C. AISWARYA, PAUL GASTIN, PRAKASH SAIVASAN ORDER-2 NESTED WORDS
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1 Chennai Mathematical Institute cm i C. AISWARYA, PAUL GASTIN, PRAKASH SAIVASAN ORDER-2 NESTED WORDS
2 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS Pushdown (stack) pop, push(a),?top(a) ababa babb
3 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS Pushdown (stack) pop, push(a),?top(a) ababa bab
4 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS Pushdown (stack) pop, push(a),?top(a) ababa baba b bb a ba
5 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(a),?top(a) Push2, Pop2 a ba b bb ababa baba
6 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(a),?top(a) Push2, Pop2 a ba b bb ababa baba ababa baba
7 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(a),?top(a) Push2, Pop2 a ba b bb ababa baba
8 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse push(a)
9 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a push(a)
10 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a push(a)push2
11 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a a push(a)push2
12 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a a push(a)push2push(b)
13 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab push(a)push2push(b)
14 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab push(a)push2push(b)push2
15 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab ab push(a)push2push(b)push2
16 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab ab push(a)push2push(b)push2push(a)
17 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba push(a)push2push(b)push2push(a)
18 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba push(a)push2push(b)push2push(a)push2
19 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba aba push(a)push2push(b)push2push(a)push2
20 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba aba push(a)push2push(b)push2push(a)push2collapse
21 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab push(a)push2push(b)push2push(a)push2collapse
22 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba aba push(a)push2push(b)push2push(a)push2
23 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba aba push(a)push2push(b)push2push(a)push2pop2
24 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba push(a)push2push(b)push2push(a)push2pop2
25 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba aba push(a)push2push(b)push2push(a)push2
26 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba aba push(a)push2push(b)push2push(a)push2pop
27 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba ab push(a)push2push(b)push2push(a)push2pop
28 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a ab aba ab push(a)push2push(b)push2push(a)push2popcollapse
29 PROLOGUE HIGHER ORDER PUSHDOWN SYSTEMS order-2 stack (stack of stacks) pop, push(b),?top(a) Push2, Pop2 Collapse a push(a)push2push(b)push2push(a)push2popcollapse
30 OUTLINE HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY under-approximations - tree interpretations - decidability
31 2-CPDS - BRIEF OVERVIEW OF RESULTS ORDER-2 COLLAPSIBLE PUSHDOWN SYSTEMS finite state machine equipped with an order-2 stack transitions perform order-2 stack operations cò có dó 1 aò s 1 2 bò s 1 0 Ò s 1 Ò 2 Ó K? Ó 1 2
32 2-CPDS - BRIEF OVERVIEW OF RESULTS ORDER-2 COLLAPSIBLE PUSHDOWN SYSTEMS 2-CPDS capture order-2 recursion schemes (Hague et al 2008) 2-CPDS strictly more powerful than 2-PDS for rec. schemes (Parys 2012) thought of as generating words, trees or graphs μ-calculus model checking of 2-CPDS-trees is decidable (Ong 2006) via parity games on configuration graphs of CPDS (Hague et al 2008) via Krivine machines (SalvatiWalukiewicz 2011) configuration graphs of 2-CPDS can encode semi grids (Hague et al 2008) μ-calculus over configuration graphs of 2-PDS decidable
