aa r X i v : . [ m a t h . L O ] M a y MOST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE
NIKOLAOS GALATOS AND GAVIN ST. JOHN
Abstract.
All known structural extensions of the substructural logic FL e , Full Lam-bek calculus with exchange/commutativity, (corresponding to subvarieties of commutativeresiduated lattices axiomatized by {∨ , · , } -equations) have decidable theoremhood; in par-ticular all the ones defined by knotted axioms enjoy strong decidability properties (such asthe finite embeddability property). We provide infinitely many such extensions that haveundecidable theoremhood, by encoding machines with undecidable halting problem. Aneven bigger class of extensions is shown to have undecidable deducibility problem (the cor-responding varieties of residuated lattices have undecidable word problem); actually withvery few exceptions, such as the knotted axioms and the other prespinal axioms, we provethat undecidability is ubiquitous. Known undecidability results for non-commutative ex-tensions use an encoding that fails in the presence of commutativity, so and-branchingcounter machines are employed. Even these machines provide encodings that fail to cap-ture proper extensions of commutativity, therefore we introduce a new variant that workson an exponential scale. The correctness of the encoding is established by employing thetheory of residuated frames. §
1. Introduction.
Substructural logics are defined as extensions of the FullLambek calculus FL and include among others classical, intuitionistic, linear,relevance, bunched-implication and many-valued logics. They find applicationsto areas as diverse as mathematical linguistics, philosophy, management of point-ers in computer architecture, engineering, theoretical physics and functional pro-gramming. Their algebraic semantics, in the sense of Blok and Pigozzi [1], are(pointed) residuated lattices (or FL-algebras) and they have an independent his-tory with connections to classical and to ordered algebra. In particular theyinclude the lattice of ideals of rings, lattice-order groups, algebras of relations,and of course Boolean and Heyting algebras. Pointed residuated lattices form avariety FL and its subvarieties correspond to extensions of FL via a dual latticeisomorphism; algebraic and logical properties are tightly linked. See [6] for anintroduction to the area.Decidability questions are at the core of the study of logical systems and herewe explore logics/varieties with structure rich enough to allow for encoding thecomputation of machines with undecidable halting problem. This yields unde-cidability results for the word problem, and hence also for the quasiequationaltheory, of these varieties (namely the deducibility relation for the logics) andsometimes even the undecidability of the equational theory of the varieties (i.e.,the theoremhood in the corresponding logics).The equational theory of FL is decidable and the same holds for many of itsstandard extensions such as FL e (with exchange/commutativity: xy = yx ), FL w NIKOLAOS GALATOS AND GAVIN ST. JOHN (with weakening/integrality: x ≤ FL ei (with exchange and weakening), FL ec (with exchange and contraction: x ≤ x ), FL em (with exchange and mingle: x ≤ x ). The equational theory of FL c is one of the few known to be undecidable [3]; aprecursor to this result is the fact that the equational theory of FL ec , even thoughdecidable, is not primitive recursive [16]. The only other known subvarieties of FL with undecidable equational theory are the ones axiomatized by x m ≤ x n (where 0 < m < n ), the one axiomatized by the modular law, and the oneaxiomatized by commutativity, involutivity and distributivity (corresponding tothe relevance logic R). In particular, the last one is the only subvariety of FL e with undecidable equational theory; actually distributivity does not correspondto a sequent structural rule, unless the syntax is expanded, so it is not evena structural extension of FL . On the contrary, prominent subvarieties of FL e ,such as (proper, non-trivial) subvarieties axiomatized by any knotted inequality x m ≤ x n (where m = n ) not only have a decidable equational theory but alsoa decidable quasiequational theory, and even the finite embeddability property[17]. (In [2] it is further shown that this remains true even under conditionsweaker than commutativity.)In contrast to the above, in this paper we construct infinitely many subvarietiesof FL e with undecidable equational theory. We also show that an even biggercollection of subvarieties of FL e have an undecidable quasiequational theory, ac-tually undecidable word problem. The encoding used for the undecidability ofthe word problem for FL e does not work for its subvarieties, so we modify itin a novel way, by storing the values in the counter machines as powers of asufficiently large constant, which depends on the subvariety.A residuated lattice R = ( R, ∨ , ∧ , · , \ , /,
1) is an algebraic structure such that( R, ∨ , ∧ ) is a lattice, ( R, · ,
1) is a monoid, and the law of residuation holds: forall x, y, z ∈ R , x · y ≤ z iff x ≤ z/y iff y ≤ x \ z, where ≤ is the induced lattice order. The residuated lattice R is called com-mutative if ( R, · ,
1) is a commutative monoid; in such a case x \ y = y/x for all x, y ∈ R and will use the notation x → y := x \ y . It is well known that (commu-tative) residuated lattices form a variety denoted ( C ) RL , see [6]. FL-algebras aredefined as expansions of residuated lattices by an arbitrary constant 0 (whichis used to define the negation operation), but we will not be making use of thisconstant in our encodings and our results remain true in the presence or absenceof this constant.For example, the extension of (0-free) FL e with the structural ruleΓ , ∆ , ∆ , Σ ⊢ Π Γ , ∆ , ∆ , ∆ , Σ ⊢ ΠΓ , ∆ , Σ ⊢ Π ( ε )corresponds to the subvariety of CRL defined by the inequality ε : x ≤ x ∨ x .More generally, structural rules, such as the above, correspond to inequalities inthe signature {∨ , · , } . We will show that theoremhood for FL e + ( ε ) and theequational theory of CRL + ε are undecidable. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE and-branching counter machines , a variant of counter ma-chines, introduced in [11]. In § § CRL and RL , extending the result that was known for just these two varieties.This is done to set the stage for the much more complicated development thatfollows, while still introducing two of the main tools: abstract machines andresiduated frames. The theory of residuated frames, developed in [5], is used toprove the completeness of the encoding, inspired by [8, 3]. The encodings usedin the latter and the standard encodings, however, do not work in the presenceof commutativity and this is why we make use of the one considered in [11]; inthe Conclusions section § § {∨ , · , } , as alluded to above, and show how theycan be viewed as conjunctions of simple inequalities. We also explain why theencoding of and-branching counter machines fails to work for subvarieties of CRL axiomatized by such equations, thus necessitating the use of a novel encoding.Parts of § § admissibility of theequation relative to the machine and also we work out some motivating examples.In § § ⋆⋆ ) emerges as a prominent condition for the givenequation so that the encoding will work. In the beginning of § ⋆⋆ ), not even the new exponential en-coding will work to establish undecidability and thus such equations are outsidethe scope of this paper. However, in § ⋆⋆ ), then it indeed defines a variety of residuated lattices with unde-cidable word problem. Actually, Theorem 7.17 shows that almost all 1-variableequations actually satisfy condition ( ⋆⋆ ), indicating that this condition is notreally restrictive. In parallel to all this, starting already from § spineless ; the class is so big that we defineit via its complement (prespinal equations), which forms a very small portion ofall equations and admits a very natural and simple definition.In § ⋆⋆ ). Therefore,the results of § § § NIKOLAOS GALATOS AND GAVIN ST. JOHN the stronger result of the undecidability of the equational theory for certain sub-varieties; this is done by using a special form of the deduction theorem, relyingon the characterization of congruences in commutative and expansive residuatedlattices. In that sense, someone who reads just the first third of the paper, upto §
5, has a full and clear grasp of all the notions and also knows the statementsof the two main results of the paper, Theorem 5.5 and Theorem 5.9.The proof, given in §
8, that these two differently-looking notions coincide isquite involved and relies on positive linear algebra, therefore § §
