Second-Order Equational Logic

Transcription

1 Second-Order Equational Logic Extended Abstract Marcelo Fiore Computer Laboratory University of Cambridge Chung-Kil Hur Laboratoire PPS Université Paris 7 Abstract We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established. Introduction The notion of algebraic structure has solid mathematical foundations. In the traditional, first order, case our understanding of the subject is as complete as to allow us to look at it from three different perspectives: universal algebra, equational logic, and categorical algebra. Of direct concern to us in this paper is the relationship between the first two. Universal algebra provides a model theory for algebraic structure and equational logic a formal deductive system for reasoning about it. These are related by Birkhoff s theorem [3] establishing the semantic soundness and completeness of equational deduction see Goguen and Meseguer [13] for the many-sorted case. Importantly, the theory of computation also plays a role here, as bidirectional term rewriting provides a sound and complete computational method for establishing equational derivability see e.g. [2]. We are interested here in extending the above fundamental theory from first to second order. The main motivation for doing so comes from the increasing need of metalanguages that support the specification of formalisms with variable binding. Such a development has also been advocated by Plotkin [22]. Syntactically, the passage from first to second order involves extending the language with both variable-binding operators and parameterised metavariables. These two concepts are orthogonal to each other, but it is with both of them in place that the language attains the required expressiveness. Variable-binding operators may bind a list of variables in each of their arguments and thereby lead to syntax up to alpha equivalence. Parameterised metavariables are, in effect, second-order variables for which substitution of a phrase also involves an instantiation of it. The consideration of such second-order syntax is far from being new. As far as we are aware, it was first put forward by Aczel [1]. A variation of it incorporating abstractions features in the CRSs of Klop [17]. A mathematical theory of second-order syntax was developed by Fiore [9], building on work of Fiore, Plotkin and Turi [12] and Hamana [14]. In it, second-order syntax is abstractly characterised by free algebras of a term monad on a suitable semantic universe. This provides initial-algebra semantics, induction principles, and structural recursion. Moreover, the crucial result that term monads are strong provides a second-order substitution calculus. Term monads for second-order syntax, being strong with respect to a biclosed action, fit into the framework of Fiore and Hur [11] for synthesising equational logics. This framework provides a canonical algebraic model theory for categorical equational presentations in the form of sets of parallel pairs of Kleisli maps together with a sound categorical equational metalogic for reasoning about them. The gist of the work in this paper is then to instantiate our framework and apply the supporting methodology to derive a universal algebra for second-order equational presentations, synthesise a sound second-order equational logic, and relate them by means of a completeness theorem. In passing, we also establish the conservativity of second-order equational logic over first-order equational logic and the completeness of bidirectional second-order term rewriting for establishing equational derivability. The perspective to first-order algebraic structure offered by Lawvere theories [19] has also been extended to secondorder by Fiore and Mahmoud. This will be presented elsewhere. Concerning related work, deductive systems for reasoning about algebraic structure with binding have already been considered in the literature. The system closest to ours is the Equational Logic for Binding Algebras of Sun [24, 25]. Some minor syntactic differences being that Sun works with 1

