 1 + pol 2 pol 1 ; pol 2!pol pol * Boolean Algebra + Kleene Algebra "Regular Expressions" + Packet Primitives S S'
20 Review: NetKAT Language pol ::= false Boolean Algebra true if p then q else r p;q +!p;r field = val field := val ` while p do q p;q*;!p pol 1 + pol 2 + Kleene Algebra "Regular Expressions" pol 1 ; pol 2!pol pol * + Packet Primitives S S'
21 Review: NetKAT Semantics pol ::= false true field = val field := val pol 1 + pol 2 pol 1 ; pol 2!pol pol * S S'
22 Review: NetKAT Semantics pol ::= Local NetKAT: input-output behavior of switches false true field = val field := val pol 1 + pol 2 pol Packet Packet Set pol 1 ; pol 2!pol pol * S S'
23 Review: NetKAT Semantics pol ::= Local NetKAT: input-output behavior of switches false true field = val field := val pol 1 + pol 2 pol Packet Packet Set pol 1 ; pol 2!pol pol * Global NetKAT: network-wide behavior S S' pol History History Set
24 Example A B
25 Local NetKAT Program A B pol A pol B
26 Local NetKAT Program A B port:=3???
27 Local NetKAT Program A B port=1; tag:=1; port:=3 + port=2; tag:=2; port:=3???
28 Local NetKAT Program A B port=1; tag:=1; port:=3 + port=2; tag:=2; port:=3 tag=1; port:=5 + tag=2; port:=6
29 Global NetKAT Program A B pol
30 Global NetKAT Program A B port=1; A B; port:=5 + port=2; A B; port:=6
31 Virtual NetKAT Program A B
32 Virtual NetKAT Program virtual "big switch" A B
33 Virtual NetKAT Program virtual "big switch" A B port=1; port:=5 + port=2; port:=6 even simpler!
34 Probabilistic NetKAT
35 Probabilistic NetKAT pol ::= false true field = val field := val pol 1 + pol 2 pol 1 ; pol 2 pol 1 r pol 2!pol pol * S S'
36 Probabilistic NetKAT pol ::= false Randomized Routing pt:=1.5 pt:=2 true field = val field := val pol 1 + pol 2 Link Failure A B.99 false pol 1 ; pol 2 pol 1 r pol 2!pol pol * S S'
37 Probabilistic NetKAT pol ::= false Randomized Routing pt:=1.5 pt:=2 true field = val field := val pol 1 + pol 2 pol 1 ; pol 2 pol 1 r pol 2!pol pol * S S' Link Failure A B.99 false Expected Congestion? Probability of Delivery? Quantitative Reasoning!
38 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History)
39 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History) Problem: Can't express correlation! pt:=1 0.5 pt:=2 (pt:=1+pt:=2) 0.5 false
40 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History Set) pol History History Set [0,1]
41 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History Set) pol History History Set [0,1] Problem: Some distributions give pr=0 to all points. p ; (S S ; p)*
42 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History Set) pol History B [0,1] where B are Borel sets of the Cantor Space
43 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History Set) pol History B [0,1] where B are Borel sets of the Cantor Space B is set of set of set of histories! uncountable set of uncountable sets!
44 Denotational Semantics [ESOP '16] Before: pol History History Set pol History Dist(History Set) Math works out, but nightmare ` to deal with! pol History B [0,1] where B are Borel sets of the Cantor Space B is set of set of set of histories! uncountable set of uncountable sets!
45 Problem: Know how to model properties mathematically Program p, Property X: 2 H -> R, Input Distribution μ E[X] = X(a) [ p ](μ, da) (Lebesgue Integral over 2 H ) but not how to compute the answer!
46 Positive Results
47 Measures μ : B->[0,1] are "finitely observable". (I.e., if μ1 μ2 there exists a finite witness!) Automata should be able to capture μ!
48 We have identified an order p q that should allow (arbitrarily close) approximation! p (0) p (1) p (2) p*
49 Wrap-Up Classical (continuous) probability theory doesn't come with algorithms out of the box Need to develop domain-specific theory & algorithms Will likely generalize to other domains
50 Questions?
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