33 OUTLINE HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY under-approximations - tree interpretations - decidability
34 ORDER-2 NESTED WORDS NESTED WORDS push(a)
35 ORDER-2 NESTED WORDS NESTED WORDS a push(a)
36 ORDER-2 NESTED WORDS NESTED WORDS a push(a)push2
37 ORDER-2 NESTED WORDS NESTED WORDS a a push(a)push2
38 ORDER-2 NESTED WORDS NESTED WORDS a a push(a)push2push(b)
39 ORDER-2 NESTED WORDS NESTED WORDS a ab push(a)push2push(b)
40 ORDER-2 NESTED WORDS NESTED WORDS a ab push(a)push2push(b)push2
41 ORDER-2 NESTED WORDS NESTED WORDS a ab ab push(a)push2push(b)push2
42 ORDER-2 NESTED WORDS NESTED WORDS a ab ab push(a)push2push(b)push2push(a)
43 ORDER-2 NESTED WORDS NESTED WORDS a ab aba push(a)push2push(b)push2push(a)
44 ORDER-2 NESTED WORDS NESTED WORDS a ab aba push(a)push2push(b)push2push(a)push2
45 ORDER-2 NESTED WORDS NESTED WORDS a ab aba aba push(a)push2push(b)push2push(a)push2
46 ORDER-2 NESTED WORDS NESTED WORDS a ab aba aba push(a)push2push(b)push2push(a)push2pop2
47 ORDER-2 NESTED WORDS NESTED WORDS a ab aba push(a)push2push(b)push2push(a)push2pop2
48 ORDER-2 NESTED WORDS NESTED WORDS a ab aba aba push(a)push2push(b)push2push(a)push2
49 ORDER-2 NESTED WORDS NESTED WORDS a ab aba aba push(a)push2push(b)push2push(a)push2pop
50 ORDER-2 NESTED WORDS NESTED WORDS a ab aba ab push(a)push2push(b)push2push(a)push2pop
51 ORDER-2 NESTED WORDS NESTED WORDS a ab aba ab push(a)push2push(b)push2push(a)push2popcollapse
52 ORDER-2 NESTED WORDS NESTED WORDS a push(a)push2push(b)push2push(a)push2popcollapse
53 ORDER-2 NESTED WORDS EXAMPLES Ò 1 Ò 1 Ò 2 Ó 1 Ò 2 Ó 1 Ó 2 Ó 2 Ó 1 Ó 1
54 ORDER-2 NESTED WORDS EXAMPLES Ò 1 Ò 1 Ò 2 Ó 1 Ò 2 Ó 1 Ó 2 Ó 2 Ó 1 Ó Ò 1 Ò 1 Ò 2 Ó 1 Ó 1 Ó 2 Ó 1 Ò 2 Ó 1 Ò 1 Ò 2 Ó 1 Ó 2 Ó 1 Ó 2 Ó 1 Ò 1 Ò 2 Ó 1 Ó 2 Ó 1
55 ORDER-2 NESTED WORDS LANGUAGE OF ORDER-2 CPDS L( ) = 0 Ò s 1 Ò 2 Ó K? Ó 1 2 { } Ò 1 Ò 1 Ò 2 Ó 1 Ò 2 Ó 1 Ó 2 Ó 2 Ó Ò 1 Ò 1 Ò 2 Ó 1 Ó 1 Ó 2 Ó 1 Ò 2 Ó 1 Ò 1 Ò 2 Ó 1 Ó 2 Ó 1 Ó 2 Ó 1 Ò 1 Ò 2 Ó 1 Ó 2 Ó 1 Ó 1
56 MOTIVATION FOR STUDYING NESTED WORDS WHAT ARE ORDER-2 NESTED WORDS? Not words Not trees Not configuration graphs Linear time behaviours extra information (matching) Words with binary relation on its positions = special graphs
57 MOTIVATION FOR STUDYING NESTED WORDS WHY ARE ORDER-2 NESTED WORDS INTERESTING? They make visible the data flow / communication Making visible the data flow is not new. Widely used in message passing distributed systems (Message Sequence Charts -MSC) also in pushdown and multi-pushdown systems (Nested words, multiply-nested words)
58 MOTIVATION FOR STUDYING NESTED WORDS MESSAGE PASSING DISTRIBUTED SYSTEMS Message Sequence Charts ITU Standard
59 MOTIVATION FOR STUDYING NESTED WORDS RECURSIVE PROGRAMS / XML q a ba c aa b a cb a a b cb a ab b aa b Nested Words Alur, Madhusudan, 2009
60 MOTIVATION FOR STUDYING NESTED WORDS MULTI-THREADED RECURSIVE PROGRAMS q Multiply Nested Words
61 MOTIVATION FOR STUDYING NESTED WORDS RECURSIVE PROCESSES COMMUNICATING VIA MESSAGE PASSING Concurrent Behaviours with Matching d 1 p a b a b d 1 d 1 a b a a a b b a d 2 d 3 d 2 d 2 q a b b a a b a b a b b a d 4 d 4 d 4 d 4
62 MOTIVATION FOR STUDYING NESTED WORDS WHY ARE ORDER-2 NESTED WORDS INTERESTING? They make visible the data flow / communication Making visible the data flow is not new. Widely used in message passing distributed systems (Message Sequence Charts -MSC) also in pushdown and multi-pushdown systems (Nested words, multiply-nested words)
63 MOTIVATION FOR STUDYING NESTED WORDS WHY NESTED WORDS yet another way of understanding the model for better specification more powerful specification formalism can make use of the nesting edges when expressing properties verification beyond reachability
64 SPECIFICATION BEYOND REACHABILITY RECURSIVE PROGRAMS / XML q a ba c aa b a cb a a b cb a ab b aa b Letter before outermost call is the same as the letter after its return
