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Checking - Calculus Structural Congruence is Graph Isomorphism Complete

Checking - Calculus Structural Congruence is Graph Isomorphism Complete. Victor Khomenko 1 and Roland Meyer 2 1 School of Computing Science, Newcastle University, UK 2 Department of Computing Science, University of Oldenburg, Germany. - Calculus Syntax.

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Checking - Calculus Structural Congruence is Graph Isomorphism Complete

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  1. 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

  2. -Calculus Syntax P ::=0| K⌊a1,…,an⌋| P + P| P | P|.P|a:P ::= a<b>| a(x)| No replication operator ‘!’ – using recursive definitions of the form K⌊a1,…,an⌋:=P instead Input prefix a(x).P and restriction x:Pbind 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

  3. 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

  4. SOS rules Not needed!

  5. 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 SCis of interest • reduction of SC to Graph Isomorphism (GI) problem allows for an efficient solution in practice, by employing a GI solver

  6. Graph isomorphism problem (GI) G1=(V1,E1) and G2=(V2,E2) are isomorphic if there is a 1-to-1 mapping :V1V2 such that {v,w}E1 iff {(v),(w)}E2 (a) = 1 (b) = 6 (c) = 8 (d) = 3 (g) = 5 (h) = 2 (i) = 4 (j) = 7 Source: Wikipedia

  7. 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

  8. GISC 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

  9. SCGI 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

  10. SCTE reduction: main ideas Problem 1: the input prefixes are different from quantifiers in TE, and the individual prefixes do not directly correspond to constants or variables in TE Solution: substitute a<b> by s(a,b) and x(y).P by ρy:r(x,y).P, where ρ isa 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)

  11. SCTE 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

  12. SCTE 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) ≡ (P), (0|) ≡ (SE|)  translation

  13. * 1 2 3 4 Gt4 Gt3 Gt2 Gt1 TELDGI 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 (t1*t2)*(t3*t4) (* is not the top-level operator of t1-t4)

  14. θ Gt TELDGI 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 for n=2 1 2 θx1:…:θxn:t (θ-quantification is not the top-level operation of t) x2 x1 x2

  15. TELDGI reduction: an example x: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

  16. TELDGI reduction: optimisation-1 • Share sub-terms whose structural congruence is easy to check (e.g. restriction-free or trivial sub-terms only)

  17. TELDGI reduction: optimisation-2 • Eliminate ρ-vertices, together with the associated auxiliary vertices (their position can always be recovered)

  18. TELDGI 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)

  19. The result of these optimisations Reduction from 60/63 down to 26/38 vertices/arcs

  20. 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)

  21. Conclusions • Showed that SCis 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 TELDGItranslation • use a GI solver

  22. Future work • Extension to the following axioms looks plausible: x:.P ≡ 0if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

  23. Thank you! Any questions?

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