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Outline N OMINAL C- UNIFICATION on , Washington L. R. de Carvalho Segundo , Mauricio Ayala-Rinc andez and Daniele Nantes Sobrinho Maribel Fern K ING S C OLLEGE L ONDON U NIVERSIDADE DE B RAS ILIA 27 th Int.

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SLIDE 1

Outline

NOMINAL C-UNIFICATION

Mauricio Ayala-Rinc´

  • n†, Washington L. R. de Carvalho Segundo†‡,

Maribel Fern´ andez‡ and Daniele Nantes Sobrinho†

†UNIVERSIDADE DE BRAS´ ILIA ‡KING’S COLLEGE LONDON

27th Int. Sym. on Logic-Based Program Synthesis and Transformation —

LOPSTR

Namur, 12 October 2017

  • Univ. de Bras´

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Outline

Outline

1

Motivation Unification problems

2

Formalisation of a nominal C-unification algorithm Termination Soundness Completeness

3

Nominal C-unification is Infinitary NP-complete

4

Conclusion and future work

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SLIDE 3

Motivation

Unification

Equations between first-order terms s ≈? t where variables {X, Y, Z, . . .} can be substituted by terms Application contexts:

logic programming type inference term rewriting theorem provers security protocol analysis information retrieval

  • Univ. de Bras´

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SLIDE 4

Motivation

Equational and binding operators

Function symbols f with basic equational properties like:

A {f(X, f(Y, Z)) ≈ f(f(X, Y ), Z)} C {f(X, Y) ≈ f(Y, X)} D {f(g(X, Y ), Z) ≈ g(f(X, Z), f(Y, Z))} U {f(X, 1) ≈ X}

Bound object-level variables [a]s ∀a∀b, P(a) ∨ Q(b) ⇒ R(a, b)

represented as

f∀[a]f∀[b]f⇒(f∨(P(a), P(b)), R(a, b))

  • Univ. de Bras´

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Motivation

Nominal basic objects

atoms {a, b, c, . . .} and variables {X, Y, Z, . . .} Freshness contexts ∇ = {a#X, b#Y, c#Z, . . .} permutations as lists of name-swappings π = (a0 b0) :: (a1 b1) :: . . . :: (an bn) :: nil The inverse of π is its reverse list π−1 = (an bn) :: (an−1 bn−1) :: . . . :: (a0 b0) :: nil

  • Univ. de Bras´

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Motivation

Nominal syntax and {α, C}-equivalence

Nominal syntax t, u, v ::= | a | [a]t | u, v | fE

k t | π.X

Freshness relation ∇ ⊢ a # t a is fresh to t under the freshness context ∇ ∇ ⊢ a # π.X only if (π−1·a)#X ∈ ∇ {α, C}-equivalence ∇ ⊢ s ≈{α,C} t {α, E}-equivalence instantiated with E = {C}

(see [Ayala-Rinc´

  • n et al., 2016a])
  • Univ. de Bras´

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Motivation

Related work: formalisation on nominal

2004 2006 2008 2010 2012 2014 2016

Nominal Unification in Isabelle/HOL [Urban et al., 2004] Nominal Reasoning Techniques in Coq [Aydemir et al., 2007] Nominal Reasoning Techniques in Isabelle/HOL [Urban, 2008] Nominal Unification in HOL4 [Kumar and Norrish, 2010] Nominal Unification (Revisited) in Isabelle/HOL [Urban, 2010] General binders in Isabelle/HOL [Urban and Kaliszyk, 2012] Nominal Unification in PVS [Ayala-Rinc´

  • n

e t a l . , 2 1 5 ]

Nominal Reasoning Techniques in Agda [Copello et al., 2015] Nominal α-A/AC in Coq [Ayala-Rinc´

  • n

e t a l . , 2 1 6 a ]

