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1 The logic-automaton connection and applications Mona project Initiated at the University of Aarhus (BRICS) Google: Mona Michael I. SchwartzbachAnders Møller Nils Klarlund

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2 Overview Introduction Pointer reasoning and start of project Verification of protocols Fast parsing with declarative constraints over trees What is WS1S? The Mona tool in use What else has been accomplished

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3 Automata Regular expressions automata –Useful, right? –Solves problems such as Expressing text patterns Expressing paths in graphs Are regular languages limiting our use of automata? –With complement operator, regular expression emulate propositional logic! –But no quantification with REs?

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4 QPL Quantified Propositional Logic is fundamental in verification Boil your problem down to QPL, then solve A compositional framework for “modeling” phenomena (albeit of limited expressive power) How to solve? –Use BDDs –But they are automata Albeit not general ones, they are acyclic

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5 Mona in Essence: Extend QPL, Tie to Automata, Solve More Problems! WS1S is the answer –It ties the class of all automata to a logic –It becomes a vehicle for the operations Cross product, Determinization, Subset construction, Projection, Complementation

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6 That Verification Problems or Data Types or Invariants Can Be Expressed As Regular Languages: Not a New Idea E.W.Dijkstra (parameterized verification) N.D. Jones & S. Muchnik (tree grammars) A. Gupta (parameterized hardware) Early 60es: logic and automata for describing temporal behavior of sequential circuits

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7 Motivation I: Pointers Pointer manipulation in program is very difficult to get right It shouldn’t be too difficult to verify that shapes in the heap stay invariant over a few operations? No dangling pointers, all allocated memory accessible, no sharing of structures supposed to be separate WF(store) –Let X reachable nodes from x Y reachable nodes from y –Intersection of X and Y is empty –Union of X and Y is all of the store xy X X

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8 Floyd-Hoare Logic of Pointers (WF(S) & S S’) => WF(S’) –Where is the transition relation that reflects pointer surgery Is this even decidable? Yes, because we can formulate it in WS1S through predicate transformations So, let’s build (1994) –A decision procedure for WS1S –A tool for translating WF predicates to WS1S –That holds takes 4 hours to calculate!

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9 Additional work Automatic Verification of Pointer Programs using Monadic Second-order Logic [PLDI ’97] Pointer Assertion Logic Engine [PLDI ’01] Related work on shape analysis We still didn’t explain WS1S

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10 Motivation II: Parameterized verification: Sliding Window Protocol (w. Mark A. Smith ) A sequence number is used as an acknowledgement The window size is the max. number of messages in transit We model –Unbounded queues –Unbounded channels –Dynamic window size

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11 Sliding window protocol

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12 We must prove: What goes out is what comes in Variables ( D is a finite domain) SendBuf: Seq[D] := {}, hSendBuf: Int := 1, W: Int := choose n where (n > 0), RetranBuf: Seq[D] := {}, hRetranBuf: Int := 1, readyToSend: Bool := false, Some variables flex in one, some even in two dimensions segment: D, seqNum: Int := 0, RcvBuf: Seq[D] := {}, hRcvB: Int := 1, sendAck: Bool := false, temp: D, transitSR: Map[Int, Mset[D]] := empty, transitRS: Map[Int, Mset[A]] := empty

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13 What kind of code? internal prepareNewSeg(d) pre readyToSend = false /\ hSendBuf <= len(SendBuf) /\ d = SendBuf[hSendBuf] /\ len(RetranBuf) < W eff RetranBuf := RetranBuf |- d; seqNum := hSendBuf; hSendBuf := hSendBuf + 1; readyToSend := true; segment := d internal sendpktSR(d) pre readyToSend = true /\ d = segment eff readyToSend := false; transitSR := update(transitSR, seqNum, insert(d, transitSR[seqNum]))

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14 We Note Operations work on both ends of linear lists We maintain pointers and length information The rest is nitty-gritty, boring stuff How is this related to regular languages? The system as it evolves over time is not a regular language! Are configurations regular languages? –Yes, if everything stretches in one dimension and –Indexing operations not ‘too complicated’ –WS1S will make this precise –Do changes to configurations, that is, operations, preserve regularity? –WS1S again can help us understand

