Checking - Calculus Structural Congruence is Graph Isomorphism Complete

Checking -Calculus Structural Congruence is

Graph Isomorphism Complete

Victor Khomenko1 and Roland Meyer2

1School of Computing Science,Newcastle University, UK

2Department of Computing Science,University of Oldenburg, Germany

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-Calculus Syntax

P ::= 0| K a⌊ 1,…,an⌋| P + P| P | P| .P| a:P

::= a<b>| a(x)|

No replication operator ‘!’ – using recursive definitions of the form K a⌊ 1,…,an :=P⌋ instead

Input prefix a(x).P and restriction x:P bind name x in P

NOCLASH assumption (can always be enforced by -conversion):

• each name is bound at most once• the sets of bound and free names are disjoint

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Structural congruence

The smallest congruence ≡ defined by the following axioms:

α-conversion of bound names is permitted (α)+ and | are associative and commutative (AC+), (AC|)0 is a neutral element for + and | (0+), (0|)x:P ≡ P if x is not a free name of P (P)x:y:P ≡ y:x:P (C)x:(P | Q) ≡ P | x:Q if x is not a free name of P (SE|)

Note: ≡ does not expand recursive calls

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SOS rules

PaPa

PPQPQP

PP

QPQPQQ

PPPxKzaPaK

QyzPNQzxMPyx

PMP

::(Res)

||(Par)

andif,(Struct)

:)~(if,}~/~{~(Const)

|}/{).(|).((React)

.(Tau)

Not needed!

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Checking structural congruence

SC – the problem of checking structural congruence ≡ of two -Calculus terms

• Repeatedly solved by -Calculus tools (e.g. the states of the system are the equivalence classes w.r.t. ≡)

hence the computational complexity of SC is of interest

reduction of SC to Graph Isomorphism (GI) problem allows for an efficient solution in practice, by employing a GI solver

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Graph isomorphism problem (GI)

Source: Wikipedia

(a) = 1 (b) = 6(c) = 8 (d) = 3(g) = 5 (h) = 2(i) = 4 (j) = 7

G1=(V1,E1) and G2=(V2,E2) are isomorphic if there is a 1-to-1 mapping :V1V2 such that {v,w}E1 iff {(v),(w)}E2

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The complexity of GI

•Trivially in NP, but not believed to be NP-complete (as Stockmeyer’s polynomial hierarchy PH would then collapse)

•No polynomial-time algorithm known

•Can be solved very efficiently in practice

•Complexity class GI – comprises problems Cook reducible to GI, e.g. Digraph Isomorphism (DGI), Labelled Digraph Isomorphism (LDGI) and many others

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GISC reduction (SC is GI-hard)• It is enough to reduce DGI to SC• Given a digraph G(V,E), where V={v1,…,vn}, build the term

• The reduction uses a very restricted -Calculus fragment: all the restrictions are in the beginning of the term no +, prefixing operator ‘.’, actions, public channels | can be replaced by + calls to process identifiers can be replaced by actions,

e.g., L v,w⌊ ⌋ can be replaced by v<w>.0• Summary: , at least one of | or +, and some means of

referring to bound names are enough to make the fragment GI-hard

EwvVvn wvLvKvv

),(1 ,|::

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SCGI reduction (SC is in GI)• Reduce SC to the Term Equality problem (TE), which is

known to be equivalent to GI [Basin’94]: Decide if two terms built using quantifiers introducing bound names; some of these

quantifiers may commute, i.e., θx:θy:t θy:θx:t associative, commutative and associative-commutative

binary operators uninterpreted functional symbols and constants the names bound by the quantifiers

are equivalent modulo associativity, commutativity and associativity-

commutativity axioms for the corresponding operators the commutativity of corresponding quantifiers α-conversion of bound names

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SCTE reduction: main ideasProblem 1: the input prefixes are different from quantifiers in TE, and the individual prefixes do not directly correspond to constants or variables in TESolution: substitute a<b> by s(a,b) and x(y).P by ρy:r(x,y).P, where ρ is a new non-commutative quantifier