  • Univ. de Bras´

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Motivation

Related work: HO correspondence and efficiency of nominal algorithms

2006 2008 2010 2012

H O P a t t e r n → N

  • m

i n a l [Cheney, 2005] N

  • m

i n a l U n i fi c a t i

  • n

i s P

  • l

y n

  • m

i a l [Calv`

es and Fern´ andez, 2008]

Q u a d r a t i c N

  • m

i n a l U n i fi c a t i

  • n [Levy and Villaret, 2010]

M a t c h i n g a n d α

  • E

q u i v a l e n c e C h e c k [Calv`

es and Fern´ andez, 2010]

Q u a d r a t i c N

  • m

i n a l U n i fi c a t i

  • n [Calv`

es and Fern´ andez, 2011]

N

  • m

i n a l → H O P a t t e r n [Levy and Villaret, 2012]

  • Univ. de Bras´

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Motivation

Related work: reasoning modulo equational theories and nominal unification extensions

2004 2006 2008 2010 2012 2014 2016

A / A C / C r e w r i t e i n I s a b e l l e / H O L [Nipkow, 1989] A C m a t c h i n g a l g

  • r

i t h m i n C

  • q [Contejean, 2004]

P e r m u t a t i

  • n
  • f

t e r m s i n C

  • q [Contejean, 2007]

A C r e w r i t e i n C

  • q [Braibant and Pous, 2011]

N

  • m

i n a l a n t i

  • u

n i fi c a t i

  • n [Baumgartner et al., 2015]

N

  • m

i n a l n a r r

  • w

i n g [Ayala-Rinc´

  • n et al., 2016b]

N

  • m

i n a l u n i fi c a t i

  • n

w i t h l e t r e c [Schmidt-Schauß et al., 2016] S

  • l

v i n g n

  • m

i n a l C fi x p

  • i

n t e q u a t i

  • n

s [Ayala-Rinc´

  • n et al., 2017]
  • Univ. de Bras´

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Motivation Unification problems

Unification problems

Unification problem ∇, P = ∇, id, P P is a finite set of equations and freshness constraints of the form s ≈? t and a#?s

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm

System ⇒#

∇, σ, P ⊎ {a#?} (#?) ∇, σ, P ∇, σ, P ⊎ {a#?¯ b} (#?a¯ b) ∇, σ, P ∇, σ, P ⊎ {a#?f t} (#?app) ∇, σ, P ∪ {a#?t} ∇, σ, P ⊎ {a#?[a]t} (#?a[a]) ∇, σ, P ∇, σ, P ⊎ {a#?[b]t} (#?a[b]) ∇, σ, P ∪ {a#?t} ∇, σ, P ⊎ {a#?π.X} (#?var) {(π−1 · a)#X} ∪ ∇, σ, P ∇, σ, P ⊎ {a#?s, t} (#?pair) ∇, σ, P ∪ {a#?s, a#?t}

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Formalisation of a nominal C-unification algorithm

System ⇒≈

∇, σ, P ⊎ {s ≈? s} (≈? refl) ∇, σ, P ∇, σ, P ⊎ {s1, t1 ≈? s2, t2} (≈? pair) ∇, σ, P ∪ {s1 ≈? s2, t1 ≈? t2} ∇, σ, P ⊎ {fE

k s ≈? fE k t}

, if E = C (≈? app) ∇, σ, P ∪ {s ≈? t} ∇, σ, P ⊎ {fC

k s ≈? fC k t}

,

  • where s = s0, s1 and t = t0, t1

v = ti, t(i+1) mod 2, i = 0, 1

  • (≈? C)

∇, σ, P ∪ {s ≈? v} ∇, σ, P ⊎ {[a]s ≈? [a]t} (≈? [aa]) ∇, σ, P ∪ {s ≈? t} ∇, σ, P ⊎ {[a]s ≈? [b]t} (≈? [ab]) ∇, σ, P ∪ {s ≈? (a b) t, a#?t} ∇, σ, P ⊎ {π.X ≈? t} let σ′ := σ{X/π−1 · t} , if X / ∈ V ar(t) (≈? inst)

