- Email: [email protected]
Erratum to “Free abelian lattice-ordered groups” [Ann. Pure Appl. Logic 134 (2–3) (2005) 265–283]
Annals of Pure and Applied Logic 167 (2016) 431–433
Contents lists available at ScienceDirect
Annals of Pure and Applied Logic www.elsevier.com/locate/apal
Erratum
Erratum to “Free abelian lattice-ordered groups” [Ann. Pure Appl. Logic 134 (2–3) (2005) 265–283] A.M.W. Glass, Angus Macintyre, Françoise Point ∗ (with Marcus Tressl)
a r t i c l e
i n f o
Article history: Received 3 April 2014 Received in revised form 24 November 2015 Accepted 25 November 2015 Available online 7 January 2016 MSC: 03B25 06F20 20F60
We realised that the statement of [1, Proposition 3.3] is incorrect. This invalidates the deductions of Corollaries 3.4 and 3.5 and Theorem 5.1. The decidability of the theory of F A(2) therefore remains open as does the theory of the free vector lattice on 2 generators. We had seek to describe a procedure that did not depend on the parameters occurring in the formula, thinking that the continuity of the parameters alone would ensure the continuity of the quantified variables. This is not the case as shown by the example given below. We hope to be able to report soon on what needs to be done. Nothing else in [1] depended on those results. We can modify the techniques of [1, Section 3] and obtain a weaker result. Let F A(2) be the free abelian lattice-ordered group on 2 generators. Theorem. Given any finite subset f¯ of F A(2) and any quantifier-free formula ψ(x, f¯) of the language L≤ , ¯ to determine whether or not ∃xψ(x, f) ¯ holds in F A(2). there is an algorithm (dependent on f) Since we can only obtain a weaker result than what was claimed in [1, Theorem 3.5], the obvious modifications must be made to Theorem 2 of [2] and also in Sections 6 and 7 of [2].
DOI of original article: http://dx.doi.org/10.1016/j.apal.2004.10.017.
* Corresponding author. E-mail address: [email protected] (F. Point). http://dx.doi.org/10.1016/j.apal.2015.11.005 0168-0072/© 2015 Elsevier B.V. All rights reserved.
432
A.M.W. Glass et al. / Annals of Pure and Applied Logic 167 (2016) 431–433
We outline the proof of Theorem A after first giving a limiting example mentioned above. Our notation is as in the original article [1]. So L := {0, +, −}, L≤ := L ∪ {≤} and L∧ := L ∪ {∧, ∨} (where x ∨ y is a shorthand for −(−x ∧ −y)). As is standard, we may effectively pass from equations between L∧ -terms to inequalities between L≤ -terms and vice versa: the formula x ≤ y of L≤ becomes x ∧y = x in L∧ and x ∧ y = z in L∧ becomes the universal formula z ≤ x & z ≤ y & (∀w)((w ≤ x & w ≤ y) −→ (w ≤ z)). The free abelian lattice-ordered group on two generators, F A(2), can be identified with the additive group of all piecewise linear continuous functions on R2 where each linear piece is homogeneous and has integer coefficients. Since f (rx, ry) = rf (x, y) for all f ∈ F A(2), (x, y) ∈ R2 , r ∈ R>0 , we may identify each element of F A(2) with its values on the square with vertical edges x = ±1 and horizontal edges y = ±1. Let C0 be the set of all α ∈ [0.2π] such that the ray through the origin at angle α meets the square at a point with both coordinates rational. For ease of notation, we will also write these functions with argument α ∈ [0, 2π] where f (2π) = f (0); so we write f (α) for the value of f at the point (a, b) that is the intersection of the ray through the origin at angle α with this square. Note that any changes in linear definition to an element of F A(2) occur at points on this square with both coordinates rational; i.e., at angle α ∈ C0 . The topology on [0, 2π] will be that induced from the interval topology on C so that the elements of F A(2) may be regarded as continuous functions on [0, 2π] which are piecewise linear with each linear piece homogeneous with integer coefficients. We begin with a limiting example to show that a solution at each point on the square is insufficient to guarantee a “global” solution in F A(2). This illustrates the difficulties in dealing with existential formulae in L∧ or equivalent ∃∀-formulae in L≤ . Example 1. Consider the existential formula ψ(f2 , f3 ) of L∧ given by ∃f t(f, f2 , f3 ) = 0 where t(a, b, c) is
[(a + c) ∨ b] ∧ [(a − c) ∨ −b].