2 abstracted terms and raw syntax, and hence needs to incorporate extensionality and alpha equivalence axioms. The major difference is however in the semantics, which Sun restricts to functional models and leads to a restricted completeness result so-called admissible completeness with respect to a satisfaction relation that cuts down to a class of so-called admissible valuations. Another early system is the abstract variable binding calculus of Pigozzi and Salibra [21]. A strong point of departure between this system and ours is that in it metavariables are treated informally in the metalanguage. This results in the deductive system not being a true equational theory. More recently, there have been the Nominal Algebra of Gabbay and Mathijssen [20] and the Nominal Equational Logic of Clouston and Pitts [4]. These systems have been shown to correspond to the Synthetic Nominal Equational Logic of Fiore and Hur [11] with metavariables that can be solely parameterised by names. 1. Syntactic theory The syntactic theory underlying Second-Order Equational Logic is presented. The development comprises second-order signatures on top of which second-order terms in context are introduced. For these the needed two-levelled substitution calculus is presented. Signatures. A second-order signature Σ = T, O, is specified by a set of types T, a set of operators O, and an arity function : O T T T. This definition is a typed version of the binding signatures of Aczel [1] see also [18, 12]. Notation. We let σ be the length of a sequence σ. For 1 i σ, we let σ i be the i th element of σ; so that σ = σ 1,..., σ σ. For an operator o O, we typically write o : σ 1 τ 1,..., σ n τ n whenever o = σ 1, τ 1... σ n, τ n, τ. The intended meaning here is that o is an operator of type τ taking n arguments each of which binds n i = σ i variables of types σ i,1,..., σ i,ni in a term of type τ i. Examples of second-order signatures arise in a wide range of subjects: category theory e.g. ends, logic e.g. quantifiers, mathematics e.g. integration, process calculi e.g. restriction, programming languages e.g. local scope, type theory e.g. lambda calculi. The paradigmatic example is given below. Example 1. The signature of the typed λ-calculus over a set of base types B has set of types B => given by β B σ, τ B => β B => σ => τ B => τ and, for σ, τ B =>, operators abs σ,τ : στ σ => τ and app σ,τ : σ => τ, σ τ. The signature of the untyped λ-calculus is as above when only one type, say D, is available. Hence, it has operators abs : DD D and app : D, D D. Contexts. We will consider terms in typing contexts. Typing contexts have two zones, each respectively typing variables and metavariables. Variable typings are types. Metavariable typings are parameterised types; the idea being that a metavariable of type [σ 1,..., σ n ]τ, when parameterised by terms of type σ 1,..., σ n, will yield a term of type τ. In accordance, we use the following representation for typing contexts M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k x 1 : σ 1,..., x l : σ l where, as usual, all metavariables and all variables are assumed distinct. Terms. Signatures give rise to terms. These are built up by means of operators from both variables and metavariables, and hence referred to as second-order. Terms are considered up the α-equivalence relation induced by stipulating that, for every operator o, in the term o..., x i t i,... the x i are bound in t i. This may be formalised in a variety of ways, but it is not necessary for us to dwell into it here. The judgement for terms in context Θ Γ is defined by the following rules. This definition is a typed version of the second-order syntax of Aczel [1]. Variables For x Γ, Θ Γ x Metavariables For M : [τ 1,..., τ n ]τ Θ, Θ Γ t i i 1 i n Θ Γ M[t 1,..., t n ] Operators For o : σ 1 τ 1,..., σ n τ n τ, Θ Γ, x i : σ i t i i 1 i n Θ Γ o x 1 t 1,..., x n t n where x : σ stands for x 1 : σ 1,..., x k : σ k. Example 2. Two sample terms for the signature of the typed λ-calculus follow: M : [σ]τ, N : σ app abs xm[x], N[ ], M : [σ]τ, N : σ M[N[ ]]. Substitution calculus. The second-order nature of the syntax requires a two-level substitution calculus, see e.g. [1, 18, 26, 9]. Each level respectively accounts for the substitution 2

