Logic and Complexity

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Roland Gregory
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2 Logic and Complexity Undecidable (FOL + LIA) Semi Decidable (FOL) NEXPTIME (EPR) PSPACE (QBF) NP (SAT)

7 Logic and Complexity Logic is The Calculus of Computer Science Zohar Manna Practical problems often have structure that can be exploited. Undecidable (FOL + LIA) Semi Decidable (FOL) NEXPTIME (EPR) PSPACE (QBF) NP (SAT)

9 Satisfiability Solution/Model x 2 + y 2 < 1 and xy > 0.1 sat, x = 1 8, y = 7 8 x 2 + y 2 < 1 and xy > 1 unsat, Proof

12 Dynamic Test Generation void top(char input[4]) { int cnt = 0; if (input[0] == b ) cnt++; if (input[1] == a ) cnt++; if (input[2] == d ) cnt++; if (input[3] ==! ) cnt++; if (cnt >= 4) crash(); } Negate each constraint in path constraint Solve new constraint new input input = good Path constraint: I 0!= b I 1!= a I 2!= d I 3!=! I 0 = b I 1 = a I 2 = d I 3 =! good bood gaod godd goo! Key technology: satisfiability solving

Automated Theorem Provier http://research.microsoft.

13 Automated Theorem Provier Leonardo de Moura and Nikolaj Bjørner SAT Simplex Rewriting Superposition Z3 is a collection of Symbolic Reasoning Engines Congruence Closure Groebner Basis eliminatio n Euclidean Solver

14 Some (MSR) Applications Functional verification: Dafny, VCC Defect detection: Corral, HAVOC, Poirot, SLAM Test generation: Pex, SAGE Design-space exploration: Formula, SpecExplorer New programming languages: F*

15 Start here Learn about Z3 and get the source code! Strategies Source code

17 Symbolic Automata and Transducers Margus Veanes, Nikolaj Bjørner (POPL 2011)

18 Core Question Can classical automata theory and algorithms be extended to work modulo large (infinite) alphabets?

Symbolic Automata: Relativized Formal Language Theory Symbolic Word Transducers Classical Word Transducers modulo Th( ) Classical Word Transducers (e.g. decoding automata, rational transductions) Classical I/O Automata (e.

19 Symbolic Automata: Relativized Formal Language Theory Symbolic Word Transducers Classical Word Transducers modulo Th( ) Classical Word Transducers (e.g. decoding automata, rational transductions) Classical I/O Automata (e.g. Mealy machine) Symbolic Word Acceptors Classical Word Acceptors (NFA, DFA) Classical Word Acceptors modulo Th( ) string transformation regex matching

20 Symbolic Finite Automaton (SFA) Alphabet is an effective Boolean Algebra A Labels are predicates over A one symbolic transition: p x. 'a' x 'd' for x 'a' x 'd' q denotes many concrete transitions: p 'a' 'c' 'b' 'd' q

21 Boolean operations over SFAs Intersection (product of transitions) A 1 : A 2 : 1 p 1 q 1 2 p 2 q 2 p 1 A 1 A 2 : q X p 2 delete when 1 2 unsat q 2

22 Intersection example 2 let k (x) ((x mod k) = 0) A: a 1 a B: b 1 3 A B: a 1 b 1 a 2 b 2 a X b b 2

23 Symbolic Finite Transducer (SFT) Classical transducer modulo a rich label theory Core Idea: represent labels with guarded transformers Separation of concerns: finite graph / theory of labels Concrete transitions: Symbolic transition: p 1920 transitions p guard \x80 / \xc2\x80 q \x7ff / \xdf\xbf q x x 7FF 16 / [C0 16 x 10,6, x 5,0 ] bitvector operations

24 Algorithms New algorithms for SFAs and SFTs Using Z3 Extensions of classical algorithms modulo Th( ) Big-O complexity matches that of classical algorithms, with factor for decision procedure

25 Analysis Example 1: x(utf8encode(x) R utf8 )? 1. E = SFT(utf8encode) 2. A = Complement(SFA(R utf8 )) 3. B = x. A(E(x)) 4. B? Does there exist an input x that causes a bad output? Example 2: x.utf8decode(utf8encode(x)) Id?

26 For More Information Bek Bex Rex Fast July 17, 2013 CIAA 2013, Halifax, Canada 26

30 Syntactic String Transformations (from Examples) Flash Fill feature in Excel 2013 Reference: Automating String Processing in Spreadsheets using Input-Output Examples, POPL 2011, Gulwani

31 Syntactic String Transformations: Language Trace Expr e := Concatenate(f 1,, f n ) Base Expr f := s SubStr(v i, p 1, p 2 ) Position Expr p := k Pos(r 1, r 2, k) 31

32 Let w = SubString(s, p, p ) Substring Operator where p = Pos(r 1, r 2, k) and p = Pos(r 1, r 2, k ) w 1 w 2 w 1 w 2 p p w r 1 matches w 1 r 2 matches w 2 r 1 matches w 1 r 2 matches w 2 s Two special cases: r 1 = r 2 = ε : This describes the substring r 2 = r 1 = ε : This describes boundaries around the substring 32

33 Too many choices for a Trace Expression Input Output Constant Constant Constant 33

34 Synthesizing Trace Expressions Number of all possible trace expressions is exponential in size of output string. To represent/learn all trace expressions, it suffices to represent/learn all base expressions for each substring of the output. # of substrings is just quadratic in size of output string! A DAG based data-structure represents all trace expressions transforming input to output, and supports efficient intersection operation 34

Too many choices for a SubStr Expression Various ways to extract 706 from 425-706-7709 : Chars after 1 st hyphen and before 2 nd hyphen.

35 Too many choices for a SubStr Expression Various ways to extract 706 from : Chars after 1 st hyphen and before 2 nd hyphen. Substr(v 1, Pos(HyphenTok,²,1), Pos(²,HyphenTok,2)) Chars from 2 nd number and up to 2 nd number. Substr(v 1, Pos(²,NumTok,2), Pos(NumTok,²,2)) Chars from 2 nd number and before 2 nd hyphen. Substr(v 1, Pos(²,NumTok,2), Pos(²,HyphenTok,2)) Chars from 1 st hyphen and up to 2 nd number. Substr(v 1, Pos(HyphenTok,²,1), Pos(²,HyphenTok,2)) 35

36 Synthesizing SubStr Expressions The number of SubStr(v,p 1,p 2 ) expressions that can extract a given substring w from v can be large! To represent/learn all SubStr expressions, we can independently represent/learn all choices for each of the two index expressions. This allows for representing and computing O(n 1 *n 2 ) choices for SubStr using size/time O(n 1 +n 2 ). 36

37 Ranking General Principles Prefer shorter programs, shorter string expression, regular expressions. Prefer programs with less number of constants. 37

Sri vidya college of engineering and technology

Sri vidya college of engineering and technology Unit I FINITE AUTOMATA 1. Define hypothesis. The formal proof can be using deductive proof and inductive proof. The deductive proof consists of sequence of statements given with logical reasoning in order