65 SPECIFICATION BEYOND REACHABILITY RECURSIVE PROGRAMS / XML Letter before outermost call is the same as the letter after its return q a ba c aa b a cb a a b cb a ab b aa b x, y ( a(x 1) x y z,z (z z z<x<z ) ) a(y +1)
66 SPECIFICATION BEYOND REACHABILITY RECURSIVE PROGRAMS / XML Relate outer most call and returns q a ba c aa b a cb a a b cb a ab b aa b Not expressible in MSO over Linear Traces even with visible alphabet
67 MOTIVATION FOR STUDYING NESTED WORDS WHAT CAN ORDER-2 NESTED WORDS MODEL? Data-flow-visible linear behaviors of order-2 CPDS Can model recursive program with stack data-structure, where stacks can be passed to subroutines (pass-by-value) Ò 1 Ò 1 Ò 2 Ò 1 Ò 2 Ó 1 Ó 1 Ó 2 Ó 1 Ò 2 Ó 1 Ó 2 Ó 2 Ó 1 Ò 2 Ó 1 Ó 2
68 MOTIVATION FOR STUDYING NESTED WORDS WHAT CAN ORDER-2 NESTED WORDS MODEL? Data flow visible linear behaviors of order-2 CPS Ò 1 Ò 2 Ò 1 Ò 2 Ó 1 Ó 2 Ó 2 Ò 1 Ò 2 Ò 2 Ó 1 Ó 2 Ó 2 Ó 1 Ó 1 Can model recursive program with stack data-structure, where stacks can be passed to subroutines (pass-by-value) Can capture branching behaviour of pushdown systems (similar to nested trees (AlurChaudhuriMadhusudhan 06)
69 OUTLINE HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY under-approximations - tree interpretations - decidability
70 NESTIFY VALID ORDER-2 NESTED WORDS Words + two binary relations. When is it a valid 2-NW?
71 NESTIFY VALID ORDER-2 NESTED WORDS Words + two binary relations. When is it a valid 2-NW? Given a sequence of order-2 stack operations is it valid? Ops = { top=a?, top=b?, push(a), push(b), pop(a), pop(b), DupStack, PopStack, Collapse } Naïvely simulate an order-2 stack May take exponential space/time
72 NESTIFY VALID ORDER-2 NESTED WORDS Words + two binary relations. When is it a valid 2-NW? Given a sequence of order-2 stack operations is it valid? Can we construct the nesting relations from such a sequence? Nestify does it in linear time/space
73 NESTIFY push1 (x) = push1(x-1) VALID ORDER-2 NESTED WORDS based on the inductive characterisation. push1(i), Push2(i) p q $ j if op j PtÒ s 1 s P Su & push 1 1pjq if op j Ptnop, Ò 2 uyts? s P Su push 1 pjq push 1 1ppush 1 1pjqq if op j Ó 1 push 1 1pPush 1 2pjqq if op j Ó 2 % push 1 1pPush 1 2ppush 1 1pjqqq if op j ó $
74 NESTIFY Push (x) = Push(x-1) VALID ORDER-2 $ NESTED WORDS & based on the inductive characterisation. push(i), Push2(i) % p p p qqq ó Push 2 pjq $ j if op j Ò 2 & Push 1 2pjq if op j Ptnop, Ó 1 uytò s 1, s? s P Su Push 1 2pPush 1 2pjqq if op j Ó 2 % Push 1 2pPush 1 2ppush 1 1pjqqq if op j ó The proof is by induction on. When 0 we proceed with a case split
75 NESTIFY VALID ORDER-2 NESTED WORDS Words + two binary relations. When is it a valid 2-NW? Given a sequence of order-2 stack operations is it valid? Can we construct the nesting relations from such a sequence? Nestify does it in linear time/space keeps an array with two pointers
76 OUTLINE HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY under-approximations - tree interpretations - decidability
77 SPECIFICATION FORMALISMS MONADIC SECOND-ORDER LOGIC NOT EXPRESSIBLE IN MSO WIHOUT NESTING : apxq x y x ñ 1 y x ñ 2 y _ Dx DX x P X P, are first-order variables and is a second-order variable. The sema EXAMPLE: CONSECUTIVE POPS LINKED TO THE SAME PUSH ãñ px, yq Dz `z ñ 1 x ^ z ñ 1 y ^ Dz 1 pz ñ 1 z 1 ^ x z 1 yq. EXAMPLES: set of all order-2 nested words language of an order-2 CPDS
78 SPECIFICATION FORMALISMS MONADIC SECOND-ORDER LOGIC : apxq x y x ñ 1 y x ñ 2 y _ Dx DX x P X P, are first-order variables and is a second-order variable. The sema SATISFIABILITY PROBLEM input: MSO sentence φ question: is there a order-2 NW that satifies φ? MODEL CHECKING PROBLEM input: MSO sentence φ, order-2 CPDS H question: do all order-2 NW from L(H) satisfy φ?