So ψ(f2 , f3 ) can be written as an equivalent ∃∀-formula in L≤ . Let f3 = |ρ1 | ∨ |ρ2 | ∈ F A(2) where ρ1 , ρ2 are the projections (the generators of F A(2)); so f3 (α) = 1 for all α ∈ [0, 2π]. Let f2 ∈ F A(2) be such that f2 (α) = 0
if α ∈ [0, π/2] ∪ {π} ∪ [3π/2, 2π],
f2 (α) < 0
if α ∈ (π/2, π),
f2 (α) > 0 if
α ∈ (π, 3π/2).
We now show that Q |= ψ(f2 (α), f3 (α)) at every α ∈ C0 . Let α ∈ [0, 2π]. If α ∈ [0, π/2] ∪ {π} ∪ [3π/2, 2π], then t(f, f2 , f3 )(α) = 0 if and only if (f + f3 )(α) ≤ 0 or (f −f3 )(α) ≤ 0; this occurs provided that f (α) ≤ 1. If α ∈ (π/2, π), then f2 (α) < 0 and [(f −f3 ) ∨−f2 ](α) > 0. Thus t(f, f2 , f3 )(α) = 0 if and only if (f +f3 )(α) = 0; i.e., f (α) = −f3 (α) = −1. Similarly, if α ∈ (π, 3π/2), then t(f, f2 , f3 )(α) = 0 if and only if f (α) = f3 (α) = 1. So there is a piecewise linear homogeneous function f : R2 → R satisfying Q |= ψ(f2 (α), f3 (α)) at every α ∈ C0 . But every element of F A(2) is continuous, so the discontinuity at π of any solution for f guarantees that F A(2) |= ¬ψ(f2 , f3 ). We will now work in L≤ . We can effectively separate the variable x0 in tj1 (x0 , x ¯) ≤ tj2 (x0 , x ¯) to obtain an equivalent formula of the form d.x0 ≤ s(¯ x) (or d.x0 ≥ s(¯ x)), for some effectively obtained L-term s(¯ x) and d ∈ Z≥0 (where x ¯ := (x1 , . . . , xn )). Hence tj1 (x0 , x ¯) = tj3 (x0 , x ¯) is effectively equivalent to d.x0 = s(¯ x) x) and d ∈ Z≥0 . We call such terms s(¯ x) derived terms and in L, for some effectively obtained L-term s(¯
A.M.W. Glass et al. / Annals of Pure and Applied Logic 167 (2016) 431–433
433
include the variables/parameters x1 , . . . , xn and the constant 0 in the list of derived terms. We will actually work on a square of sides bounded by x = ±D, y = ±D where D is the l.c.m. of the set of such d arising from the finite set of terms that are used to form ϑ, but continue to use α ∈ [0, 2π] to indicate the requisite points. This will allow us to work in Z2 instead of Q2 . There is no change in C0 for the two squares. We now outline the proof of Theorem A. ¯ be a quantifier-free formula in L≤ . We seek an Proof. Let f1 , . . . , fn be the set of parameters and ψ(g, f) algorithm to determine whether or not F A(2) |= (∃g)ψ(g, f¯). Let t1 (f¯), . . . , tm (f¯) be the set of terms appearing in ψ(g, f¯) (including the term 0 (if it is not already included) and s1 (f¯), . . . , sk (f¯) be the set of derived terms appearing in ψ(g, f¯) (also including 0). Now ψ(g, f¯) is equivalent to a finite set of inequalities, equalities and non-equalities (partial data) between some subset of the ti (f¯) and sj (f¯), together with derived inequalities, equalities and non-equalities of the form dr g ≤ sr (f¯), (or ≥, <, >, =, or ¬) where each dr ∈ Z+ . Since there are only a finite number of parameters and terms and derived terms, the set of all inequalities, etc. is finite and can be obtained algorithmically from ψ(x, f¯) and f¯. Each parameter, term and derived term is (finite) piecewise linear. By solving the equations, we can algorithmically determine where any of these parameters, terms and derived terms cross each other. So we can algorithmically subdivide [0, 2π] into intervals I1 , . . . , IM such that on each Ij (j ∈ {1, . . . , M }) each parameter, term and derived term appearing is linear and the inequalities (strict or otherwise) and equalities hold throughout Ij . For each j ∈ {1, . . . , M }, let I¯j be the closure of Ij . Since all crossings occur at rational points on the square, the end points βj , γj of Ij are rational. Fix j ∈ {1, . . . , M }. Any relation ≤ or = between fi (α), ti (f¯(α)) and sr (f¯(α)) which holds at βj and γj must hold at αj := (βj +γj )/2 and is unchanged if we replace αj by any other point α of Ij . Moreover, if