3 of variables and metavariables, with the latter operation depending on the former. The operation of capture-avoiding simultaneous substitution of terms for variables maps to Θ x 1 : σ 1,..., x n : σ n t, Θ Γ t i : σ i 1 i n Θ Γ t[t i /x i] 1 i n according to the following definition: x j [t i /x i] 1 i n = t j M[..., s,...] [ [t i /x i] 1 i n = M..., s[ t i /x i] 1 i n,... ] o..., y 1,..., y k s,... [t i /x i] 1 i n = o..., z 1,..., z k s[t i /x i, z j /y j] 1 i n,1 j k,... with z j domγ for all 1 j k. The operation of metasubstitution of abstracted terms for metavariables maps to M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k Γ t, Θ Γ, x i : σ i t i i 1 i k Θ Γ t{m i := x i t i } 1 i k according to the following definition: x{m i := x i t i } 1 i k = x M j [s 1,..., s m ] {M i := x i t i } 1 i k = t j [s j/x i,j] 1 j m where, for 1 j m, s j = s j{m i := x i t i } 1 i k o..., xs,... {M i := x i t i } 1 i k = o..., xs{m i := x i t i } 1 i k,... The syntactic theory can be completely justified on model-theoretic grounds. We turn to this next. 2. Abstract syntactic theory Having developed some second-order syntactic theory, our purpose in this section is to show how this arises from a model theory. This is important for several reasons: it provides an abstract characterisation of syntax by free constructions and thereby supports initial-algebra semantics and definitions by structural recursion; it encompasses and guarantees all the necessary properties of the substitution calculus; and it opens up the development of an algebraic model theory for second-order equational presentations together with an associated equational logic. We now introduce the semantic universe and, within it, constructions for modelling substitution and algebraic structure leading to a notion of model for signatures [12, 8]. The syntactic nature of free models is then explained [9]. Notation. For a sequence σ, we let [ σ] = {1,..., σ }. Semantic universe. For a set T, we write F[T ] for the free cocartesian category on T. Explicitly, it has set of objects T and morphisms σ τ given by functions ρ : [ σ] [ τ] such that σ i = τ ρi for all i [ σ]. For a set of types T, we will work within and over the semantic universe Set F[T ] T of T -sorted sets in T -typed contexts [6]. We write y for the Yoneda embedding F[T ] op Set F[T ]. Substitution. We recall the substitution monoidal structure in semantic universes [7]. It has tensor unit and tensor product respectively given by V τ = yτ, X Y τ = X τ Y where P Y = σ F[T ] P σ i [ σ] Y σ i. A monoid V A A A for the substitution monoidal structure equips A with substitution structure. In particular, the unit V A provides an embedding A y σ τ i [ σ] A σ i A τ A which together with the multiplication yields a substitution operation A τ y σ i [ σ] A σ i A τ. The category of monoids for the substitution tensor product is isomorphic to that of T -sorted Lawvere theories and maps not translations. The Lawvere theory L A associated to A has objects T and homsets L A σ, τ = i [ τ] A τ i σ, with identities and composition provided by the monoid structure. On the other hand, for every cartesian category C and assignment C C T consider the functor C, : C T Set F[T ] T 1 defined as C, D τ = C, D τ with C, c τ σ = C 1 i σ C σ i, c. Then, C, C has a canonical monoid structure given by projections and composition. Of relevance in what follows see [9], there are canonical natural isomorphisms i I X i Y = i I X i Y 2 1 i n X i Y = 1 i n X i Y 3 and, for all points ν : V Y, a natural extension map ν # : P y σ Y P Y y σ. 4 Algebras. Every signature Σ over a set of types T induces a signature endofunctor on Set F[T ] T given by 3