  • ∇, σ′, P {X/π−1 · t} ∪
  • Y ∈dom(σ′),

a#Y ∈∇

{a#?Y σ′}

  • ∇, σ, P ⊎ {π.X ≈? π′.X}

, if π′ = id (≈? inv) ∇, σ, P ∪ {π ⊕ (π′)−1.X ≈? X}

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm

Derivation tree for ∇, P

Nodes are labelled w.r.t. each ⇒≈(resp. ⇒#)-derivation step

1

The root node is labelled with P = ∇, id, P

2

P is reduced by ⇒≈ (for each branch), until reach Qi (a normal form w.r.t. ⇒≈)

3

For each Qi = ∇i, δi, Qi. If Qi contains only fixpoint equations and freshness contraints then it is reduced until reach ¯ Q (a normal form w.r.t. ⇒#)

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm

Derivation tree for ∇, P

Nodes are labelled w.r.t. each ⇒≈(resp. ⇒#)-derivation step

1

The root node is labelled with P = ∇, id, P

2

P is reduced by ⇒≈ (for each branch), until reach Qi (a normal form w.r.t. ⇒≈)

3

For each Qi = ∇i, δi, Qi. If Qi contains only fixpoint equations and freshness contraints then it is reduced until reach ¯ Q (a normal form w.r.t. ⇒#)

  • Univ. de Bras´

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SLIDE 15

Formalisation of a nominal C-unification algorithm

Derivation tree for ∇, P

Nodes are labelled w.r.t. each ⇒≈(resp. ⇒#)-derivation step

1

The root node is labelled with P = ∇, id, P

2

P is reduced by ⇒≈ (for each branch), until reach Qi (a normal form w.r.t. ⇒≈)

3

For each Qi = ∇i, δi, Qi. If Qi contains only fixpoint equations and freshness contraints then it is reduced until reach ¯ Q (a normal form w.r.t. ⇒#)

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm

Definition of a solution

Definition (Solution for a triple or a problem) A solution for a triplet P = ∇, δ, P is given by a pair ∆, σ that satisfies

1

∀ a#X ∈ ∇, ∆ ⊢ a # Xσ

2

if a#?t ∈ P then ∆ ⊢ a # tσ

3

if s ≈? t ∈ P then ∆ ⊢ sσ ≈{α,C} tσ

4

∃λ s.t. ∀X ∈ dom(δλ) ∪ dom(σ), ∆ ⊢ Xδλ ≈{α,C} Xσ The solution set for a problem or triple P is denoted by UC(P).

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Formalisation of a nominal C-unification algorithm

Example

P =∅, {[a][b]X ≈? [b][a]X} . . .

Nominal unification [Urban et al., 2004]

∅, {X ≈? (a b).X} | {a#X, b#X}, id

In general ∅, {π.X ≈ X} = ⇒ dom(π)#X, id

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SLIDE 18

Formalisation of a nominal C-unification algorithm

Example

P =∅, {[a][b]X ≈? [b][a]X} . . .

Nominal unification [Urban et al., 2004]

∅, {X ≈? (a b).X} | {a#X, b#X}, id

In general ∅, {π.X ≈ X} = ⇒ dom(π)#X, id

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm

Example

P = ∅, {[e](a b).X ∗ Y ≈? [f](a c)(c d).X ∗ Y }

∅, id, { (a b).X ∗ Y ≈? (a c)(c d)(e f).X ∗ (e f).Y , e#?(a c)(c d).X ∗ Y } branch 2 branch 1 ⇒(≈?C) ⇒(≈?C) ⇒(≈?[ab])

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Formalisation of a nominal C-unification algorithm

Example: branch 1

∅, id, { (a b).X ≈? (a c)(c d)(e f).X , Y ≈? (e f).Y , e#?(a c)(c d).X ∗ Y } ∅, id, {(a b)[(a c)(c d)(e f)]−1.X ≈? X, [(e f)]−1.Y ≈? Y, e#?(a c)(c d).X ∗ Y } ∅, id, {(a b)(e f)(c d)(a c).X ≈? X, (e f).Y ≈? Y, e#?(a c)(c d).X , e#?Y }

e#X , e#Y , id, { (a b)(e f)(c d)(a c).X ≈? X , (e f).Y ≈? Y } = Q1

⇒(#?var)(2×) ⇒ (#?app), (#?pair) ⇒(≈?inv)(2×)