2. Preliminaries.
We denote the sets of natural numbers, positive inte-gers, and real numbers by N , Z + , and R , respectively. We denote the powerset of a set X by ℘ ( X ). Given a set X and a binary operation symbol · , we de-note by ( X ∗ , · ,
1) the free commutative monoid generated by X with unit 1. A substitution on X is a monoid homomorphism σ : X ∗ → X ∗ ; substitutions aredetermined by their restriction to X . For x ∈ X ∗ , we write x n to denote 1,if n = 0, and the term x · · · · · x consisting of n copies of x for n >
0. Forsubsets
A, B of X , we define A · B = { a · b : a ∈ A, b ∈ B } , and if a ∈ X then a · B = { a } · B .Let L be an algebraic language, i.e., containing no relational symbols. Givena set of variables X , T ( X ) denotes the set of terms over X and L and T ( X ) theabsolutely free algebra of terms. A quasiequation is (the universal closure of) aformula of the form s = t & . . . & s n = t n = ⇒ s = t , (1)where s , t , ..., s n , t n ∈ T ( X ) are terms and n ∈ N . If n = 0 then the left-handside is empty and we obtain an equation.For a class of algebras K in the language L , we say that (1) holds in K (i.e. K | = (1)) if for every algebra A ∈ K and homomorphism h : T ( X ) → A , ( ∀ i ∈ { , ..., n } )( A , h | = s i = t i ) = ⇒ A , h | = s = t . Here A , h | = s = t means h ( s ) = h ( t ).A presentation is a pair h X, E i where X is a set of generators and E is a setof equations over T ( X ). A presentation h X, E i is said to be finite iff both X and E are finite. We denote the conjunction of equations in E by & E . For a varietyof algebras V in the language L , we say V has an undecidable word problem ifthere exists a finite presentation h X, E i such that there is no algorithm, whichon input s, t ∈ T ( X ) decides whether the quasiequation& E = ⇒ s = t (2)holds in V . Note that if V has undecidable word problem then its quasiequationaland universal theories are undecidable as well. The word problem is also referredto as the local word problem and the quasiequational theory as the global word OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE {∨ , · , } -reduct, and therefore theundecidability results apply to all reducts that contain it. §
3. Counter Machines.
For proving undecidability we use a type of ab-stract machine known as an
And-branching k -Counter Machine ( k -ACM), intro-duced in [11], as they have an undecidable halting problem. A k -ACM is a tuple M = ( R k , Q , P , q f ) representing a type of parallel-computing counter machine,where • R k := { r , ..., r k } is a set of k registers , each able to store a nonnegativeinteger (representing the number of tokens in that register), • Q is a finite set of states with a designated final state q f , and • P is a finite set of instructions (to be formalized below) that indicatewhether to, given a certain state of the machine, increment a register or decrement a nonzero register, as well as a “branching” instruction knownas forking , with no instruction applicable to the state q f .A configuration C of a k -ACM is a tuple consisting of a single state and, foreach register, a nonnegative integer indicating the contents of that register. Wecan imagine a configuration being a box labeled by a state and containing tokenseach labelled by an element of the set R k . In essence, a configuration is specifiedby the state label and the multiset of register labels of the tokens. Since theorder of the symbols is irrelevant, we represent a configuration C as a term inthe free commutative monoid generated by Q ∪ R k , and canonically arranged as q r n r n · · · r n k k , where q ∈ Q is the state of the configuration and n i is the number stored inthe register r i , for each i = 1 , .., k ; if n i = 0, we say the register r i is empty .Since C contains precisely one state, we may define the set of configurations by Conf ( M ) := Q · R ∗ k .The instructions of a k -ACM replace a single configuration by a new config-uration (via increment and decrement), or by two configurations (via forking).An increment instruction can be understood as “if a box is labeled by state q ,add one register- r i token and relabel the box with state q ′ ,” decrement as “ifa box is labeled by state q , and r i is not empty, remove one register- r i tokenand relabel the box with state q ′ ,” and forking as “if a box is labeled by state q ,duplicate the box and its contents, resulting in two boxes relabeled by q ′ and q ′′ ,respectively.” As a consequence of the forking instruction, the machine can beoperating on multiple configurations, i.e. branches, in parallel and is inherentlynondeterministic. The status of a machine at a given moment in a computation,called an instantaneous description (ID), is represented by the configurationsthat are present. Formally, an ID is an element C ∨ · · · ∨ C m , of the free commutative semigroup ID ( M ) generated by Conf ( M ); we denote theassociated binary operation by ∨ . NIKOLAOS GALATOS AND GAVIN ST. JOHN
In this way, we view ID ( M ) as a subset of the commutative semiring A M =( A M , ∨ , · , ⊥ ,
1) generated by R k ∪ Q , where • ( A M , ∨ , ⊥ ) is a commutative monoid with additive identity ⊥ , • ( A M , · ,
1) is a commutative monoid with multiplicative identity 1, and • multiplication distributes over join.Even though ∨ in A M is not defined to be idempotent, we will consider ho-momorphisms from A M that will map ∨ to a semilattice operation and for ourapplications it would not hurt to define ∨ to be idempotent. However, the non-idempotent status of ID’s matches better the intuition of computation and thisis the reason for our choice.Since multiplication fully distributes over ∨ in A M , each element of A M canbe written as a finite join W i ∈ I m i , where I is a finite (possibly empty) set, ofmonoid terms m i ∈ ( Q ∪ R k ) ∗ , for all i ∈ I ; recall that the join of the empty setis the bottom element ( ⊥ = W ∅ ). As usual, each element of A M , which is theequivalence class [ t ] of a term t in the absolutely free algebra over {∨ , · , } and Q ∪ R k , will be identified with the term t itself, when no confusion arrises.Formally, an instruction p of a k -ACM is an expression of the form q ≤ q ′ r i , q r i ≤ q ′ , or q ≤ q ′ ∨ q ′′ , where q, q ′ , q ′′ ∈ Q and r i ∈ R k , representing increment r i , decrement r i , and fork , respectively. We will often write p : C ≤ u to indicatethe instruction p is given by C ≤ u , where C ∈ Conf ( M ) and u ∈ ID ( M ). For astate q ∈ Q , we say p is a q -instruction if p : qx ≤ u for some x ∈ R ∗ k . Note thata machine M with final state q f contains no q f -instructions by definition.The computation relation ≤ for the machine M = ( R K , Q , P , q f ) is defined to bethe smallest {· , ∨} -compatible preorder containing P , and will be denoted by ≤ M .For a given instruction p : C ≤ u , it will be useful to define the relation ≤ p to bethe closure of p under the inference rules v ≤ p wvx ≤ p wx [ · ] and v ≤ p wv ∨ t ≤ p w ∨ t [ ∨ ] , for all v, w, x, t ∈ A M (in that order, without loss of generality, due to the dis-tributivity of · over ∨ ). Consequently, v ≤ p w if and only if v = C x ∨ t and w = u x ∨ t , for some x, t ∈ A M ; these equalities are understood inside A M , so theterms v and C x ∨ t need not be identical. We therefore conclude that if v ≤ p w ,then v ∈ ID ( M ) if and only if w ∈ ID ( M ).It is easily verified that ≤ M is equivalent to the smallest preorder generatedby S {≤ p : p ∈ P } . Therefore, if s ≤ M t , then there exist n ∈ N , a sequence of A M -terms t , . . . , t n and a sequence of instructions p , . . . , p n from P , collectivelycalled a computation in M of length n witnessing s ≤ M t , such that s = A M t ≤ p t ≤ p · · · ≤ p n t n = A M t. Clearly, if there is a computation witnessing s ≤ M t , then there is a computationof minimal length, the value of which we simple call the computation length . Thefollowing result is an easy consequence of the definitions. Lemma . Let s, t, t ′ ∈ A M . If s ≤ M t , then there exists a computation witnessing it and furthermore, s ∈ ID ( M ) iff t ∈ ID ( M ) . OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE t ∨ t ′ ≤ M s if and only if there exist s ′ , s ′′ ∈ A M such that s = s ′ ∨ s ′′ , t ≤ M s ′ and t ′ ≤ M s ′′ . Furthermore, the sum of the computation lengths of t ≤ M s ′ and t ′ ≤ M s ′′ is less than or equal to the computation length of t ∨ t ′ ≤ M s . An ID consisting entirely of configurations labeled by the state q f with allregisters empty is called a final ID and we denote the set of final ID’s by Fin ( M ) := { W ni =1 q f : n ≥ } . Our choice to not assume the idempotency of ∨ in A M explainsthe necessity of treating W ni =1 q f as a final ID. We say that a term t ∈ A M is accepted by M if there exists u f ∈ Fin ( M ) such that t ≤ M u f , and we define the set accepted terms to be Acc( M ) := { t ∈ A M : ∃ u f ∈ Fin ( M ) , t ≤ M u f } . Theorem . ([10, 13, 11]) There exists a 2-ACM ˜ M such that membershipin Acc(˜ M ) is undecidable. Example . Consider the 1-ACM M even = ( R , Q even , P even , q f ), where Q even = { q , q , q f } and P even = { p , p , p f } is given by p : q r ≤ q p : q r ≤ q p f : q ≤ q f ∨ q f . For example, q r ≤ p q r ≤ p q ≤ p f q f ∨ q f is a computation showing that q r is accepted. On the other hand q r ≤ p q and q r ≤ p f ( q f ∨ q f ) r = q f r ∨ q f r are the only maximal computations starting with q r and none ofthem ends in a final configuration, so q r is not accepted. In general, it is easyto see that q r n ∈ Acc( M even ) if and only if n is even. Lemma . Let M be an ACM.