any two terms are equal at the endpoints βj , γj or divisible by dr , dr there, then the same is true at every point in between. Hence if sr (f¯(α)) < sr (f¯(α)) for some α ∈ Ij , then sr (f¯(αj )) < sr (f¯(αj )) and we can define g(αj ) ∈ Z to lie between them on a square of side 2D if we wish (strictly if desired) since we may make g piecewise linear on I¯j without being linear. We replace any inequality of the form dr g(αj ) ≤ sr (f¯(αj )) or sr (f¯(αj )) ≥ dr g(αj ) by, respectively, g(αj ) ≤ [sr (f¯(αj ))/dr ] or [sr (f¯(αj ))/dr ] ≥ g(αj ) where [x] is the greatest integer less than or equal to x. For equality, we need only determine if sr (f¯(αj )) is a multiple of dr , and for not equal, we only need decide if sr (f¯(αj )) is not a multiple of dr . Thus we can determine if Z |= (∃x)ψ(x, f¯(αj )) at the witness αj ∈ Ij on the square of side 2D, where D ∈ Z+ is the l.c.m. of all denominators. If this is not the case, then F A(2) |= ¬(∃g)ψ(g, f¯). However, as we have seen from Example 1, this is not sufficient. We further require that if Ij and Ij share a common endpoint, say γj = βj , then the restrictions on g(γj ) and g(βj ) imposed by those on the parameters, terms and derived terms at γj from Ij and at βj from Ij must be equal. Since Pressburger arithmetic is decidable, we can decide whether or not the original set of conditions as well as this finite extra set of conditions all hold. If they all hold then F A(2) |= (∃g)ψ(g, f¯)); otherwise F A(2) |= ¬(∃g)ψ(g, f¯)). This algorithm completes the proof. 2 References [1] A.M.W. Glass, A. Macintyre, F. Point, Free abelian lattice-ordered groups, Ann. Pure Appl. Logic 134 (2005) 265–283. [2] A.M.W. Glass, F. Point, Finitely presented abelian lattice-ordered groups, in: S. Aguzzoli, et al. (Eds.), Algebraic and Proof-Theoretic Aspects of Non-classical Logics, in: LNAI, vol. 4460, 2007, pp. 160–193.
Contents lists available at ScienceDirect
Annals of Pure and Applied Logic www.elsevier.com/locate/apal
Erratum
Erratum to “Free abelian lattice-ordered groups” [Ann. Pure Appl. Logic 134 (2–3) (2005) 265–283] A.M.W. Glass, Angus Macintyre, Françoise Point ∗ (with Marcus Tressl)
a r t i c l e
i n f o
Article history: Received 3 April 2014 Received in revised form 24 November 2015 Accepted 25 November 2015 Available online 7 January 2016 MSC: 03B25 06F20 20F60
We realised that the statement of [1, Proposition 3.3] is incorrect. This invalidates the deductions of Corollaries 3.4 and 3.5 and Theorem 5.1. The decidability of the theory of F A(2) therefore remains open as does the theory of the free vector lattice on 2 generators. We had seek to describe a procedure that did not depend on the parameters occurring in the formula, thinking that the continuity of the parameters alone would ensure the continuity of the quantified variables. This is not the case as shown by the example given below. We hope to be able to report soon on what needs to be done. Nothing else in [1] depended on those results. We can modify the techniques of [1, Section 3] and obtain a weaker result. Let F A(2) be the free abelian lattice-ordered group on 2 generators. Theorem. Given any finite subset f¯ of F A(2) and any quantifier-free formula ψ(x, f¯) of the language L≤ , ¯ to determine whether or not ∃xψ(x, f) ¯ holds in F A(2). there is an algorithm (dependent on f) Since we can only obtain a weaker result than what was claimed in [1, Theorem 3.5], the obvious modifications must be made to Theorem 2 of [2] and also in Sections 6 and 7 of [2].
DOI of original article: http://dx.doi.org/10.1016/j.apal.2004.10.017.
* Corresponding author. E-mail address: [email protected] (F. Point). http://dx.doi.org/10.1016/j.apal.2015.11.005 0168-0072/© 2015 Elsevier B.V. All rights reserved.
