Proof Systems for Inductive Reasoning in the Logic of Bunched - - PowerPoint PPT Presentation

Proof Systems for Inductive Reasoning in the Logic of Bunched Implications

James Brotherston

• Dept. of Computing, Imperial College London

J.Brotherston@imperial.ac.uk HAV workshop 25 March 2007

Overview

The logic of bunched implications BI, offers a convenient means

• f expressing properties of programs that access and modify

some shared resource (e.g., the heap). Our main contributions are:

• to extend BI with a general framework for inductive

definitions;

• to give two proof systems in sequent calculus style for two

different inductive reasoning techniques in the extended logic:

• 1. explicit rule induction over definitions;
• 2. cyclic proof embodying a notion of proof by infinite descent

for inductively defined relations.

The logic of bunched implications (BI)

• our structures M contain a notion of resource, given by a

partial commutative monoid R, ◦, e;

• in addition to the standard first-order logic connectives BI

introduces the new binary connectives ∗ and — ∗;

• satisfaction of a formula F is given by the relation

M, r | =ρ F, where r ∈ R is the “current resource state”: M, r | =ρ Qt ⇔ QM(r, ρ(t)) M, r | =ρ F1 ∗ F2 ⇔ r = r1 ◦ r2 and M, r1 | =ρ F1 and M, r2 | =ρ F2 for some r1, r2 ∈ R M, r | =ρ F1 — ∗ F2 ⇔ M, r′ | =ρ F1 implies M, r′ ◦ r | =ρ F2 for all r′ ∈ R

BI with inductive definitions (BIID)

• two types of predicate symbol: ordinary Q1, Q2, . . . and

inductive P1, . . . , Pn;

• our inductive definitions are given by a finite set Φ of

productions which are rules of the form: C(x) i ∈ {1, . . . , n} Pit(x) C(x) ::= I | Qt(x) | Pit(x) (i ∈ {1, . . . , n}) | C(x) ∧ C(x) | C(x) ∗ C(x)

• from Φ we generate an n-ary monotone operator ϕΦ.

Example: list segments

I ls x x x → x′ ∗ ls x′ y ls x y where → is an ordinary predicate. In separation logic, ls is a predicate representing (possibly cyclic) list segments. The

• perator for Φls is defined by:

ϕΦls(X) = {(e, (d, d)) | d ∈ D} ∪ {(r1 ◦ r2, (d, d′)) | (r1, (d, d′′)) ∈ →M and (r2, (d′′, d′)) ∈ X}

Example: tree segments

I ts x x x → (x′, y′) ∗ tree x′ ∗ ts y′ y ts x y x → (x′, y′) ∗ tree y′ ∗ ts x′ y ts x y In separation logic, ts is a predicate representing (possibly cyclic) tree segments. We have: ϕΦts(X) = {(e, (d, d)) | d ∈ D} ∪ {(r1 ◦ r2 ◦ r3, (d, d′)) | (r1, (d, (d1, d2))) ∈ →M and (r2, d1) ∈ tree M and (r3, (d2, d′)) ∈ X} ∪ {(r1 ◦ r2 ◦ r3, (d, d′)) | (r1, (d, (d1, d2))) ∈ →M and (r2, d2) ∈ tree M and (r3, (d1, d′)) ∈ X}

Standard models of BIID

• From the n-ary monotone operator ϕΦ we construct a

sequence (ϕα

Φ) of approximants by iteratively applying ϕΦ

to the empty set.

• standard result:

α ϕα Φ is the least prefixed point of ϕΦ.

Definition M is a standard model if for all inductive predicates Pi we have P M

i

= πn

i ( α ϕα Φ).

Proposition For any definition set Φ we have

α ϕα Φ = ϕω Φ.