1. Acc( M ) ⊆ ID ( M ) and the terms in any computation ending in a final ID areall ID’s. For all u, v ∈ A M , u ∨ v ∈ Acc( M ) if and only if u ∈ Acc( M ) and v ∈ Acc( M ) . Proof.
The first claim follows from Lemma 3.1(1) by induction on the com-putation length since
Fin ( M ) ⊆ ID ( M ) by definition. The second claim followsfrom Lemma 3.1(2) since Fin ( M ) is exactly the set of finite non-empty joins ofthe same configuration q f . ⊣ Lemma 3.4 shows that in a computation witnessing the acceptance of an ID allconfigurations are ID’s and therefore for those cases the inference rule [ · ] couldhave been restricted to x ∈ R ∗ k . §
4. Machines and Residuated Frames.
As a demonstration of our generaltechnique, we will use counter machines and residuated frames to show that anyvariety between
CRL and RL has an undecidable word problem.Let M = ( R k , Q , P , q f ) be an ACM. For each u ∈ ID ( M ), formally viewed as an {∨ , · , } -term in T ( R k ∪ Q ), we define the quasiequation acc M ( u ) to be& P com ⇒ u ≤ q f , where q f is the final state of M , and P com := P ∪ { xy ≤ yx : x, y ∈ R k ∪ Q } is the(finite) set of instructions P together with a finite set encoding commutativityfor letters (and hence also all words) over the set R k ∪ Q ; for our purposes wecould actually restrict the x in P com to only state variables. NIKOLAOS GALATOS AND GAVIN ST. JOHN
The following lemma shows that computations in machines can be performedalso in the theory of residuated lattices.
Lemma . Let M be an ACM and u an ID. If u ∈ Acc( M ) then RL | = acc M ( u ) . Proof.
Let M = ( R k , Q , P , q f ) be an ACM and suppose u ∈ ID ( M ) is acceptedin M . We proceed by induction on the length n of the computation witnessing theacceptance of u in M . If the length is zero then u ∈ Fin ( M ). Since ∨ is idempotentin residuated lattices, RL | = u f ≤ q f for any u f ∈ Fin ( M ). Hence RL | = acc M ( u ) a fortiori . Now, suppose the claim holds for all accepted ID’s with computationlength 0 ≤ k < n . By Lemma 3.4(1), there is an instruction p ∈ P such that u ≤ p u ′ ∈ Acc( M ) for some u ′ ∈ ID ( M ), where the acceptance computation of u ′ has length less than n . Formally viewing u ′ as an element in T ( X ) where X = R k ∪ Q , RL | = acc M ( u ′ ) by the induction hypothesis.Now, suppose that for a residuated lattice R and for a homomorphism f : T ( X ) → R we have R , f | = P com . Hence f ( u ′ ) ≤ R f ( q f ) since RL | = acc M ( u ′ ). As ≤ R is transitive, we need only show f ( u ) ≤ R f ( u ′ ) to establish R , f | = acc M ( u ).Let S ( X ) be the free algebra over {∨ , · , } and X . As R has a semiring reductand f ( a ) f ( b ) = R f ( b ) f ( a ), for all a, b ∈ X , the restriction of f on S ( X ) factorsthrough A + M := A M \ {⊥} as f : S ( X ) ν → A + M h → R as a semiring homomorphism,where ν is the natural surjective homomorphism and h is a semiring homomor-phism. So, h ( a ) h ( b ) = R h ( b ) h ( a ), for all a, b ∈ X , and h ( C ) ≤ R h ( v ) where p : C ≤ v . By definition of ≤ p , u = A M C x ∨ w ≤ p v x ∨ w = A M u ′ , for some x ∈ X ∗ and w ∈ A M , where v x ∨ w = A M v x if w = ⊥ . Using theproperties above and the fact that h is a semiring homomorphism we obtain h ( u ) = R h ( C x ∨ w ) ≤ R h ( v x ∨ w ) = R h ( u ′ ) . It follows that h ( u ) ≤ R h ( u ′ ) and therefore that f ( u ) ≤ R f ( u ′ ) . ⊣ To show the converse of Lemma 4.1, we will need to show that given an ACM M , if u Acc( M ) then there is a residuated lattice W + M (which will actually evenbe commutative) that falsifies acc M ( u ); actually, in the proof we proceed by con-traposition. We will further prove that every subvariety of RL that contains W +˜ M has undecidable word problem (and thus undecidable quasiequational theory).The construction of W + M is based on residuated frames [5], structures that willalso be used later in the paper, so we define them briefly here.For the purposes of this paper, a commutative residuated frame is a structure W = ( W, W ′ , N , · , W, · ,
1) is a commutative monoid, W ′ is a set, and N is a subset of W × W ′ , such that there exists a function (cid:12) : W ′ × W → W ′ with: ∀ x, y ∈ W z ∈ W ′ , x · y N z iff x N z (cid:12) y . Given such a residuated frame,for X ⊆ W , x ∈ W , Y ⊆ W ′ and y ∈ W ′ , we define X N y to mean x N y forall x ∈ X , and x N Y to mean x N y for all y ∈ Y . For X ⊆ W and Y ⊆ W ′ ,we define X ⊲ := { y ∈ W ′ : X N y } , Y ⊳ := { x ∈ W : x N Y } . The pair ( ⊲ , ⊳ )forms what is known as a Galois connection , and we will make use of the factthat X ⊲⊳ ⊆ X ⊲⊳ if and only if X ⊲ ⊆ X ⊲ for any X , X ⊆ W .Define γ ( X ) = X ⊲⊳ . We write γ ( x ) = γ ( { x } ) for x ∈ W , and ℘ ( W ) γ = γ [ ℘ ( W )]. It follows from [5] that the algebra W + := ( ℘ ( W ) γ , ∩ , ∪ γ , · γ , → , γ (1)) OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE X ∪ γ Y := γ ( X ∪ Y ), X · γ Y := γ ( X · Y ),and X → Y := { z ∈ W : X · { z } ⊆ Y } .Inspired by [8], given an ACM M = ( R k , Q , P , q f ) we define the tuple W M =( W M , W M , N M , · , W M := ( Q ∪ R k ) ∗ ⊆ A M and x N M y if and only if xy ∈ Acc( M ), for all x, y ∈ W M . Lemma . W M is a residuated frame and therefore W + M ∈ CRL . Proof.