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Formalisation of a nominal C-unification algorithm

Example: branch 2

∅, id, (a b).X ≈? (e f).Y , Y ≈? π1.X, e#?(a c)(c d).X ∗ Y } ∅, {X/(e f)(a b).Y }, { Y ≈? (a c)(c d)(e f)(e f)[(a b)]−1.Y , e#?(a c)(c d)(e f)[(a b)]−1.Y ∗ Y }} ∅, {X/(e f)(a b).Y }, {[(a c)(c d)(e f)(e f)(a b)]−1.Y ≈? Y, e#?(a c)(c d)(e f)(a b).Y ∗ Y } ∅, {X/(e f)(a b).Y }, {(a b)(e f)(e f)(c d)(a c).Y ≈? Y, e#?(a c)(c d)(e f)(a b).Y , e#?Y }

e#Y , f#Y , {X/(e f)(a b).Y }, { (a b)(e f)(e f)(c d)(a c).Y ≈? Y } = Q2

⇒(#?var)(2×) ⇒ (#?app), (#?pair) ⇒(≈?inv) ⇒(≈?inst)

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Formalisation of a nominal C-unification algorithm Termination

Termination

Lemma (Termination of ⇒≈ and ⇒#) There is no infinite chain of reductions ⇒≈ (or ⇒#) starting from an arbitrary triple P = ∇, σ, P. The proof is by well-founded induction on P

P =

  • s≈?t ∈ P≈

|s| + |t| +

  • a#?u∈P#

|u|

1

P≈ = |V ar(P≈)|, P≈, |Pnfp≈|

2

P# = P#

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm Soundness

Soundness

Theorem (Soundness of T∇,P) If P′ = ∇′, σ, P ′ is the label of a leaf in T∇,P, then

1

UC(P′) ⊆ UC(∇, id, P) and

2

if P ′

fp≈ = P ′ then UC(P′) = ∅

Induction on ⇒≈ and ⇒# Non trivial cases: ⇒(≈?[ab]) and ⇒(≈?inst)

  • Univ. de Bras´

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Formalisation of a nominal C-unification algorithm Completeness

Completeness

Theorem (Completeness of T∇,P) Let ∇, P and T∇,P be a unification problem and its derivation tree. Then UC(∇, id, P) =

  • Q∈ST(T∇,P )

UC(Q). Induction on ⇒≈ and ⇒# Non trivial cases: ⇒(≈?C), ⇒(≈?[ab]) and ⇒(≈?inst)

  • Univ. de Bras´

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Nominal C-unification is Infinitary

Atom permutations as k-cycles

Q1 = e#X, e#Y, id, { (a b)(e f)(c d)(a c).X ≈? X , (e f).Y ≈? Y } Q2 = e#Y, f#Y, {X/(e f)(a b).Y }, { (a b)(e f)(e f)(c d)(a c).Y ≈? Y }

  • (a b)(e f)(c d)(a c) −

→ (a b c d)(e f)

  • (a b)(e f)(e f)(c d)(a c) −

→ (a b c d)

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Nominal C-unification is Infinitary

Pseudo-cycles

κ = (a0 a1 . . . ak−1) ∈ π

1

κ = (a0 · · · ak−1) is a (trivial) pseudo-cycle w.r.t. κ

2

κ′ = (A0 ... Ak′−1) is a pseudo-cycle w.r.t. κ, if

1

each element of κ′ is of the form Bi ∗ Bj, Bi, Bj are different elements of κ′′, a pseudo-cycle w.r.t. κ

2

Ai ≈α,C Aj for i = j, 0 ≤ i, j ≤ k′ − 1

3

for each 0 ≤ i < k′ − 1, κ · Ai ≈{α,C} A(i+1)mod k′

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Nominal C-unification is Infinitary

Example: κ = (a b c d)

(a b c d)

4

((a ∗ b) (b ∗ c) (c ∗ d) (d ∗ a))

4

(((a ∗ b) ⊗ (b ∗ c)) . . . ((b ∗ c) ⊗ (d ∗ a)))

4

. . . . . .