79 SPECIFICATION FORMALISMS NOTICE : ORDER-2 STACK IS BOUNDED, NO COLLAPSE MONADIC SECOND-ORDER LOGIC - UNDECIDABILITY 1 Ò s Ò 2 Ó 1 K? 5 6 Ó 2 Ò s Ò 2 Ó 1 Ó 1 K? Ó 2 Ò s Ò 2 Ó 1 Ó 1 Ó 1 K? Ó 2 Ò s Ò 2 Ó 1 Ó 1 Ó 1 Ó 1 K? 26 Ó 2 3 Ò s 1 Ò 2 1 Ó Ó 2 3 K?
80 SPECIFICATION FORMALISMS PROPOSITIONAL DYNAMIC LOGIC WITH LOOP AND CONVERSE Ï : a Ï _ Ï Ï xfiyï Looppfiq fi : tïu? Ñ ñ 1 apple 1 ñ 2 apple 2 fi fi fi ` fi fi 1 1 q p here P. The semantics is given on -labelled graphs p G, i ù xfiyï if G, i, j ù fi and G, j ù Ï for some j G, i ù Looppfiq if G, i, i ù fi G, i, j ù tïu? if i j and G, i ù Ï G, i, j ù Ñ if i Ñ j in the graph G G, i, j ù ñ 1 if i ñ 1 j in the graph G G, i, j ù apple 2 if j ñ 2 i in the graph G. An LCPDL sentence is a boolean combination of atomic se
81 SPECIFICATION FORMALISMS PROPOSITIONAL DYNAMIC LOGIC WITH LOOP AND CONVERSE Why LCPDL? MSO powerful, but high complexity LTL less expressive, but low complexity LCPCL offers the best of both worlds (expressiveness, complexity)
82 SPECIFICATION FORMALISMS PROPOSITIONAL DYNAMIC LOGIC WITH LOOP AND CONVERSE EXAMPLE: PATH THAT GOES TO THE PUSH WHICH PUSHED THE CURRENT TOPMOST ELEMENT ptò 2 _? _ nopu? ` apple 1 ` apple 2 t ptò _ Ó _? _ nopu? ` apple 2 t Lo q tò 1 u? tò u? Ò 1 Ò 1 Ò 2 Ò 1 Ò 2 Ó 1 Ó 1 Ó 2 Ó 1 Ò 2 Ó 1 Ó 2 Ó 2 Ó 1 Ò 2 Ó 1 Ó 2
83 SPECIFICATION FORMALISMS PROPOSITIONAL DYNAMIC LOGIC WITH LOOP AND CONVERSE EXAMPLE: PATH THAT GOES TO THE PUSH WHICH PUSHED THE CURRENT TOPMOST ELEMENT ptò 2 _? _ nopu? ` apple 1 ` apple 2 t t Loopp ` ñ 2 apple 2 qu? ptò _ Ó _? _ nopu? ` apple 2 t Loopp ` ñ 2 apple 2 qu? q q tò 1 u? tò u? 1 2 cò 2 aò s cò 2 aò s cò 2 aò s cò 2 bò s dó 1 dó 1 có cò 2 aò s cò 2 bò s dó 1 dó 1 có
84 SPECIFICATION FORMALISMS PROPOSITIONAL DYNAMIC LOGIC WITH LOOP AND CONVERSE LCPDL SENTENCE Boolean combinations of the form Eφ EXAMPLES: set of all order-2 nested words language of an order-2 CPDS
85 SPECIFICATION FORMALISMS PROPOSITIONAL DYNAMIC LOGIC WITH LOOP AND CONVERSE LCPDL SENTENCE Boolean combinations of the form Eφ SATISFIABILITY PROBLEM input: LCPDL sentence φ question: is there a order-2 NW that satifies φ? MODEL CHECKING PROBLEM input: LCPDL sentence φ, order-2 CPDS H question: do all order-2 NW from L(H) satisfy φ?