We define z (cid:12) y = yz . Clearly, for x, y, z ∈ W M , xy N M z iff xyz ∈ Acc( M ) iff x N M yz. ⊣ For an ACM M = ( R k , Q , P , q f ), we define the assignment e : Q ∪ R k → W + M via e ( a ) := { a } ⊲⊳ and its homomorphic extension ¯ e : T ( Q ∪ R k ) → W + M .We will need to make use of the following technical lemma. Lemma . If M = ( R k , Q , P , q f ) is an ACM, then W + M , ¯ e | = P com . Further-more, ¯ e ( x ∨ y ) = { x, y } ⊲⊳ for any x, y ∈ W M . Proof.
In [5] it is shown that the map γ satisfies the properties γ ( γ ( X ) · γ ( Y )) = γ ( X · Y ) and γ ( γ ( X ) ∪ γ ( Y )) = γ ( X ∪ Y ), for all X, Y ⊆ W . Using thefirst one, for each a, b ∈ Q ∪ R k we have¯ e ( ab ) = ¯ e ( a ) · γ ¯ e ( b ) = e ( a ) · γ e ( b ) = γ ( γ ( a ) · γ ( b )) = γ ( ab ) . It follows by induction that ¯ e ( x ) = γ ( x ) for each x ∈ W M .Now, let x, y ∈ W M . Then¯ e ( x ∨ y ) = ¯ e ( x ) ∪ γ ¯ e ( y ) = γ ( x ) ∪ γ γ ( y ) = γ ( γ ( x ) ∪ γ ( y )) = γ ( { x, y } )where the last equality follows from the second property of γ above.Since W + M is commutative, we need only show W + M | = P . Let p : C ≤ C ∨ C be in P . By the calculation above, ¯ e ( C ) = { C } ⊲⊳ and ¯ e ( C ∨ C ) = { C , C } ⊲⊳ .So, to show W + M , ¯ e | = C ≤ C ∨ C , we need to show that { C } ⊲⊳ ⊆ { C , C } ⊲⊳ ,or equivalently that { C , C } ⊲ ⊆ { C } ⊲ . Suppose x ∈ { C , C } ⊲ , then C N M x and C N M x , so C x, C x ∈ Acc( M ) . Now C ≤ p C ∨ C implies C x ≤ p ( C ∨ C ) x , thusby Lemma 3.4(2) C x ≤ p ( C ∨ C ) x = C x ∨ C x ∈ Acc( M ) , and it follows that C N M x , or equivalently x ∈ { C } ⊲ . ⊣ Lemma . Let V be a subvariety of RL containing W + M for some ACM M .Then for all u ∈ ID ( M ) , u ∈ Acc( M ) if and only if V | = acc M ( u ) . Proof.
Let M = ( R k , Q , P , q f ) be a k -ACM. The forward direction follows fromLemma 4.1. For the reverse direction note that from W + M ∈ V we have W + M | =acc M ( u ). By Lemma 4.3, W + M , ¯ e | = P com and so W + M , ¯ e | = u ≤ q f . Let t , ..., t n ∈ ( Q ∪ R k ) ∗ be given so that u = t ∨ · · · ∨ t n , so ¯ e ( t ∨ · · · ∨ t n ) ⊆ ¯ e ( q f ), which yields { t , . . . , t n } ⊲⊳ ⊆ q f ⊲⊳ by Lemma 4.3. This is equivalent to { q f } ⊲ ⊆ { t , ..., t n } ⊲ . Since ≤ M is reflexive, q f ∈ Acc( M ) and thus q f N M , so 1 ∈ { q f } ⊲ . Therefore,1 ∈ { t , ..., t n } ⊲ , that is { t , ..., t n } N M
1, so t N M , . . . , t n N M
1. Hence t , ..., t n ∈ Acc( M ) and by Lemma 3.4(2) we conclude u ∈ Acc( M ). ⊣ The argument that follows clearly works for v = C as well. NIKOLAOS GALATOS AND GAVIN ST. JOHN
As a consequence of Lemma 4.4, if
V ⊆ RL is a variety containing W + M , forsome ACM M , then { acc M ( u ) : u ∈ Acc( M ) } = { acc M ( u ) : V | = acc M ( u ) } . Since h Q ∪ R k , P com i is a finite presentation and all equations in acc M ( u ) have a commonantecedent & P com , the following is immediate: Theorem . Let M be an ACM and W + M ∈ V ⊆ RL for a variety V . Thendeciding the word problem of V is at least as hard as deciding membership in Acc( M ) . Corollary . If V is a subvariety of RL containing W + M , where M is anACM such that membership in Acc( M ) is undecidable, then V has an undecidableword problem. In particular, any variety in the interval CRL to RL has undecid-able word problem since W +˜ M ∈ CRL , where ˜ M is the machine from Theorem 3.2. The above results hold even for the {∨ , · , } reducts of these varieties. Since { acc M ( u ) : V | = acc M ( u ) } ⊆ { ξ : ξ is a quasieq. such that V | = ξ } , we thereforealso obtain the undecidability of the quasiequational theory. The quasiequationaltheories of RL and CRL alone were known to be undecidable; see [9] and [11],respectively. §
5. Equations in the signature {∨ , · , } and machine admissibility. Our goal is to find proper subvarieties of ( C ) RL for which Theorem 4.5 willbe applicable, as well as strengthening this result to the undecidability of theequational theory for some proper subvarieties of CRL . Since structural rulescorrespond to equations in the signature {∨ , · , } (see [5]), we will restrict ourattention to varieties axiomatized by such equations.Since in residuated lattices multiplication distributes over joins, every equationover {∨ , · , } is equivalent to an equality between finite joins of monoid terms.This equality can in turn be written as two inequalities and in each one of themthe joins on the left-hand side of the inequality yield a conjunction of inequalitiesof the form t ≤ t ∨ · · · ∨ t l , where t , ..., t l ∈ X ∗ are monoid terms. We callequations of this form basic equations (or basic inequalities), if it is furthertrue that the variable sets on the two sides of the inequality are the same. Itcan be easily shown that joinands on the right-hand side containing variablesthat do not appear on the left can be safely omitted, resulting in an equivalentequation. (In the case where all the joinands are of this form, the equation implies1 ≤ x , so it defines the trivial variety). Furthermore, if there are variables thatappear on the left and not on the right, then the inequality implies integrality( x ≤ {∨ , · , } -equations have the FEP, hencedecidable universal (and quasiequational) theory. Therefore the restriction ofthe variables appearing on both sides does not leave out any unknown cases of(un)decidability. Via a process of linearization [5] any basic equation is furtherequivalent to one where the term t is linear , namely to an equation of the form x · · · x n ≤ W ki =1 m i , where k ≥ m , ..., m k ∈ X ∗ and x , . . . , x n are distinctelements of X . (For example the equation x ≤ x is equivalent to the linearizedequation x x ≤ x ∨ x ). Such equations are called simple in [5], but in this OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE simple equations of RL for the subclass where thevariable sets on the left and the right-hand sides of the equation are equal.When writing simple equations, we will be using the set of variables { x i : i ∈ Z + } , and we will assume implicitly that this set is ordered by the naturalorder of the indices. We will informally use variables like x, y, z in some of theexamples, as well. We also define x n := ( x , . . . , x n ), for all n ∈ Z + and for atuple a ∈ N n of natural numbers, we define x n a = x a (1)1 · · · x a ( n ) n ; we also define x n1 = x · · · x n . For reasons that will be clear later, we will actually think of a as a column vector (as opposed to a row vector).For a simple equation ε , we will be interested in its commutative version ε C ,obtained from ε by rearranging the variables within each monoid term accordingto the natural ordering of their indices and removing any resulting duplicatejoinands on the right-hand side. In particular, CRL + ε | = ε C , and we callequations of the form ε C simple equations of CRL . As the encodings for theundecidability are harder for commutative varieties than for arbitrary ones, byproving the results in the commutative case we obtain as corollaries results forgeneral subvarieties of RL . Therefore, we restrict ourselves to simple equationsof CRL and we will refer to them simply as simple equations . Such equations areof the form [D] : x n1 ≤ _ d ∈ D x n d , where D is a finite nonempty set of n -column vectors with entries in N , such thatthe variable sets on the two sides are the same, namely there is no row such thatall column vectors in D are zero on that row. Note that because of the equalityof the variable sets on the left and on the right and due to the idempotency ofjoin in residuated lattices, every simple equation is fully determined by the setof joinands on its right-hand side. Our notation is chosen so that if D is a set of n -columns, then [D] denotes the simple equation displayed above (the exponentsof the joinands on the right-hand side come from D). [D] Di . x ≤ x { } ii . x ≤ ∨ x { , } iii . x ≤ x ∨ x { , } iv . xy ≤ ∨ x y ∨ x y (cid:26) (cid:27) [D] Dv . xyz ≤ x y ∨ y z ∨ xz vi . xyz ≤ yz ∨ xz Table 1.