4 ΣX τ = o: σ 1τ 1,..., σ nτ n τ 1 i n X τ i y σ i. Σ-algebras provide interpretations for the operators of Σ. By virtue of 2 4 every signature endofunctor comes equipped with a pointed strength [9] ϖ X,V Y : ΣX Y ΣX Y. Models. The models that we are interested in referred to as Σ-monoids in [12, 9] are algebras equipped with a compatible substitution structure. For a signature Σ over a set of types T, we let Σ-Mod be the category of Σ-models with objects A Set F[T ] T equipped with a Σ-algebra and a monoid structure ΣA V A A A that are compatible in the sense that the diagram ΣA A A A ϖ A,V A ΣA A ΣA commutes. Morphisms are maps that are both Σ-algebra and monoid homomorphims. Term monad. The forgetful functor Σ-Mod Set F[T ] T is monadic [9, 10]. Hence, writing M for the induced monad and M-Alg for its category of Eilenberg-Moore algebras, we have a canonical isomorphism Σ-Mod = M-Alg. Carriers of free models can be explicitly described as initial algebras: A MX = µz. V + X Z + ΣZ. 5 Thereby free models on objects arising from meta-variable contexts have a syntactic description [14, 9]. Indeed, for a metavariable context Θ = M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k let where Θ = 1 i k y σ α = in Set F[T ] T { P, if α = τ 0, if α τ Then, we have the following syntactic characterisation MΘ τ σ = { t Θ x : σ t }. 6 We thus refer to M as the term monad. For a metavariable context Θ, the monoid multiplication MΘ MΘ MΘ amounts to the operation of capture-avoiding simultaneous substitution [9]. The term monad comes equipped with a strength MX P MX P where X P τ = X τ P, see [9]. It follows that every model MA A admits an interpretation map: MX [X, A] M X [X, A] MA A where [X, Y ] = τ T Y τ Xτ. For metavariable contexts Θ and Ξ, the interpretation map MΘ [Θ, MΞ] MΞ amounts to the operation of metasubstitution [9]. 3. Equational metalogic Our aim now is to use the above monadic model theory for second-order syntax to synthesise a Second-Order Equational Logic. This development, which is presented in Section 4, depends on a general theory and methodology of the authors [11, 5, 15]. An outline of the framework follows. To every strong monad T with respect to a biclosed action [16] we associate an Equational Metalogic. This is a deductive system for reasoning about the equality of the interpretation of Kleisli maps X T Y in Eilenberg- Moore algebras T A A as captured by the following satisfaction relation: A = f g : X T Y iff [f ] = [g ] : X [Y, A] A where [h] = X [Y, A] h id T Y [Y, A] A. 7 Equational metalogic is parameterised by a set of axioms E given by parallel pairs of Kleisli maps. The rules assert the derivability of judgements of the form E f g : X and are given in Figure 1. Example 3. The rule T Y Local parameterised composition E f g : X T i I Y i E f i g i : Y i P T Z i I E f P [ [f i ] i I ] g P [ [gi ] i I ] : X P T Z is derivable, and may be used instead of the Composition and Parameterisation rules. The category T, E-Alg of T, E-algebras is defined as the full subcategory of the category T -Alg of Eilenberg- Moore algebras that satisfy the axioms E. We have the following two important results. Soundness If E f g then A = [[f ] = [g ] for all T, E-algebras A. 8 4

5 Axiom f, g : X T Y E E f g : X T Y Equivalence f : X T Y E f f : X T Y E f g : X T Y E g f : X T Y E f g : X T Y E g h : X T Y E f h : X T Y Composition E f 1 g 1 : X T Y E f 2 g 2 : Y T Z E f 1 [f 2 ] g 1 [g 2 ] : X T Z where, for f : X T Y and g : Y T Z, f[g] is the Kleisli composite X f T Y T g T T Y T Y Parameterisation E f g : X T Y E f P g P : X P T Y P where, for h : X T Y, h P = X P h id T Y P T Y P Local character E f e i g e i : X i T Y i I E f g : X T Y {ei : X i X} i I jointly epi Figure 1. Equational Metalogic Internal completeness If every object Z admits a free T, E-algebra T E Z, then we have a quotient map q : T T E and the following are equivalent: 1 A = f g : X T Y for all T, E-algebras A 2 T E Y = f g : X T Y 3 q Y f = q Y g : X T E Y 4. Second-Order Equational Logic Second-Order Equational Logic is now