86 SPECIFICATION FORMALISMS LCPDL UNDECIDABILITY CANNOT GO TO NEXT-POP IN LCPDL Ò s Ò 2 Ó 1 K? 5 6 Ó 2 Ò s Ò 2 Ó 1 Ó 1 K? Ó 2 Ò s Ò 2 Ó 1 Ó 1 Ó 1 K? Ó 2 Ò s Ò 2 Ó 1 Ó 1 Ó 1 Ó 1 K? 26 Ó 2 xãñyï :: Looppptispop 1 u? Ñq` Ñ tispop 2 u? Ñ Ñ pñ tispop 1 u?q` tïu? apple 1 ñ 1 q With this, we can write an LCPDL formula to encode the computation of a Turing machine
87 OUTLINE HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY Eliminating collapse under-approximations - tree interpretations - decidability
88 ELIMINATING COLLAPSE ELIMINATING COLLAPSE Ò 1 Ò 2 Ò 1 Ò 1 Ò 2 Ò 1 Ò 2 Ò 1 Ò 2 Ó 1 aó Ò 1 Ò 2 Ó 1 Ó 1 bó Ò 1 Ò 2 Ò 1 Ò 1 Ò 2 Ò 1 Ò 2 Ò 1 Ò 2 Ó 1 a #Ó 2 #Ó 2 #Ó 2 Ò 1 Ò 2 Ó 1 Ó 1 b #Ó 2 #Ó 2 Figure 2 A 2-NW (top) and its encoding in 2-NW without collapse (below). We show the labels a a _ a Ï 1 _ Ï 2 Ï 1 _ Ï 2 Ï Ï xfiyï xfiyï Looppfiq Looppfiq tïu? tïu? fi 1 fi 2 fi 1 fi 2 fi 1 ` fi 2 fi 1 ` fi 2 fi fi ñ 1 ñ 1 apple 1 apple 1 Ñ Ñ pt#u? Ñq t #u? pt#u? q t #u? apple 2 pñ t#u?q apple 2 ñ 2 ñ 2 pt#u? q t #u? MSO to MSO without collapse. Translation of MSO is similar in spirit to that of LCPDL
89 OUTLINE HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY under-approximations - tree interpretations - decidability
90 TREE INTERPRETATIONS TREE-INTERPRETATIONS since the behaviors are graphs, we can use graph-based underapproximations. for example, tree-width. decidability follows if the graph relations can be interpreted in a tree, and if the class is MSO definable. (Courcelle, also ParlatoMadhu, AGastin) in our case they are even LCPDL definable (non-trivial, uses the characterisation) can work on the tree domain and show that MSO SAT/MC is decidable PDL SAT/MC is ExpTime Complete
91 TREE INTERPRETATION AN EXAMPLE : BOUNDED POP A pushed element can be popped at most k times does not bound the size of the stacks in either level definable in LCPDL has bounded split-width (implies bounded tree-width)
92 CONCLUSION HIGHER ORDER PUSHDOWN SYSTEMS (OF ORDER 2) prologue 2CPDS - a brief overview of results 2NW: a different way of understanding CPDS behaviours Nestify Spec languages: MSO, PDL, - UNDECIDABALITY Eliminating collapse under-approximations - tree interpretations - decidability
93 PERSPECTIVES CONCLUSIONS AND PERSPECTIVES Yet another way of understanding HOPDS The power of data-flow edges. How to harness that? under-approximations applications lifting to higher orders
94 QUESTIONS? THANK YOU cmi.ac.in
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