(((a ∗ b) ⋆ (c ∗ d)) ((d ∗ a) ⋆ (a ∗ b)))

2

  • 1

((a ⊕ c) (b ⊕ d))

2

((a ⊕ c) ∗ (b ⊕ d))

1

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Nominal C-unification is Infinitary

Results

Theorem (Power of two cycles) A pseudo-cycle κ generates unitary pseudo-cycles iff |κ| = 2p. Theorem (Combinatory solutions) Let P = ∅, {π.X ≈? X} be a fixpoint problem. P has a combinatory solution iff there exists a unitary pseudo-cycle κ w.r.t. π.

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Nominal C-unification is Infinitary

Unitary pseudo-cycles as combinatory solutions

If (s) is a unitary pseudo-cycle of κ then κ · s ≈{α,C} s In our example:

e#X, e#Y, {X/(a ⊕ c) ∗ (b ⊕ d)} ∈ UC(Q1) e#X, f#Y, {X/(e f)(a b).Y }{Y/(a ⊕ c) ∗ (b ⊕ d)} ∈ UC(Q2)

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Nominal C-unification is Infinitary

Extended pseudo-cycles

In [Ayala-Rinc´

  • n et al., 2017] we defined extended pseudo-cycles to

encompass all feasible combinatory solutions Example epc’s w.r.t (a b c d) include other syntactic elements s′ =(h a⊕h c)∗(h b⊕h d) and t′ =([i]a⊕[i]c)∗([i]b⊕[i]d) So that {X/s′} and {X/t′} (resp. {Y/s′} and {Y/t′}) are solutions of (a b c d)(e f).X ≈? X (resp. (a b c d).Y ≈? Y )

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Nominal C-unification is NP-complete

NP-completeness

1

Decide, for a given P, if UC(P) = ∅ Guessing non-deterministically a path to a successful leaf

2

NP-completeness: positive 1-in-3-SAT reduces to nominal C-unification C = {Ci|1 ≤ i ≤ n} where Ci = pi ∨ qi ∨ ri ⊕ commutative a → True b → False Each clause Ci = pi ∨ qi ∨ ri in C is polynomially translated into ((Xpi ⊕ Xqi) ⊕ Xri) ⊕ Yi ≈? ((b ⊕ b) ⊕ a) ⊕ ((b ⊕ a) ⊕ b)

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SLIDE 32

Conclusion and future work

Conclusion

Formalisation of a nominal C-unification algorithm

1

termination

2

soundness

3

completeness

Nominal C-unification is

1

Infinitary

2

NP-Complete

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SLIDE 33

Conclusion and future work

Future work

1

Implementantion

2

Nominal    A / AC / C matching AC unification AC / C narrowing

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SLIDE 34

Conclusion and future work

Future work

1

Implementantion (in progress)

2

Nominal    A / AC / C matching (in progress) AC unification AC / C narrowing

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Conclusion and future work

THANK YOU

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SLIDE 36

Conclusion and future work

References I

Ayala-Rinc´

  • n, M., Carvalho-Segundo, W., Fern´

andez, M., and Nantes-Sobrinho, D. (2016a). A Formalisation of Nominal Equivalence with Associative-Commutative Function Symbols. In Proc. of the 11th Workshop on Logical and Semantic Frameworks with Applications (LSFA), volume 332 of ENTCS, pages 21–38. Elseviser. Ayala-Rinc´