synthesised from Equational Metalogic. This is done by 1 considering the term monad M; 2 restricting attention to Kleisli maps of the form y MΘ, which by 6 amount to secondorder terms of type τ in context Θ x : σ; and 3 rendering the rules in syntactic form. Presentations. An equational presentation is a set of axioms each of which is a pair of terms in context. Without loss of generality see Appendix A one may restrict to axioms containing an empty variable context, as in the CRSs of Klop [18]. There is no need for us to do so here. Example 4. The equational presentation of the typed λ-calculus follows. β M : [σ]τ, N : [ ]σ app abs zm[z], N[ ] [ M N[ ] ] η F : [ ]σ => τ abs xappf[ ], x F[ ] : σ => τ Logic. The rules of Second-Order Equational Logic are given in Figure 2. The Metasubstitution rule is a syntactic rendering of the Local parameterised composition rule. The syntactic counterpart of the Local character rule is derivable and hence omitted. We illustrate the expressive power of the system by giving some sample derivable rules. Example 5 Structural rules. For ρ : σ τ, Θ x : σ s t : γ Θ y : τ s[ρ] t[ρ] : γ where u[ρ] = u[y ρi /x i] 1 i σ. For f : {1,..., k} {1,..., l} and ρ i : 1 i k, σ fi M 1 : [ α 1 ]β 1,... M k : [ α k ]β k Γ s t : γ N 1 : [ σ 1 ]τ 1,... N l : [ σ l ]τ l Γ s{f/ρ} t{f/ρ} : γ where, for n j = σ j 1 j l, α i 5

6 Axiom Θ Γ s t E Θ Γ s t Equivalence Θ Γ t Θ Γ t t Θ Γ s t Θ Γ t s Θ Γ s t Θ Γ s u Θ Γ t u Metasubstitution M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k Γ s t Θ, x i : σ i s i t i i 1 i k Θ Γ, s{m i := x i s i } 1 i k t{m i := x i t i } 1 i k Figure 2. Second-Order Equational Logic u{f/ρ} = u{m i := x i N fi [x i,ρi1,..., x i,ρin fi ]} 1 i k. Example 6. Substitution Θ x 1 : σ 1,..., x n : σ n s t Θ Γ s i t i : σ i 1 i n Θ Γ s[s i /x i] 1 i n t[t i /x i] 1 i n Example 7. Extension M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k Γ s t M 1 : [ σ 1, σ]τ 1,..., M k : [ σ k, σ]τ k Γ, x : σ s # t # where u # = u{m i := x i M i [ x i, x]} 1 i k. 5. Model theory The model theory of Second-Order Equational Logic is presented and exemplified. The soundness of deduction is a by-product of our methodology. As an application, a conservative extension result is established. Semantics. The interpretation of a term Θ x : σ t in a model A is that of its associated Kleisli map y MΘ see 6 according to the general definition 7. Explicitly, for Θ = M 1 : [ α 1 ]β 1,..., M k : [ α k ]β k and Γ = x : σ, we have that and that [Θ Γ] A = 1 i k A β i y α i y σ [Θ Γ t ] A : [Θ Γ] A A τ is given inductively on the structure of terms as follows: [Θ Γ x j : σ j ] A is the composite [Θ Γ] A π 2 y σ yσ j A σj. [Θ Γ M i [t 1,..., t mi ] : β i ] A is the composite [Θ Γ] A π i π 1,f A βi y α i 1 j m i A αi,j A βi where f = [Θ Γ t j : α i,j ] A 1 j m i. For o : γ 1 τ 1,..., γ n τ n τ, is the composite [Θ Γ o y 1 t 1,..., y n t n ] [Θ Γ] A f j 1 j n 1 j n A τ j y γ j where f j is the exponential transpose of 1 i k A y α i β i y σ y γ j = 1 i k A β i y α i y σ, γ j A τ [Θ Γ, y j: γ j t j:τ j ] A A τj. Models. A model A is said to satisfy Θ Γ s t, written A = Θ Γ s t, iff [Θ Γ s ] A = [Θ Γ t ] A. For an equational presentation E over a signature Σ, we write Σ, E-Mod for the full subcategory of Σ-Mod consisting of the Σ-models A that satisfy the axioms E. Thus, Σ, E-Mod = M, E-Alg where E is the set of parallel pairs of Kleisli maps corresponding to the pairs of terms in E. Example 8. For the signature of the typed λ-calculus over a set of base types B Example 1, a model A τ yσ A σ A σ=>τ, A σ=>τ A σ A τ yτ A τ A τ A satisfies the β and η axioms Example 4 iff the following diagrams commute. 6