Some simple equations viewed as sets of column vectors.For the bigger class of basic equations of
CRL (which may not be linearized),if D is again a nonempty set of column vectors and f a column vector over thepositive integers, we denote by [ f, D], the basic equation[ f, D] : x n f ≤ _ d ∈ D x n d . NIKOLAOS GALATOS AND GAVIN ST. JOHN
The following theorem provides a link between simple equations that hold in W + and conditions that hold in W .Let ( W, · ,
1) be a commutative monoid and [D] a n -variable simple equationgiven by D = { d j : 1 ≤ j ≤ m } . If W = ( W, W ′ , N , · ,
1) is a residuatedframe then we write W | = (D) iff for all u n ∈ W n and v ∈ W ′ , the followingimplication is satisfied (the premises above the line are understood conjunctivelyand the vertical line denotes the implication to the conclusion below): u n d N v · · · u n d m N v u n1 N v (D)For example if [D] is x x ≤ x ∨ x , then (D) is: ∀ x , x ∈ W, v ∈ W ′ , x N v & x N v = ⇒ x x N v. Theorem . Let [D] be a simple equation and suppose W is a residu-ated frame. Then W + | = [D] iff W | = (D) . CRL . Recall from Example 3.3, thatthe computations of the 1-ACM M even leading to a final state are faithfully repre-sented by the inequality relation of CRL , in the sense that
CRL | = (& P ⇒ u ≤ q f )iff u ∈ Acc( M ). If we consider the inequality relation in CRL s , where s is the simpleequation x ≤ x ∨ x , we observe that for the computation relation of a machineto be faithfully represented by the associated inequality relation it must furtheradmit the “ambient instruction” given by t ≤ s t ∨ t , for all t ∈ ( Q even ∪ R ) ∗ in addition to being closed under the inference rules [ · ]and [ ∨ ]. Let ≤ s M even be the smallest compatible preorder generated by P even ∪ ≤ s ,and define Acc( s M even ) to be the set of accepted ID’s under the relation ≤ s M even .It is clear that Acc( M even ) ⊆ Acc( s M even ) since ≤ M even ⊆ ≤ s M even , and since thereare no instructions (nor instances of ≤ s ) that remove state variables we obtainAcc( s M even ) ⊆ ID ( M even ). However, while q r Acc( M even ), we have q r ∈ Acc( s M even ) since q r ≤ s q r ∨ q r ∈ Acc( M even ) . It is clear that the expansion of the machine by the ambient instruction (neededfor representing the inequality relation in
CRL s ) does not have the same computa-tion relation, or put differently the machine M even is not suitable for representingthe inequality relation in CRL s because these ambient instructions are not already admissible in it.Likewise, there is no guarantee that there is a machine that has an undecidableacceptance problem (for example the machine ˜ M ) and in which these ambientinstructions are available/admissible. For that reason we cannot use the sameargumentation to show that CRL s has undecidable word problem.Exactly the same issue occurs if the simple equation is contraction c : x ≤ x .Actually, for the case of contraction not only does this particular encoding failto be faithful, but there is no faithful encoding of an undecidable machine: theword problem for CRL c is actually decidable [17]. However, we will show thateven though for the equation s above the current encoding is problematic (asis with contraction), surprisingly, unlike with contraction, there is a different OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE s ; this will allow us to prove that the word problemfor CRL s is undecidable. We present the idea of this new encoding by showingthat it at least faithfully encodes the machine M even . As we will see, what makesit work is that the new encoding is such that, even if they were available, theambient instructions would not contribute to any more accepted configurations;this is a rephrasing of what we referred to as: the given equation is admissible in the particular machine.The idea is to construct a new machine M K , for an appropriate integer K , as amodification of M even that works at an exponential scale (with base K ) comparedto that of M even . In particular, M K manages to replace the decrement instructions p : q r ≤ q and p : q r ≤ q by programs (sets of instructions) P and P ,respectively, that divide the contents of register r by the fixed constant K . Forexample, the general effect for p being q r m ≤ p q r n iff m = n + 1 is mirroredby P in the sense that q r M ≤ P q r N iff M = K · N , and consequently q r K n +1 ≤ P q r K n ; therefore computations in M even are simulated in M K bystoring the contents of r by K n instead of n . In this case, we will say a termis accepted if it computes a join of configurations of the form q f r K (i.e. q f r ),so q r n ∈ Acc( M K ) iff n = K m for some m ≥
0. Thereupon an additionalnecessary condition for acceptance in M K is demanded for configurations labeledby a state q ∈ Q even (independently from the conditions of acceptance in M even )namely that if q r N is accepted in M K then N must be a power of K . For theequation s , if we choose K > s is applied q r n r m ≤ s q r n ( r m ∨ r m ) = q r n +2 m ∨ q r n +4 m , the only way n + 2 m and n + 4 m are both powers of K is if m = 0. In suchan instance, the configuration on the left-hand side of the equation appears as ajoinand on the right-hand side. Consequently we see that, with respect to beingaccepted, instances of ≤ s in a computation are superfluous, and we obtain q r n r m ∨ q r n r m ∈ Acc( M K ) = ⇒ q r n r m ∈ Acc( M K ) , thus Acc( M K ) = Acc( s M K ). So, the equation s is admissible in the machine M K .The reason why this works is that the effect of the inequality s , even whenapplied repeatedly, is to modify the register values in a linear or polynomial way,but when these values are encoded on an exponential scale the applications ofthe inequality do not produce modifications on the same scale and thus do notlead to final configurations.More generally, consider an n -variable simple equation [D] : x n1 ≤ W d ∈ D x n d .For [D] to be admissible, and viewing [D] as an ambient instruction, we need toconsider all the substitution instances t n1 ≤ W d ∈ D t n d of [D], where the tupleof terms t n = ( t , . . . , t n ) is given by a substitution σ : x i t i , for all i . Thenfor any term s , s t n1 ≤ D s W d ∈ D t n d is part of the computation that includes the This definition of acceptance for the machine M K is for heuristic convenience. In Section 7,to properly define programs to multiply/divide by K , we will need to add new states andinstructions to carry out such computations, as well a fresh variable q F , acting as a new finalstate, and a set of instructions that guarantee q f r ≤ M K q F . Indeed, if n + 2 m = K a and n + 4 m = K a + b , for some a ≥ b ≥