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15 Motivation III: YakYak---A Fast Parser With Constraints on Parse Trees Logic notations for parsing > 69 different Yacc-like parsers available… So what’s new: a concise, declarative way of specifying constraints on parse trees That also yields a fast parser

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16 YakYak Consider HTML –An a element denotes a text anchor –Text is in bold if inside a bold element Here are two constraints –“For all positions p with p an a element there is not a position q below p that is an a element” –“If any part of a text within an a element is in bold then all anchor texts must be in bold”

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17 How to Turn Such Constraints Into Automata? Note we need tree automata Xpath formulation possible (if parse tree was XML) XML parse tree would be slow Xpath query evaluation would be slow Goal: one transition per production per constraint in a pre-computed automaton that works bottom- up We need to go from formulas to tree automata!

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18 What Is WS1S Weak Second-order theory of 1 Successor First-order terms t –0, p, t’ + 1 Second-order terms T –Empty, P, T union T’, T intersection T’ Formulas ’, ~ v ’ -t = t’, t < t’, t in T, b -ex2 P: -ex1 p: -ex0 b:

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19 A. Meyer’s Result Deciding WS1S is non-elementary No finite stack of exponentials can limit the growth Each quantifier bumps you up one exponential Recall: first experiment for very simple example took 4 hours to complete Some people have suggested that we should have given up at this point For more on this viewpoint – Google: Klarlund madman

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20 Example var2 P,Q; P\Q = {0,4} union {1,2}; var1 x; var0 A; ex2 Q: x in Q & (all1 q: (0 (q in Q => q - 1 notin Q) & (q notin Q => q - 1 in Q)) & 0 in Q; A & x notin P;

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21 Mona Output A counter-example of least length (1) is: P X X Q X X x X 1 A 0 X P = {}, Q = {}, x = 0, A = false A satisfying example of least length (7) is: P X 1110100 Q X 000X0XX x X 0000001 A 1 XXXXXXX P = {0,1,2,4}, Q = {}, x = 6, A = true

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22 What Was Calculated

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23 A BDD Represents a Boolean Function of Boolean Variables BDD = Boolean Decision Diagram x 1 or (x 2 iff x 3 ) Often the diagram is very sparse

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24 The Central Trick in Mona Is Similar

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25 Now Formulate Algorithms Keep automata determinized and minimal Cross product (for & and v) Projection for existential quantification Subset construction for determinization Minimization

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26 Three and Six-valued Logic To really make Mona work, we had to overcome spurious state space explosions They were a direct consequences of working in an only two-valued logic! The problem: say you want to model {green, blue, red}. You need two bits, say X and Y 00=green, 01=blue, 10=red. Then, what is the truth status of formulas when XY=11? For more, see [J. HOSC, to appear] Many more tricks in [IJFCS 2002]

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27 Applications of Mona Debian/GNU package; also AIX package Integrated with PVS, a leading theorem proving environment, SRI Used as essential tool in Ph.D. theses and other research such as natural language processing (Ohio) duration calculus verifier (Mumbai) Mona as decision procedure for description logics (Dresden) verification of parameterized systems (Kiel) verification and reachability (Upsala) multimedia applications (Kent) automata-based representations for arithmetic (Santa Barbara) Presburger arithmetic (Synopsis) automata in control synthesis (Aarhus) acceleration of counter automata (Cachan) verification of structures in imperative programs (Tel Aviv) high-level language for verification (Toulouse) a WS2S specification language (Freiburg) YacYac parser generator (Aarhus) Pale pointer engine (Aarhus) Google Mona for home page with many papers online

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28 Explain Automatic Pointer Reasoning I points-to(a,b) iff cell at a contains a pointer to b This predicate is definable for a wf store (because of list/tree assumptions) Assume we want to verify {P}S{Q} S is straight-line code, say p^.next := x The store after is the same as before except that the predicate points-to(a,b) has changed for a= p

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29 Explain Automatic Pointer Reasoning II Let Q’ be Q, rewritten to account for a= p situation The WF property can be expressed using least-fixed points in WS1S (or WS2S) based on the points-to predicate WF is assumed in initial store by storage layout model So, we need to verify P Q’ & WF’

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30 Explain Automatic Pointer Reasoning III Sometimes we need an invariant x is only empty if y is empty, and p points to the last element of z ( x = nil => y = nil ) & z p & ( z nil => p^.next = nil )

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