432
A.M.W. Glass et al. / Annals of Pure and Applied Logic 167 (2016) 431–433
We outline the proof of Theorem A after first giving a limiting example mentioned above. Our notation is as in the original article [1]. So L := {0, +, −}, L≤ := L ∪ {≤} and L∧ := L ∪ {∧, ∨} (where x ∨ y is a shorthand for −(−x ∧ −y)). As is standard, we may effectively pass from equations between L∧ -terms to inequalities between L≤ -terms and vice versa: the formula x ≤ y of L≤ becomes x ∧y = x in L∧ and x ∧ y = z in L∧ becomes the universal formula z ≤ x & z ≤ y & (∀w)((w ≤ x & w ≤ y) −→ (w ≤ z)). The free abelian lattice-ordered group on two generators, F A(2), can be identified with the additive group of all piecewise linear continuous functions on R2 where each linear piece is homogeneous and has integer coefficients. Since f (rx, ry) = rf (x, y) for all f ∈ F A(2), (x, y) ∈ R2 , r ∈ R>0 , we may identify each element of F A(2) with its values on the square with vertical edges x = ±1 and horizontal edges y = ±1. Let C0 be the set of all α ∈ [0.2π] such that the ray through the origin at angle α meets the square at a point with both coordinates rational. For ease of notation, we will also write these functions with argument α ∈ [0, 2π] where f (2π) = f (0); so we write f (α) for the value of f at the point (a, b) that is the intersection of the ray through the origin at angle α with this square. Note that any changes in linear definition to an element of F A(2) occur at points on this square with both coordinates rational; i.e., at angle α ∈ C0 . The topology on [0, 2π] will be that induced from the interval topology on C so that the elements of F A(2) may be regarded as continuous functions on [0, 2π] which are piecewise linear with each linear piece homogeneous with integer coefficients. We begin with a limiting example to show that a solution at each point on the square is insufficient to guarantee a “global” solution in F A(2). This illustrates the difficulties in dealing with existential formulae in L∧ or equivalent ∃∀-formulae in L≤ . Example 1. Consider the existential formula ψ(f2 , f3 ) of L∧ given by ∃f t(f, f2 , f3 ) = 0 where t(a, b, c) is
[(a + c) ∨ b] ∧ [(a − c) ∨ −b].
So ψ(f2 , f3 ) can be written as an equivalent ∃∀-formula in L≤ . Let f3 = |ρ1 | ∨ |ρ2 | ∈ F A(2) where ρ1 , ρ2 are the projections (the generators of F A(2)); so f3 (α) = 1 for all α ∈ [0, 2π]. Let f2 ∈ F A(2) be such that f2 (α) = 0
if α ∈ [0, π/2] ∪ {π} ∪ [3π/2, 2π],
f2 (α) < 0
if α ∈ (π/2, π),
f2 (α) > 0 if
α ∈ (π, 3π/2).
We now show that Q |= ψ(f2 (α), f3 (α)) at every α ∈ C0 . Let α ∈ [0, 2π]. If α ∈ [0, π/2] ∪ {π} ∪ [3π/2, 2π], then t(f, f2 , f3 )(α) = 0 if and only if (f + f3 )(α) ≤ 0 or (f −f3 )(α) ≤ 0; this occurs provided that f (α) ≤ 1. If α ∈ (π/2, π), then f2 (α) < 0 and [(f −f3 ) ∨−f2 ](α) > 0. Thus t(f, f2 , f3 )(α) = 0 if and only if (f +f3 )(α) = 0; i.e., f (α) = −f3 (α) = −1. Similarly, if α ∈ (π, 3π/2), then t(f, f2 , f3 )(α) = 0 if and only if f (α) = f3 (α) = 1. So there is a piecewise linear homogeneous function f : R2 → R satisfying Q |= ψ(f2 (α), f3 (α)) at every α ∈ C0 . But every element of F A(2) is continuous, so the discontinuity at π of any solution for f guarantees that F A(2) |= ¬ψ(f2 , f3 ). We will now work in L≤ . We can effectively separate the variable x0 in tj1 (x0 , x ¯) ≤ tj2 (x0 , x ¯) to obtain an equivalent formula of the form d.x0 ≤ s(¯ x) (or d.x0 ≥ s(¯ x)), for some effectively obtained L-term s(¯ x) and d ∈ Z≥0 (where x ¯ := (x1 , . . . , xn )). Hence tj1 (x0 , x ¯) = tj3 (x0 , x ¯) is effectively equivalent to d.x0 = s(¯ x) x) and d ∈ Z≥0 . We call such terms s(¯ x) derived terms and in L, for some effectively obtained L-term s(¯
A.M.W. Glass et al. / Annals of Pure and Applied Logic 167 (2016) 431–433
433