Problem 2: some axioms in the definition of ≡ have no analogs in TE, viz. (0+), (0|), (P), (SE|)Solution: translate the terms into the following normal form:

enforce the NOCLASH assumption use (0+), (0|) and (P) to simplify the terms until none of

these axioms applies maximise the scope of restrictions using (SE|) (in the

reverse direction)This normal form does not require these axioms to prove structural congruence (long and tedious proof in the paper)

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SCTE reduction (cont’d)The resulting terms comprise an instance of TE, where:

• + and | are associative-commutative operators

• s(_,_), r(_,_), the prefixing operator ‘.’ and the process identifiers are uninterpreted functional symbols

is a commutative quantifier and ρ is a non-commutative quantifier

• public channels, and 0 are constants (since all the axioms for 0 no longer apply, it can be regarded as uninterpreted)

• the names introduced by the restriction and input prefixes are the names bound by the quantifiers and ρ

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SCTE reduction: an example

x:a<x>.b(z).z<x>.0 | y:a(p).b<y>.0 | q:.0 | t:0

x:a<x>.b(z).z<x>.0 | y:a(p).b<y>.0 | .0

x:y:(a<x>.b(z).z<x>.0 | a(p).b<y>.0 | .0)

x:y:(s(a,x).ρz:r(b,z).s(z,x).0 | ρp:r(a,p).s(b,y).0 | .0)

≡ (SE|)

≡ (P), (0|)

translation

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TELDGI reduction [Basin’94]• Build the parse tree of the TE term

• Compound the vertices corresponding to associative and associative-commutative operations into vertices with larger out-degrees

• Drop the arc labels for commutative operators

1 2 3 4

*

Gt4Gt3Gt2Gt1

(t1*t2)*(t3*t4)

(* is not the top-level operator of t1-t4)

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TELDGI reduction (cont’d)• Translating the quantifiers

• Erase the names of bound variables (to express that they can be changed by α-conversion)

• Drop the arc labels for commutative quantifiers

1 2

θ

Gt

θx1:…:θxn:t

(θ-quantification is not the top-level operation of t)

x1 x2 x2

for n=2

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TELDGI reduction: an examplex:y:s(a,x).ρz:r(x,z).s(z,y).K(a,x) | .s(a, b).K(a,b) + .0 +

.K(a,b) | ρp:r(a,p).s(p,c).ρq:r(c,q).s(q, a).0

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TELDGI reduction: optimisation-1• Share sub-terms whose structural congruence is easy to

check (e.g. restriction-free or trivial sub-terms only)

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TELDGI reduction: optimisation-2• Eliminate ρ-vertices, together with the associated auxiliary

vertices (their position can always be recovered)

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TELDGI reduction: optimisation-3• After the common sub-terms are shared (and parallel arcs

removed), the auxiliary vertices for quantifiers have the in- and out-degree one, and can be contracted

• Adjacent vertices corresponding to the prefixing operator ‘.’ can be compounded

• The 0 vertex (unique after sharing common sub-terms) can be eliminated

• The unlabelled vertices corresponding to the variables can be labelled by either ρ or (depending on the type of the binding quantifier)

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The result of these optimisationsReduction from 60/63 down to 26/38 vertices/arcs

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Summary and extensions

These results are not affected if either or both of the following axioms are added:

x:(P + Q) ≡ P + x:Q if x is not a free name of P (SE+)

x:.P ≡ .x:P if x does not occur in (SE)

-Calculus fragment Complexity of SC full -Calculus GI-complete, at least one of + or |, and some means of referring to restricted channels (i/o prefixes, process identifiers)

GI-complete

without both + and | in Pwithout in P

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Conclusions

• Showed that SC is a GI-complete problem• The result is robust:

holds for restricted fragments of -Calculus holds for alternative definitions of ≡, viz. with

(SE+) and/or (SE) -Calculus fragments for which SC is in P have

been identified• Practical algorithm for solving SC:

reduce to TE use the optimised TELDGI translation use a GI solver

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Future work

• Extension to the following axioms looks plausible:

x:.P ≡ 0 if has the form x<·> or x(·) (P)

x:(P + Q) ≡ x:P + x:Q (D+)• Also generalisation of (P) to an axiom replacing any

process that has no behaviour in any context by 0

Related work• Engelfriet and Gelsema• Gadducci• Romanel and Priami

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Thank you!Any questions?