1, then K a ≥ m = K a + b − K a ≥ K a ( K − , and hence K ≤ NIKOLAOS GALATOS AND GAVIN ST. JOHN ambient instructions coming from [D]. It is shown in Lemma 6.5 that if [D] doesnot have instances equivalent to a k -mingle equation and s W d ∈ D t n d is acceptedin an ACM M , then the term s contains precisely one state variable and no term t i contains any state variable. In the case where M is a 1-ACM, for example M = M even , this implies that s = q r C , for some state q and number C , and σ is a (1-variable) substitution with σ ( x i ) = t i = r σ ( i )1 , where σ is an n -tuple ofnatural numbers. Using the equality relation for A M , this is equivalently writtenas q r C + σ ≤ D _ d ∈ D q r C + σd , (3)where σd = σ (1) d (1) + · · · σ ( n ) d ( n ) and σ = σ (1) + · · · σ ( n ).Admissibility of [D] in such a 1-ACM M is the demand that if the right-hand sideof the above inequality is accepted in M then the left-hand side is also accepted(thus making every instance of [D] superfluous). The most naive and obviousway to ensure this is to ask that the left-hand side already appears as one ofthe joinands on the right-hand side; that is σ = σ ¯ d for some ¯ d ∈ D, hencerendering the substitution instance of [D] by σ trivial. Recall that in the case ofthe machine M K constructed from M even , if a configuration q r N is accepted in M K then N must be some power of K . So, for [D] to be admissible in M K the mostobvious condition to require is:If the exponents in the right-hand side of [D] produced by a 1-variablesubstitution are translated powers of K (by the same constant), then thesubstitution instance is trivial.In symbolic terms this can be written asIf for some σ ∈ N n and C ∈ N , every C + σd is a power of K , where d ∈ D,then there exists ¯ d ∈ D such that σ ¯ d = σ ( ⋆K )in which case we say that [D] satisfies ( ⋆K ). We also consider the condition ( ⋆ ):there exists K > ⋆K ) holds.In the following sections we will make rigorous the notion of admissibility andcarefully construct the machines M K . We will now define a class of simple equations,for which their defining subvarieties will have an undecidable word problem.The class is so vast that it is easier to define its complement. We motivate thedefinition with the following observation.Consider the machine M even from Example 3.3, and the simple equation d : x ≤ ∨ x . As before, it is easy to see that q r ∈ Acc( d M even ) \ Acc( M even ).However, this behavior cannot be remedied by M K for any K >
1. E.g., let Note that if there is an instance of [D] that is equivalent to k -mingle ( x k ≤ x ) for some k >
1, [D] cannot be admissible for any ACM M : from q kf ≤ D q f we would obtain that q kf isaccepted, a contradiction. Actually, then the variety of CRL +[D] has a decidable word problem.More generally, we note that k -mingle, as well as contraction, are examples of knotted equations :equations of the form x k ≤ x l , where k = l . It is known [17] that all knotted subvarieties of CRL have decidable universal theories, and therefore so do the subvarieties of
CRL axiomatizedby any set of simple equations Γ for which
CRL + Γ | = x k ≤ x l by [5]. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE n = ( K − K ) /
2, then q r K + n Acc( M K ) since K + n = K m for any m ∈ N .However, q r K + n = q r K r n ≤ d q r K r ∨ q r K r n = q r K ∨ q r K ∈ Acc( M K ) . By setting C = K and σ = ( n ) ∈ N , this also demonstrates that ( ⋆K ) is notsatisfied by d for any K >
1, i.e., d fails ( ⋆ ). In fact, we will show in Lemma 8.1that this failure occurs not just for functions of the form n K n but actuallyfor any function on N with infinite range (in particular those that are actuallycomputable).The equation d is an example of a spinal equation and as explained above,unfortunately it cannot be handled by our work. In general, a basic equation[ f, V] is called spinal if it is of the form:[ f, V] : x f (1)1 · · · x f ( k ) k | {z } f ≤ (1 ∨ ) | {z } v x v (1)1 | {z } v ∨ x v (1)1 x v (2)2 | {z } v ∨ · · · ∨ x v k (1)1 · · · x v k ( k ) k | {z } v k , where f V, v j ( j ) = 0 and v i ( j ) = 0 for each 0 ≤ i < j ≤ k , and (1 ∨ ) is meantto signify that 1 may or may not be included in the join. Note that if the columnvectors of the set V = { ( v , ) v , . . . , v k } are listed in the above order, V becomesan upper-right triangular matrix whose diagonal contains only positive entries.In Corollary 8.2 we establish that spinal equations falsify the condition ( ⋆K ) forevery K > Definition . We say that a basic equation [ f, V] is spinal if f V, v i ( i ) = 0 and v i ( j ) = 0, for all j > i , for all v i in V that are not constantlyzero. In this case, we will refer to the set V as a spine . We say that a simpleequation is prespinal if it has a spinal equation as the image under some monoidalsubstitution.From now on we will only consider monoidal substitutions (the image of everyvariable is a monoidal term, i.e. a product of variables) and we will refer to themsimply as substitutions.Note that the only one-variable spinal equations are the knotted inequalities x n ≤ x m (for which we know that they define subvarieties of CRL with decidablequasiequational theory) and their variants x n ≤ ∨ x m , where n = m , for whichdecidability results are still open. Also, their equivalent simple equations areprespinal, as verified below in Lemma 5.4. As a consequence of the definition, asimple equation [D] is prespinal if and only if [D ∪ { } ] is prespinal.From Table 1, we see that (i)-(ii) are spinal. The simple equation (vi) isprespinal via the 1-variable substitution σ given by σ ( x ) = x , σ ( y ) = x and σ ( z ) = 1. i.e., CRL + (vi) | = x ≤ x . On the other hand, no trivial equationsare prespinal. The general characterization of whether a simple equation isprespinal will be addressed in §
8, where it is verified that (iii)-(v) in Table 1 arenot prespinal right after Theorem 8.6.
Definition . A simple equation is called spineless if it is not prespinal. Asimple equation ε for RL is called spineless if ε C is spineless. The only exception being the case where n > m = 1, where equations of this form havethe finite model property by Theorem 3.15 in [5]. NIKOLAOS GALATOS AND GAVIN ST. JOHN
To demonstrate the vastness of the collection of spineless equations, we willfocus our attention only on 1-variable basic equations below. Note that everyone-variable basic equation has the form x n ≤ _ p ∈ P x p for some n > P of N such that P = { } ; we denotesuch an equation by [ n, P ]. Also, note that [ n, P ] is trivial iff n ∈ P . Lemma . Let [ n, P ] be a -variable basic equation. Then the linearizationof [ n, P ] is a spineless simple equation iff [ n, P ] is trivial or P contains at leasttwo distinct positive integers. Proof.