7 A τ yσ A σ abs id β subst A σ=>τ A σ app A τ A σ=>τ λapp id var η yσ A τ abs id A σ=>τ The Lawvere theory L A associated to such a model is a B => -sorted Lawvere theory equipped with cartesian closed structure L γ σ, τ = L γ, σ => τ. On the other hand, for every cartesian closed category C, an assignment C : B C canonically gives rise to a model on C #, C # satisfying the β and η axioms; where C # σ => τ = C # σ C # τ and, for β B, C # β = C β. Models of the untyped λ-calculus are briefly discussed in Appendix B. Soundness. The soundness of Second-Order Equational Logic follows as a direct consequence of that of Equational Metalogic. Soundness For an equational presentation E over a signature Σ, if Θ Γ s t is derivable from E then A = Θ Γ s t for all Σ, E-models A. Conservativity. Appendix C applies the model theory to establish the following result. Conservative extension Second-Order Equational Logic is a conservative extension of First-Order Equational Logic. This also follows from our proof of completeness, to which we turn next. 6. Completeness We establish the semantic completeness of equational derivability. Our proof is then used to show the derivability completeness of bidirectional second-order term rewriting. Free algebras. It follows from our theory of free constructions, see [11] and [10], that the forgetful functor Σ, E-Mod Set F[T ] T is monadic. Hence, writing M E for the induced monad, we have a quotient map q : M M E. In fact, since M is finitary and preserves epimorphisms 1, we have the following construction of free algebras again see [11] and [10]. 1 For E = {Θ i Γ i l i r i i } i I we take the joint coequaliser 1 As can be seen directly from the explicit description 5 or using our theory of free constructions [10]. [Θi Γ i ] MΘ for i i [Θ i Γ i l i:τ i [Θ i Γ i r i:τ i MΘ q1 coeq MΘ 1 2 We perform the following inductive construction MMΘ µ Θ µ 1 MΘ q1 MΘ 1 where MΘ 0 = MΘ. Mq n+1 MMΘ n MMΘ n+1 pushout µ n+1 3 We obtain M E Θ as the colimit of µ n+2 MΘ n+1 qn+2 MΘ n+2 MΘ q1 MΘ 1 MΘ n q n, and hence have an epimorphic quotient map q Θ : MΘ M E Θ. 9 In the light of the Internal completeness result, our method for establishing completeness is to show that, for Θ x : σ s and Θ x : σ t, if q Θ,τ, σ s = q Θ,τ, σ t then Θ Γ s t is derivable from E. This will be evident from a rule-based characterisation of the quotient 9 that we will extract from the above constructions. Explicit substitutions. We start by analising the syntactic structure of MMΘ. We need consider first syntax with explicit substitutions as abstractly introduced by the following construction where T X = µz. V + X Z + ΣZ X Y τ = X τ Y, P Y = σ T P σ i [ σ] Y σ i. Indeed, we have that T Θ τ σ = {u Θ x : σ u } where the judgement Θ Γ is given by the following rules. Variables For x Γ, Explicit substitutions Θ Γ x Θ x 1 : σ 1,..., x n : σ n t Θ Γ u i : σ i 1 i n Θ Γ t[x 1,..., x n u 1,..., u n ] Operators For o : σ 1 τ 1,..., σ n τ n τ, Θ Γ, x i : σ i u i i 1 i n Θ Γ o x 1 u 1,..., x n u n 7

8 Θ Γ u Θ Γ u 0 v Θ Γ u 0 v Θ Γ v 0 w Θ Γ u 0 u Θ Γ v 0 u Θ Γ u 0 w Θ y : τ s[y ρi/x i] 1 i σ = t Θ Γ u i 0 v ρi : σ i 1 i σ Θ Γ s[ x u] 0 t[ y ρ : σ τ v] Θ Γ, x i : σ i u i 0 v i i 1 i n Θ Γ o..., x i u i,... 0 o..., x i v i,... o : σ 1 τ 1,..., σ n τ n τ Figure 3. Rules of 0 Here the α-equivalence relation is induced by considering that in t[ x u] the x are bound in t and that in o..., x i t i,... the x i are bound in t i. The canonical quotient q 0 : X MX X MX gives rise to a quotient q # 0 : T X MX characterised as the unique map such that V + X MX + ΣMX V+X q V+q 0+ΣX # 0 +Σq# 0 V + X T X + ΣT X V + X MX + ΣMX = T X q # 0 = MX It follows that MMΘ = T MΘ / 0 where 0 is given by the rules in Figure 3. The composite T MΘ MMΘ MΘ maps Θ Γ u to Θ Γ u according to the following definition: x = x t[x 1,..., x n u 1,..., x n ] = t[u 1/x 1,..., u n/x n] o..., x i u i,... = o..., x i u i,... Syntactic model. We describe equivalence relations n on MΘ and n on T MΘ respectively giving rise to the quotients q n : MΘ MΘ n and q # n : T MΘ MMΘ n, where q n = q n q 1 and q # n = Mq n q # 0 ; so that MΘ n = MΘ/ n and MMΘ n = T MΘ/ n. As base cases, we let 0 be syntactic equality and 1 be given by the rules in Figure 4; whilst we have already defined 0 by the rules in Figure 3. For the inductive step we note that we have the following two situations: V + MΘ n MMΘ n + ΣMMΘ n V+q n q V+q # n +Σq# n 0+ΣMMΘ n V + MΘ T MΘ V + MΘ n MMΘ n + ΣT MΘ + ΣMMΘ n = T MΘ q # n = MMΘ n T MΘ MMΘ n µ n+1 pushout II MΘ q 1 MΘ 1 MΘ n+1 q # n q n+1 It follows from I that, for n 1, the equivalence relation n on T MΘ is given by the rules in Figure 5. We have from II that, for n 1, the equivalence relation n+1 on MΘ is generated by 1 and the -projection n of n. Since n is a reflexive and symmetric relation that contains 1, we have that n+1 is the transitive closure of n. Further analysis gives the characterisation of n+1 by the rules of Figure 6. Finally, the equivalence relation on MΘ describing the quotient 9 is given by the rules in Figure 7. Since derivability in this deductive system can be mimicked in Second-Order Equational Logic we have: Completeness For an equational presentation E over a signature Σ, if A = Θ Γ s t for all Σ, E-models A then Θ Γ s t is derivable from E. Furthermore, since can be characterised as the equivalence relation generated by the second-order term rewriting relation given in Figure 8, we have: Completeness of second-order term rewriting Θ Γ s t iff Θ Γ s t. I 8

9 Θ Γ t Θ Γ t 1 t Θ Γ s 1 t Θ Γ t 1 s Θ Γ r 1 s Θ Γ r 1 t Θ Γ s 1 t M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k x : α l r E ρ : α β, Θ y : β, xj : σ j t j j 1 j k Θ y : β l[y ρi/x i] 1 i α {M j := x j t j } 1 j k 1 r[y ρi/x i] 1 i α {M j := x j t j } 1 j k Figure 4. Rules of 1. Θ Γ u Θ Γ u n v Θ Γ u n v Θ Γ v n w Θ Γ u n u Θ Γ v n u Θ Γ u n w Θ y : τ s[y ρi/x i] 1 i σ n t Θ Γ u i n v ρi : σ i 1 i σ Θ Γ s[ x u] n t[ y ρ : σ τ v] Θ Γ, x i : σ i u i n v i i 1 i k Θ Γ o..., x i u i,... n o..., x i v i,... o : σ 1 τ 1,..., σ k τ k τ Figure 5. Rules of n for n 1. References [1] P. Aczel. A general Church-Rosser theorem. Typescript, [2] F. Baader and T. Nipkow. Term Rewriting and All That. CUP, [3] G. Birkhoff. On the structure of abstract algebras. Proc. of the Cambridge Philosophical Society, 31: , [4] R. Clouston and A. Pitts. Nominal equational logic. ENTCS, 172: , [5] M. Fiore. Algebraic meta-theories and synthesis of equational logics. Research Programme, Available from mpf23/.. [6] M. Fiore. Semantic analysis of normalisation by evaluation for typed lambda calculus. In PPDP 2002, pages 26 37, [7] M. Fiore. Mathematical models of computational and combinatorial structures. In FOSSACS 2005, volume 3441 of LNCS, pages 25 46, [8] M. Fiore. A mathematical theory of substitution and its applications to syntax and semantics. Invited tutorial ICMS, Available from mpf23/. [9] M. Fiore. Second-order and dependently-sorted abstract syntax. In LICS 08, pages 57 68, [10] M. Fiore and C.-K. Hur. On the construction of free algebras for equational systems. TCS, 410: , [11] M. Fiore and C.-K. Hur. Term equational systems and logics. In MFPS XXIV, volume 218, pages , [12] M. Fiore, G. Plotkin, and D. Turi. Abstract syntax and variable binding. In LICS 99, pages , [13] J. Goguen and J. Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11: , [14] M. Hamana. Free Σ-monoids: A higher-order syntax with metavariables. In APLAS 2004, LNCS 3202, pages , [15] C.-K. Hur. Categorical Equational Systems: Algebraic Models and Equational Reasoning. PhD thesis, Computer Laboratory, University of Cambridge, [16] G. Janelidze and G. Kelly. A note on actions of a monoidal category. Theory and Applications of Categories, 94:61 91, [17] J. Klop. Combinatory Reduction Systems. PhD thesis, Mathematical Centre Tracts 127, CWI, Amsterdam, [18] J. Klop, V. van Oostrom, and F. van Raamsdonk. Combinatory reduction systems: introduction and survey. TCS, 121: , [19] F. Lawvere. Functorial semantics of algebraic theories. Reprints in TAC, No. 5, pp , [20] M.J. Gabbay and A. Mathijssen. A formal calculus for informal equality with binding. In WoLLIC 07, [21] D. Pigozzi and A. Salibra. The abstract variable-binding calculus. Studia Logica, 55: , [22] G. Plotkin. An algebraic framework for logics and type theories. Invited talk ICMS, [23] A. Sangalli. On the structure and representation of clones. Algebra Universalis, 25: , [24] Y. Sun. A Framework for Binding Operators. PhD thesis, LFCS, The University of Edinburgh, [25] Y. Sun. An algebraic generalization of Frege structures binding algebras. TCS, 211: , [26] F. van Raamsdonk. Higher-order rewriting. In Term Rewriting Systems, pages ,

