Nominal Rewriting and Unification Theory Maribel Fern´andez FoPSS 2019 Maribel Fern´andez Nominal Rewriting and Unification Theory Nominal Rewriting and Unification Theory Introduction First-order languages • Languages with binding operators • Specifying binders: ↵-equivalence • Nominal terms • Nominal unification (unification modulo ↵-equivalence) • Nominal matching (matching modulo ↵-equivalence) • Nominal rewriting Extending first-order rewriting to specify binding operators • Closed rewriting • Confluence • Typed Rewriting Systems • Equational Axioms: AC operators • Maribel Fern´andez Nominal Rewriting and Unification Theory Further reading C. Urban, A. Pitts, M.J. Gabbay. Nominal Unification. • Theoretical Computer Science 323, pages 473-497, 2004. C. Calv`es, M. Fern´andez. Matching and Alpha-Equivalence • Check for Nominal Terms. Journal of Computer and System Sciences, 2010. M. Ayala-Rinc´on, M. Fern´andez, D. Nantes-Sobrinho. • Fixed-Point Constraints for Nominal Equational Unification. Proceedings of FSCD 2018, LIPICS. M. Fern´andez, M.J. Gabbay. Nominal Rewriting. Information • and Computation 205, pages 917-965, 2007. J. Dominguez, M. Fern´andez. Nominal Syntax with Atom • Substitution: Matching, Unification, Rewriting. Proceedings of FCT 2019, Lecture Notes in Computer Science, Springer. E. Fairweather, M. Fern´andez. Typed Nominal Rewriting. • ACM Transactions on Computational Logic, 2018. Maribel Fern´andez Nominal Rewriting and Unification Theory First-order languages vs. languages with binders Most programming languages support first-order data structures and first-order operators. Examples of first-order data structures: numbers, lists, trees, etc. First-order operator on lists: append(nil, x) x ! append(cons(x, z), y) cons(x, append(z, y)) ! Very few programming languages support data structures with binding constructs. However, in many situations, we need to manipulate data with bound names. Example: compilers, type checkers, code optimisation, etc. Maribel Fern´andez Nominal Rewriting and Unification Theory β and ⌘-reductions in the λ-calculus: • (λx.M)N M[x/N] ! (λx.Mx) M (x fv(M)) ! 62 ⇡-calculus: • P ⌫a.Q ⌫a.(P Q)(a fv(P)) | ! | 62 Logic equivalences: • Pand( x.Q) x(PandQ)(x fv(P)) 8 ,8 62 Binding operators: Examples Some concrete examples of binding constructs (informally): Operational semantics: • let a = N in M (fun a.M)N ! Maribel Fern´andez Nominal Rewriting and Unification Theory ⇡-calculus: • P ⌫a.Q ⌫a.(P Q)(a fv(P)) | ! | 62 Logic equivalences: • Pand( x.Q) x(PandQ)(x fv(P)) 8 ,8 62 Binding operators: Examples Some concrete examples of binding constructs (informally): Operational semantics: • let a = N in M (fun a.M)N ! β and ⌘-reductions in the λ-calculus: • (λx.M)N M[x/N] ! (λx.Mx) M (x fv(M)) ! 62 Maribel Fern´andez Nominal Rewriting and Unification Theory Logic equivalences: • Pand( x.Q) x(PandQ)(x fv(P)) 8 ,8 62 Binding operators: Examples Some concrete examples of binding constructs (informally): Operational semantics: • let a = N in M (fun a.M)N ! β and ⌘-reductions in the λ-calculus: • (λx.M)N M[x/N] ! (λx.Mx) M (x fv(M)) ! 62 ⇡-calculus: • P ⌫a.Q ⌫a.(P Q)(a fv(P)) | ! | 62 Maribel Fern´andez Nominal Rewriting and Unification Theory Binding operators: Examples Some concrete examples of binding constructs (informally): Operational semantics: • let a = N in M (fun a.M)N ! β and ⌘-reductions in the λ-calculus: • (λx.M)N M[x/N] ! (λx.Mx) M (x fv(M)) ! 62 ⇡-calculus: • P ⌫a.Q ⌫a.(P Q)(a fv(P)) | ! | 62 Logic equivalences: • Pand( x.Q) x(PandQ)(x fv(P)) 8 ,8 62 Maribel Fern´andez Nominal Rewriting and Unification Theory Binding operators - ↵-equivalence Terms are defined modulo renaming of bound variables,i.e., ↵-equivalence. Example: In x.P the variable x can be renamed (avoiding name capture) 8 x.P = y.P x y 8 ↵ 8 { 7! } How can we formally define (or program) binding operators? There are several alternatives. Maribel Fern´andez Nominal Rewriting and Unification Theory Efficient matching and unification algorithms (+) • No binders (-) • We need to ’implement’ ↵-equivalence and non-capturing • substitution from scratch (-) For example, we can encode a system with binders such as the • lambda-calculus using numbers to represent bound variables and operators such as “lift” and “shift” to encode non-capturing substitution (cf. De Bruijn’s notation) First-order frameworks We can encode ↵-equivalence in a first-order specification or programming language. Simple notion of substitution (first-order) (+) ) Maribel Fern´andez Nominal Rewriting and Unification Theory No binders (-) • We need to ’implement’ ↵-equivalence and non-capturing • substitution from scratch (-) For example, we can encode a system with binders such as the • lambda-calculus using numbers to represent bound variables and operators such as “lift” and “shift” to encode non-capturing substitution (cf. De Bruijn’s notation) First-order frameworks We can encode ↵-equivalence in a first-order specification or programming language. Simple notion of substitution (first-order) (+) • Efficient matching and unification algorithms (+) ) Maribel Fern´andez Nominal Rewriting and Unification Theory We need to ’implement’ ↵-equivalence and non-capturing • substitution from scratch (-) For example, we can encode a system with binders such as the • lambda-calculus using numbers to represent bound variables and operators such as “lift” and “shift” to encode non-capturing substitution (cf. De Bruijn’s notation) First-order frameworks We can encode ↵-equivalence in a first-order specification or programming language. Simple notion of substitution (first-order) (+) • Efficient matching and unification algorithms (+) • No binders (-) ) Maribel Fern´andez Nominal Rewriting and Unification Theory For example, we can encode a system with binders such as the • lambda-calculus using numbers to represent bound variables and operators such as “lift” and “shift” to encode non-capturing substitution (cf. De Bruijn’s notation) First-order frameworks We can encode ↵-equivalence in a first-order specification or programming language. Simple notion of substitution (first-order) (+) • Efficient matching and unification algorithms (+) • No binders (-) • We need to ’implement’ ↵-equivalence and non-capturing ) substitution from scratch (-) Maribel Fern´andez Nominal Rewriting and Unification Theory First-order frameworks We can encode ↵-equivalence in a first-order specification or programming language. Simple notion of substitution (first-order) (+) • Efficient matching and unification algorithms (+) • No binders (-) • We need to ’implement’ ↵-equivalence and non-capturing • substitution from scratch (-) For example, we can encode a system with binders such as the ) lambda-calculus using numbers to represent bound variables and operators such as “lift” and “shift” to encode non-capturing substitution (cf. De Bruijn’s notation) Maribel Fern´andez Nominal Rewriting and Unification Theory Logical frameworks based on Higher-Order Abstract Syntax • also work modulo ↵-equivalence. let a = N in M(a) (fun a M(a))N ! ! Higher-order frameworks Higher-order rewrite systems (CRS, HRS, etc.) include a • general binding construct and terms are defined modulo ↵-equivalence. Example: β-rule app(lam([a]Z(a)), Z 0) Z(Z 0) ! One step of rewriting: app(lam([a]f (a, g(a)), b) f (b, g(b)) ! using (a restriction of) higher-order matching. Maribel Fern´andez Nominal Rewriting and Unification Theory Higher-order frameworks Higher-order rewrite systems (CRS, HRS, etc.) include a • general binding construct and terms are defined modulo ↵-equivalence. Example: β-rule app(lam([a]Z(a)), Z 0) Z(Z 0) ! One step of rewriting: app(lam([a]f (a, g(a)), b) f (b, g(b)) ! using (a restriction of) higher-order matching. Logical frameworks based on Higher-Order Abstract Syntax • also work modulo ↵-equivalence. let a = N in M(a) (fun a M(a))N ! ! Maribel Fern´andez Nominal Rewriting and Unification Theory Implicit ↵-equivalence (+) • We targeted ↵ but now we have to deal with β too (-) • Substitution is a meta-operation using β (-) • Unification is undecidable in general (-) • Leaving name dependencies implicit is convenient, e.g. • let a = N in M vs. let a = N in M(a) app(lambda[a]Z, Z 0)vs. app(lam([a]Z(a)), Z 0). Higher-order frameworks The syntax includes binders (+) ) Maribel Fern´andez Nominal Rewriting and Unification Theory We targeted ↵ but now we have to deal with β too (-) • Substitution is a meta-operation using β (-) • Unification is undecidable in general (-) • Leaving name dependencies implicit is convenient, e.g. • let a = N in M vs. let a = N in M(a) app(lambda[a]Z, Z 0)vs. app(lam([a]Z(a)), Z 0). Higher-order frameworks The syntax includes binders (+) • Implicit ↵-equivalence (+) ) Maribel Fern´andez Nominal Rewriting and Unification Theory Substitution is a meta-operation using β (-) • Unification is undecidable in general (-) • Leaving name dependencies implicit is convenient, e.g. • let a = N in M vs. let a = N in M(a) app(lambda[a]Z, Z 0)vs. app(lam([a]Z(a)), Z 0). Higher-order frameworks The syntax includes binders (+) • Implicit ↵-equivalence (+) • We targeted ↵ but now we have to deal with β too (-) ) Maribel Fern´andez Nominal Rewriting and Unification Theory Unification is undecidable in general (-) • Leaving name dependencies implicit is convenient, e.g. • let a = N in M vs. let a = N in M(a) app(lambda[a]Z, Z 0)vs. app(lam([a]Z(a)), Z 0). Higher-order frameworks The syntax includes binders (+) • Implicit ↵-equivalence (+) • We targeted ↵ but now we have to deal with β too (-) • Substitution is a meta-operation using β (-) ) Maribel Fern´andez