Sequent calculus rules for BI

We write sequents Γ ⊢ F where F is a formula and Γ is a bunch: Γ ::= F | Γ; Γ | Γ, Γ where ; is equivalent to ∧ and , is equivalent to ∗. The rules for the multiplicative connectives ∗ and — ∗ are: ∆ ⊢ F1 Γ(∆′, F2) ⊢ F (— ∗L) Γ(∆, ∆′, F1 — ∗ F2) ⊢ F Γ(F1, F2) ⊢ F (∗L) Γ(F1 ∗ F2) ⊢ F Γ, F1 ⊢ F2 (— ∗R) Γ ⊢ F1 — ∗ F2 Γ ⊢ F1 ∆ ⊢ F2 (∗R) Γ, ∆ ⊢ F1 ∗ F2

LBIID: a sequent calculus for induction in BIID

Extend sequent calculus for BI by adding introduction rules for inductively defined predicates. E.g. the right-introduction rules for ls are: Γ ⊢ I (lsR1) Γ ⊢ ls t t Γ ⊢ t1 → t ∗ ls t t2 (lsR2) Γ ⊢ ls t1 t2 The left-introduction rules embody rule induction over definitions, e.g. for ls: ∆; I ⊢ Hxx ∆; x → x′ ∗ Hx′y ⊢ Hxy Γ(∆; Htu) ⊢ F (Ind ls) Γ(∆; ls t u) ⊢ F where H is the induction hypothesis associated with ls and x, x′, y are fresh. (NB. mutual definitions give rise to mutual induction rules.)

CLBIω

ID: a cyclic proof system for BIID

• Rules are as for LBIID except the induction rules are

replaced by weaker case-split rules, e.g. for ls: Γ(t1 = t2; I) ⊢ F Γ(t1 → x, ls x t2) ⊢ F (Case ls) Γ(ls t1 t2) ⊢ F where x is fresh. ls x t2 is called a case-descendant of ls t1 t2;

• pre-proofs are finite derivation trees in which every bud

(node to which no proof rule is applied) is assigned a companion (an identically labelled interior node);

• for soundness we need to impose a global condition on

pre-proofs.

Traces

A trace following a path (Γi ⊢ Fi) in an CLBIω

ID pre-proof D is

a sequence Ψi(τi) such that for each i:

• Γi = Ψi(τi) and τi = Pt for some inductive predicate P;
• if a case-split rule is applied at i, then τi+1 may be a

case-descendant of τi (and we have progress at i);

• otherwise, τi+1 is the occurrence in Γi+1 ⊢ Fi+1

corresponding to the occurrence of τi in Γi ⊢ Fi, modulo any splitting of Γi and any substitutions applied by the rule. Key point: if Γ ⊢ F is false in some r, ρ, then there is an

• rdinal associated with any trace element in Γ.

Definition An CLBIω

ID pre-proof D is a proof if for every infinite path in D

there is a trace following some tail of the path that progresses infinitely often.

Example: cyclic proof of list segment concatenation

(Id) ls x y ⊢ ls x y (≡) I, ls x y ⊢ ls x y (=L) (x′ = x; I), ls x′ y ⊢ ls x y (Id) x → z ⊢ x → z (†) ls x x′, ls x′ y ⊢ ls x y (Subst) ls z x′, ls x′ y ⊢ ls z y (∗R) x → z, ls z x′, ls x′ y ⊢ x → z ∗ ls z y (lsR2) x → z, ls z x′, ls x′ y ⊢ ls x y (Case ls) (†) ls x x′, ls x′ y ⊢ ls x y (∗L) ls x x′ ∗ ls x′ y ⊢ ls x y

A progressing trace following the cycle given by (†) is

• highlighted. One can build an infinitely progressing trace on the
• nly infinite path by concatenating copies of this trace. So this

pre-proof is a proof.

Discussion of CLBIω

ID

• it is decidable whether a CLBIω

ID pre-proof is a proof;

• some cyclic proofs seem to avoid the need for generalisation

in inductive proof;

• work is in progress to develop cyclic proof systems for

proving termination of simple programs.

• J. Brotherston.

Cyclic proofs for first-order logic with inductive definitions. In Proceedings of TABLEAUX 2005.

• J. Brotherston and A. Simpson.

Complete sequent calculi for induction and infinite descent. To appear at LICS 2007.

• J. Brotherston.

Sequent Calculus Proof Systems for Inductive Definitions. PhD thesis, University of Edinburgh, November 2006.