10 Θ Γ t Θ Γ s n+1 t Θ Γ r n+1 s Θ Γ s n+1 t Θ Γ t n+1 t Θ Γ t n+1 s Θ Γ r n+1 t Θ x : σ s n t Θ Γ s i n+1 t i : σ i 1 i σ Θ Γ s[s i /x i] 1 i σ n+1 t[t i /x i] 1 i σ Θ Γ, x i : σ i s i n+1 t i i 1 i k Θ Γ o..., x i s i,... n+1 o..., x i t i,... o : σ 1 τ 1,..., σ k τ k τ Figure 6. Rules of n+1 for n 1. Θ Γ t Θ Γ t t Θ Γ s t Θ Γ t s Θ Γ r s Θ Γ r t Θ Γ s t M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k x : α l r E ρ : α β, Θ y : β, xj : σ j t j j 1 j k Θ y : β l[y ρi/x i] 1 i α {M j := x j t j } 1 j k r[y ρi/x i] 1 i α {M j := x j t j } 1 j k Θ x : σ s t Θ Γ s i t i : σ i 1 i σ Θ Γ s[s i /x i] 1 i σ t[t i /x i] 1 i σ Θ Γ, x i : σ i s i t i i 1 i k Θ Γ o x 1 s 1,..., x k s k o x1 t 1,..., x k t k o : σ1 τ 1,..., σ k τ k τ Figure 7. Rules of. M 1 : [ σ 1 ]τ 1,..., M k : [ σ k ]τ k x : α l r E ρ : α β, Θ y : β, xj : σ j t j j 1 j k Θ Γ s l : β l 1 l β Θ Γ l[y ρi/x i] 1 i α {M j := x j t j } 1 j k [s l /y l ] 1 l β r[y ρi/x i] 1 i α {M j := x j t j } 1 j k [s l /y l ] 1 l β Θ Γ s i Θ Γ M[..., s i,...] t i : σ i M[..., t i,...] 1 i n, M : [σ1,..., σ n ]τ Θ Θ Γ, x i : σ i s i t i i Θ Γ o..., x i s i,... o..., x i t i,... 1 i k, o : σ1 τ 1,..., σ k τ k τ Figure 8. Rules of. 10

11 A. Abstraction Every term Θ Γ t can be abstracted to yield a term Θ, Γ t where, for Γ = x 1 1,..., x n n, and Γ = X 1 : [ ]τ 1,..., X n : [ ]τ n t = t[ X1[ ] /x 1,..., Xn[ ] /x n]. Performing the operation on a set of equations E to obtain a set of abstracted equations Ê, we have that the following are equivalent: Θ Γ E s t, Θ, Γ E ŝ t, Θ Γ be s t, Θ, Γ be ŝ t. Example 9. An alternative equational presentation of the typed λ-calculus follows. β T : [σ]τ x : σ η f : σ => τ app abs zt[z], x T[x] abs xappf, x f : σ => τ B. Untyped λ-calculus models A model of the syntax of the untyped λ-calculus is given by an abstract clone see e.g. [23] for this concept A n n N, x n i A n 1 i n N, k A k A n A n s, t 1,..., t k k, n N s[t 1,..., t k ] together with natural families of operations A n+1 A n : t λt, A n 2 A n : s, t st such that, for all s, s A k and t 1,..., t k A n, ss [t 1,..., t k ] = s[t 1,..., t k ] s [t 1,..., t k ] and, for all s A n+1 and t A n, λs [t 1,..., t k ] = λs[t 1,..., t k, xn+1 n+1 ] where t = t[x n+1 1,..., x n+1 n ]. See [12]. A model satisfies β-conversion if β for all s A n+1 and t A n. λst = s[x n+1 1,..., x n+1 n, t] Traditional models of the λ-calculus fit our framework as follows. Every D D D in a cartesian closed category gives rise to a model satisfying β-conversion with clone D, D and operations D, D V = D, D D D, D, D, D 2 = D, D 2 D, D D D D, D. C. Conservativity We show that Second-Order Equational Logic is a conservative extension of First-Order Equational Logic. Extension. Every first-order signature can be regarded as a second-order signature, and every first-order term as the second-order term Γ t Γ t. It follows that, for a set of first-order equations E, if Γ s t is derivable in first-order equational logic, then Γ s t is derivable in second-order equational logic. We turn to establishing the converse. 10 Conservation. Since C, preserves limits, it follows that, for every first-order signature Ω, an Ω-algebra structure on C C T 1 i n C τ i C τ τ1,..., τ n τ in Ω yields an Ω-algebra structure 1 i n C, C τ i = C, 1 i n C τ i C, C τ on C, C Set F[T ] T. The Ω-algebra and monoid structures are compatible, and we thus obtain an Ω-model Ω C, C V C, C C, C C, C The interpretations of first-order terms are related as follows: 1 i n C, C τ i = C, 1 i n C τ i [[ Γ b bt ]] C,C C, [[Γ t ]] C C, C τ and we have that C = Γ s t iff C, C = Γ ŝ t. Thus, if A = Γ ŝ t for all Ω, Ê-models A Set F[T ] T, then C = Γ s t for all Ω, E-algebras C C T and the converse of 10 holds. 11