include the variables/parameters x1 , . . . , xn and the constant 0 in the list of derived terms. We will actually work on a square of sides bounded by x = ±D, y = ±D where D is the l.c.m. of the set of such d arising from the finite set of terms that are used to form ϑ, but continue to use α ∈ [0, 2π] to indicate the requisite points. This will allow us to work in Z2 instead of Q2 . There is no change in C0 for the two squares. We now outline the proof of Theorem A. ¯ be a quantifier-free formula in L≤ . We seek an Proof. Let f1 , . . . , fn be the set of parameters and ψ(g, f) algorithm to determine whether or not F A(2) |= (∃g)ψ(g, f¯). Let t1 (f¯), . . . , tm (f¯) be the set of terms appearing in ψ(g, f¯) (including the term 0 (if it is not already included) and s1 (f¯), . . . , sk (f¯) be the set of derived terms appearing in ψ(g, f¯) (also including 0). Now ψ(g, f¯) is equivalent to a finite set of inequalities, equalities and non-equalities (partial data) between some subset of the ti (f¯) and sj (f¯), together with derived inequalities, equalities and non-equalities of the form dr g ≤ sr (f¯), (or ≥, <, >, =, or ¬) where each dr ∈ Z+ . Since there are only a finite number of parameters and terms and derived terms, the set of all inequalities, etc. is finite and can be obtained algorithmically from ψ(x, f¯) and f¯. Each parameter, term and derived term is (finite) piecewise linear. By solving the equations, we can algorithmically determine where any of these parameters, terms and derived terms cross each other. So we can algorithmically subdivide [0, 2π] into intervals I1 , . . . , IM such that on each Ij (j ∈ {1, . . . , M }) each parameter, term and derived term appearing is linear and the inequalities (strict or otherwise) and equalities hold throughout Ij . For each j ∈ {1, . . . , M }, let I¯j be the closure of Ij . Since all crossings occur at rational points on the square, the end points βj , γj of Ij are rational. Fix j ∈ {1, . . . , M }. Any relation ≤ or = between fi (α), ti (f¯(α)) and sr (f¯(α)) which holds at βj and γj must hold at αj := (βj +γj )/2 and is unchanged if we replace αj by any other point α of Ij . Moreover, if any two terms are equal at the endpoints βj , γj or divisible by dr , dr there, then the same is true at every point in between. Hence if sr (f¯(α)) < sr (f¯(α)) for some α ∈ Ij , then sr (f¯(αj )) < sr (f¯(αj )) and we can define g(αj ) ∈ Z to lie between them on a square of side 2D if we wish (strictly if desired) since we may make g piecewise linear on I¯j without being linear. We replace any inequality of the form dr g(αj ) ≤ sr (f¯(αj )) or sr (f¯(αj )) ≥ dr g(αj ) by, respectively, g(αj ) ≤ [sr (f¯(αj ))/dr ] or [sr (f¯(αj ))/dr ] ≥ g(αj ) where [x] is the greatest integer less than or equal to x. For equality, we need only determine if sr (f¯(αj )) is a multiple of dr , and for not equal, we only need decide if sr (f¯(αj )) is not a multiple of dr . Thus we can determine if Z |= (∃x)ψ(x, f¯(αj )) at the witness αj ∈ Ij on the square of side 2D, where D ∈ Z+ is the l.c.m. of all denominators. If this is not the case, then F A(2) |= ¬(∃g)ψ(g, f¯). However, as we have seen from Example 1, this is not sufficient. We further require that if Ij and Ij share a common endpoint, say γj = βj , then the restrictions on g(γj ) and g(βj ) imposed by those on the parameters, terms and derived terms at γj from Ij and at βj from Ij must be equal. Since Pressburger arithmetic is decidable, we can decide whether or not the original set of conditions as well as this finite extra set of conditions all hold. If they all hold then F A(2) |= (∃g)ψ(g, f¯)); otherwise F A(2) |= ¬(∃g)ψ(g, f¯)). This algorithm completes the proof. 2 References [1] A.M.W. Glass, A. Macintyre, F. Point, Free abelian lattice-ordered groups, Ann. Pure Appl. Logic 134 (2005) 265–283. [2] A.M.W. Glass, F. Point, Finitely presented abelian lattice-ordered groups, in: S. Aguzzoli, et al. (Eds.), Algebraic and Proof-Theoretic Aspects of Non-classical Logics, in: LNAI, vol. 4460, 2007, pp. 160–193.