  • n, M., Carvalho-Segundo, W., Fern´

andez, M., and Nantes-Sobrinho, D. (2017). On Solving Nominal Fixpoint Equations. In Proc. of the 11th Int. Symp. on Frontiers of Combining Systems (FroCoS), volume 10483 of LNCS, pages 209–226. Springer. Ayala-Rinc´

  • n, M., Fern´

andez, M., and Nantes-Sobrinho., D. (2016b). Nominal Narrowing. In Proc. of the 1st Int. Conf. on Formal Structures for Computation and Deduction (FSCD), volume 52 of LIPIcs, pages 11:1–11:17. Ayala-Rinc´

  • n, M., Fern´

andez, M., and Rocha-oliveira, A. C. (2015). Completeness in PVS of a Nominal Unification Algorithm. In Proc. of the 10th Workshop on Logical and Semantic Frameworks with Applications (LSFA), volume 323 of ENTCS, pages 57–74. Elseviser. Aydemir, B., Bohannon, A., and Weirich, S. (2007). Nominal Reasoning Techniques in Coq. ENTCS, 174(5):69–77. Baumgartner, A., Kutsia, T., Levy, J., and Villaret, M. (2015). Nominal Anti-Unification. In Proc. of the 26th Int. Conf. on Rewriting Techniques and Applications, (RTA), volume 36 of LIPics, pages 57–73.

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Conclusion and future work

References II

Braibant, T. and Pous, D. (2011). Tactics for Reasoning Modulo AC in Coq. In In Proc. of the 1st. Int. Conf. on Certified Programs and Proofs (CPP), volume 7086 of LNCS, pages 167–182. Springer. Calv` es, C. F. and Fern´ andez, M. (2008). A Polynomial Nominal Unification Algorithm. Theoretical Computer Science, 403(2-3):285–306. Calv` es, C. F. and Fern´ andez, M. (2010). Matching and Alpha-Equivalence Check for Nominal Terms.

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Conclusion and future work

References III

Contejean, E. (2007). Modeling permutations in coq for coccinelle. In Rewriting, Computation and Proof, Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60th Birthday, LNCS, pages 259–269. Springer. Copello, E., Tasistro, A., Szasz, N., Bove, A., and Fern´ andez, M. (2015). Principles of Alpha-Induction and Recursion for the Lambda Calculus in Constructive Type Theory. In Logical and Semantic Frameworks with Applications, pages 51–66. Kumar, R. and Norrish, M. (2010). (Nominal) Unification by Recursive Descent with Triangular Substitutions. In Proc. of Interactive Theorem Proving, 1st Int. Conf. (ITP), volume 6172 of LNCS, pages 51–66. Springer. Levy, J. and Villaret, M. (2010). An Efficient Nominal Unification Algorithm. In Proc. of the 21st Int. Conf. on Rewriting Techniques and Applications (RTA), volume 6 of LIPIcs, pages 209–226. Levy, J. and Villaret, M. (2012). Nominal unification from a higher-order perspective. ACM Trans. Comput. Log., 13(2):10:1–10:31. Nipkow, T. (1989). Equational Reasoning in Isabelle. Science of Computer Programming, 12(2):123–149.

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SLIDE 39

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References IV

Schmidt-Schauß, M., Kutsia, T., Levy, J., and Villaret, M. (2016). Nominal Unification of Higher Order Expressions with Recursive Let. In Proc. of the 26th Int. Sym. on Logic-Based Program Synthesis and Transformation (LOPSTR), volume 10184 of LNCS, pages 328–344. Springer. Urban, C. (2008). Nominal Techniques in Isabelle/HOL.

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Urban, C. (2010). Nominal Unification Revisited. In Proc. of the 24th Int. Work. on Unification (UNIF), volume 42 of EPTCS, pages 1–11. Urban, C. and Kaliszyk, C. (2012). General Bindings an Alpha-Equivalence in Nominal Isabelle. Logical Methods in Computer Science, 8:1–35. Urban, C., Pitts, A. M., and Gabbay, M. J. (2004). Nominal Unification. Theoretical Computer Science, 323(1-3):473–497.

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