The simple equation resulting from the linearization over
CRL of[ n, P ] is [D] : x n1 ≤ _ ( x n d : d ∈ N n , n X i =1 d ( i ) ∈ P ) . (4)and is equivalent over CRL to [ n, P ]. We prove the contraposition for each direction. For the forward direction, if P = { p } , where 0 < p = n , then [ n, P ] is spinal by definition and hence [D] isprespinal by its obvious substitution to [ n, P ]: x i x , for all i .For the reverse direction, suppose that [ n, P ] is nontrivial and P containsdistinct positive numbers p > q . Then for each i ≤ n , the terms x pi and x qi appearas joinands on the right-hand side of [D]. Let σ be a monoidal substitution thatis non-trivializing for [D]. Then for some i ≤ n , σ : x i w for some monoidterm w = 1. So both w p and w q appear as joinands on the right-hand side of σ [D]. Since p > q > w p = 1 = w q and CRL = w p = w q , so σ [D] is not spinal,as w p and w q contain the same variables. Since σ was arbitrary, it follows that[ n, P ] is spineless. ⊣ In the following sections we undertake a deep analysis of spineless equations,culminating in Corollary 8.14. To highlight this result, we (re)state it here andwe use it right afterwards, in Section 5.3, to obtain results (Theorem 5.9) aboutthe equational theory.
Theorem . Let Γ be a finite set of spineless simple equations, then anyvariety between CRL + Γ and RL has an undecidable word problem. CRL . In this section we ex-ploit the fact that in certain varieties certain quasiequations are equivalent toequations to show that even their equational theory is undecidable, making useof Theorem 5.5.The negative cone of a residuated lattice A is the set A − = { a ∈ A : a ≤ } .We will say that a subvariety V of CRL is negatively n -potent if the negative coneof each algebra in V is n -potent, i.e., V | = ( x ∧ n = ( x ∧ n +1 (or equivalently, V | = ( x ∧ n ≤ ( x ∧ n +1 ). By setting x := x ∨ · · · x n in [ n, P ], we obtain W { x n a : P ni =1 a ( i ) = n } ≤ W (cid:8) x n d : d ∈ N n , P ni =1 d ( i ) ∈ P (cid:9) . Since x n1 ≤ W { x n a : P ni =1 a ( i ) = n } , we obtain [D].Conversely, by setting x i := x , for all i ≤ n , in [D], we obtain [ n, P ]. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE t be a term and S be a finite set of terms in the language of CRL . It canbe easily verified that ( ∃ m ∈ N )( ∃ s , ..., s m ∈ S ) CRL | = Q mi =1 (1 ∧ s i ) ≤ t if and only if ( ∃ k ∈ N ) CRL | = (1 ∧ V S ) k ≤ t. (5)If V ⊆
CRL is a negatively n -potent variety, then we obtain ( ∃ m ∈ N )( ∃ s , ..., s m ∈ S ) V | = m Y i =1 (1 ∧ s i ) ≤ t ⇐⇒ V | = (1 ∧ ^ S ) n ≤ t. (6)We consider the following quasiequation and equation: ξ S ( t ) : & s ∈ S ≤ s = ⇒ ≤ t ε nS ( t ) : (1 ∧ ^ S ) n ≤ t. In this way we establish the fact that satisfaction of a quasiequation in anegatively n -potent subvariety of CRL is equivalent to the satisfaction of a cor-responding equation.
Lemma . If V is a negatively n -potent subvariety of CRL and S ∪ { t } afinite set of terms in the language of V , then V | = ξ S ( t ) ⇐⇒ V | = ε nS ( t ) . Proof.
Let F V be the free algebra for V , and define the congruence C :=Cg( { (1 ∧ s, s ) : s ∈ S } ). We denote the quotient algebra by F V /C . For a subset X of F −V , we denote by M ( X ) the convex normal submonoid of F −V generatedby X . Observe that
V | = & s ∈ S ≤ s ⇒ ≤ t ⇐⇒ in F V /C , [1 ∧ t ] C = [1] C ⇐⇒ in F V , (1 ∧ t ) ∈ M ( { ∧ s : s ∈ S } ) [6] ⇐⇒ in F V , ( ∃ m ∈ N )( ∃ s , ..., s m ∈ S ) Q mi =1 (1 ∧ s i ) ≤ t [6] ⇐⇒ ( ∃ m ∈ N )( ∃ s , ..., s m ∈ S ) V | = Q mi =1 (1 ∧ s i ) ≤ t ⇐⇒ V | = (1 ∧ V S ) n ≤ t Eq. (6) . ⊣ For an inequality p : s ≤ t , define the term p → := s → t . Let M = ( R k , Q , P , q f )be an ACM. Define P → := { p → : p ∈ P } . Then for u ∈ A M , the quasiequationacc M ( u ) is equivalent to ξ P → ( u → q f ). By Lemma 5.6 and Theorem 4.5, we obtainthe following: Theorem . Let V be a subvariety of CRL containing W + M , for some k -ACM M and satisfying ( x ∧ n ≤ ( x ∧ n +1 for some n ≥ . Then deciding membershipin the equational theory of V is at least as hard as deciding membership in Acc( M ) . The forward direction is trivial, taking k = m , since V S ≤ s , for all s ∈ S . The reversedirection holds by setting m = k · | S | , and observing that Q s ∈ S ( s ∧ ≤ ∧ V S . The reverse direction follows from (5), while the forward direction uses the fact that(1 ∧ x ) n ≤ (1 ∧ x ) k , if k ≤ n , and (1 ∧ x ) n = (1 ∧ x ) k , if k > n , by the negative n -potency of V . See Theorem 3.47 in [6]. NIKOLAOS GALATOS AND GAVIN ST. JOHN
We say a simple equation ε is expansive if it has, as a substitution instance,an equation of the form x n ≤ m _ j =1 x n + c j , (7)for some n, m ≥ c , ..., c n ≥
1. It is easy to verify that if ε is expansivethen CRL + ε is negatively n -potent. We say a variety is expansive if it satisfiesan expansive equation. As a consequence of Lemma 5.4, if a simple equation isthe equivalent linearization of an expansive basic equation where m ≥ Corollary . Let V be an expansive subvariety of CRL containing W + M ,for some ACM M . Then deciding membership in the equational theory of V is atleast as hard as deciding membership in Acc( M ) . In particular, we prove the following theorem as a consequence of Theorem 5.5and the corollary above.
Theorem . If Γ is a finite set spineless simple equations containing anexpansive equation then variety CRL + Γ has an undecidable equational theory. §
6. Admissibility.
We now begin investigating the required features that amachine should have, in order to achieve the exponential encoding. We begin byformalizing the notion of admissibility and its two natural parts.Let M = ( R k , Q , P , q f ) be a k -ACM and [D] a n -variable simple equation. Wedefine the relation ≤ D to be the smallest relation containing t n1 ≤ _ d ∈ D t n d , for all t n ∈ (( Q ∪ R k ) ∗ ) n , and closed under the inference rules [ · ] and [ ∨ ]. We definethe computation relation ≤ D M as the smallest compatible preorder generated by P ∪ ≤ D , and set Acc(D M ) := { u ∈ A M : ∃ u f ∈ Fin ( M ) , u ≤ D M u f } . The construction of ≤ D M enjoys an analogue to Lemma 3.1(2), and thereforethe following analogue to Lemma 3.4(2): Lemma . Let M be an ACM and [D] be a simple equation. For all u, v ∈ A M , u ∨ v ∈ Acc(D M ) if and only if u ∈ Acc(D M ) and v ∈ Acc(D M ) . The frame W D M is defined as W M , but the nuclear relation is defined withrespect to Acc(D M ) instead of Acc( M ). Lemma . If M is an ACM and [D] a simple equation, then W +D M ∈ CRL +[D] . Proof.
Let [D] be an n -variable simple equation where D = { d , . . . , d m } .It is enough to show that W +D M | = [D]. By Theorem 5.1, this is equivalent toshowing W D M | = (D), i.e., for all s ∈ W , t n ∈ W n , t n d N D M s · · · t n d m N D M s t n1 N D M s (D) . OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE N D M and ≤ D M and by Lemma 6.1 we obtain( ∀ d ∈ D) t n d N D M s ⇐⇒ ( ∀ d ∈ D) s t n d ∈ Acc(D M ) ⇐⇒ _ d ∈ D s t n d ∈ Acc(D M ) . Now, by the definition of ≤ D M , s t n ≤ D M s _ d ∈ D t n d = _ d ∈ D s t n d , hence s t n ∈ Acc(D M ) and t n N D M s . Therefore W +D M | = [D] . ⊣ Since ≤ M ⊆ ≤ D M , it follows that Acc( M ) ⊆ Acc(D M ). We say a simple equation[D] is admissible in a machine M if Acc( M ) = Acc(D M ). As the only differencebetween W M and W D M is Acc( M ) and Acc(D M ), if [D] is admissible in M then W + M = W +D M . Therefore by Lemma 6.2 we obtain the following lemma: Lemma . If a simple equation [D] is admissible in M , then W + M ∈ CRL +[D] . As we will see, admissibility in M depends on the machine M as well as theequation [D]. We define the intermediate notions register-admissibility and state-admissibility to make this distinction clear.For a given ACM M = ( R k , Q , P , q f ) and n -variable simple equation [D], wedefine ≤ D R to be the smallest relation containing, x n1 ≤ _ d ∈ D x n d , for all x n ∈ ( R ∗ k ) n , and closed under the inference rules [ · ] and [ ∨ ]. Define thenew computation relation ≤ D RM to be the smallest compatible preorder generatedby P ∪ ≤ D R , and set Acc(D RM ) := { u ∈ A M : ∃ u f ∈ Fin ( M ) , u ≤ D RM u f } . Since theeffect of [D] is restricted to only register terms in ≤ D R , by the same argument asLemma 3.4(1), it follows that Acc(D RM ) ⊆ ID ( M ).It is clear then that Acc(D RM ) ⊆ Acc(D M ), since ≤ D R is merely a restriction of ≤ D . In total, we obtain Acc( M ) ⊆ Acc(D RM ) ⊆ Acc(D M ) . If Acc( M ) = Acc(D RM ), then we say [D] is register-admissible in M . If Acc(D RM ) =Acc(D M ), we say [D] is state-admissible in M . Hence [D] is admissible in M iff [D]is both register and state-admissible in M . Due to the property that instructionsin an ACM replace a single state-variable of a configuration by precisely onestate-variable, we show in Lemma 6.5 that state-admissibility is a property ofthe equation [D] independent of the machine M . We say that a simpleequation [D] is mingly if there exists a one-variable substitution σ such that σ [D] : x λ ≤ W d ∈ D x for some λ >
1. That is, if [D] is an equation in n -variables, σ ( x n1 ) = x λ and σ ( x n d ) = x , for all d ∈ D.Of course, this is equivalent over RL to x λ ≤ x , but since we do not assumeidempotency of ∨ in A M , we write x λ ≤ W d ∈ D x so as to be explicit about theimplementation of the equation in computations.By definition, mingly equations are prespinal. The equation (vi) from Table 1is mingly, using the substitution witnessing it is prespinal, while the remaining0 NIKOLAOS GALATOS AND GAVIN ST. JOHN equations can easily be verified to be not mingly. Since mingly equations areprespinal by definition, we obtain the following result.
Lemma . A spineless equation is non-mingly.
As we will be dealing only with spineless equations, the equations we willconsider are not mingly. The following lemma shows that only mingly equationsinvalidate state-admissibility and also that state-admissibility is independent ofthe choice of the machine.
Lemma . The following are equivalent for any M and simple equation [D] .
1. [D] is state-admissible in M .
2. Acc(D M ) ⊆ ID ( M ) .
3. [D] is not mingly.
Proof.
Assume M = ( R k , Q , P , q f ) and let [D] be an n -variable simple equation.(1 ⇒
2) We have that Acc(D M ) = Acc(D RM ) ⊆ ID ( M ).(2 ⇒
3) Proceeding by contraposition, suppose [D] is mingly. Then x λ ≤ W d ∈ D x is a direct substitution image of [D], for λ >
1. For x = q f , we have q λf ≤ D _ d ∈ D q f ∈ Fin ( M ) ⊆ Acc(D M ) . Since λ >
1, it follows that q λf ID ( M ) . (3 ⇒
1) Proceeding by contraposition, suppose Acc(D RM ) is a proper subset ofAcc(D M ) and let t ∈ Acc(D M ) \ Acc(D RM ) be a witness with minimal computation t = u ≤ p u ≤ p · · · ≤ p N u N = u f ∈ Fin ( M ) , for some u , u , ..., u N ∈ A M and p , ..., p N ∈ P ∪ { D } . Since Fin ( M ) ⊆ Acc(D RM ),we have that t Fin ( M ) and so N >
1. By Lemma 6.1, we may assume t ∈ ( Q ∪ R k ) ∗ . Since N is minimal, it follows that p = D and u ∈ Acc(D RM ) ⊆ ID ( M ).So, t = s t n1 ≤ D _ d ∈ D s t n d = u . where s ∈ ( Q ∪ R k ) ∗ and t n ∈ (( Q ∪ R k ) ∗ ) n ; here we used the fact that the rule[ ∨ ] is not applicable, as t ∈ ( Q ∪ R k ) ∗ .Since u ∈ ID ( M ), it follows that s t n d ∈ Conf ( M ) for all d ∈ D. Also, because p = D R , there is some t i that contains at least one state variable. As that t i has to also appear on the right-hand side and s t n1 ∈ Conf ( M ), it follows that s cannot contain any state variable, so s ∈ R ∗ k . Consequently, every joinand in theright-hand side has a unique t i containing a state variable.Therefore, applying the substitution σ defined by: σ ( x i ) = x if t i contains astate variable and σ ( x i ) = 1 otherwise, yields x λ ≤ x , where λ is the number of t i ’s containing a state variable.We will show that λ >
1. If, by way of contradiction, there was a unique t j containing a state variable, then it would have the form t j = qx for some Note that the remaining equations are such that each variable appears with degree at least2 on the right-hand side, so any 1-variable non-trivializing substitution instance will result ina joinand of degree at least 2.
OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE q ∈ Q and x ∈ R ∗ k (with t i ∈ R ∗ k for all other t i ’s) and t j would appear on all thejoinands on the right-hand side. So, we have t = sqx Y i = j t i ≤ D R _ d ∈ D sqx n Y i = j t id ( i ) = u ∈ Acc(D RM ) , and thus t ∈ Acc(D RM ) , a contradiction. ⊣ Therefore, in our search for an appropriate machine for spineless equations,state-admissibility will be automatic and will not restrict the type of possiblemachines. §
7. The exponential encoding.
Given a 2-ACM M = ( R , Q , P , q f ) (we willlater choose as M a machine with undecidable halting problem) and simple equa-tion [D], our ultimate goal is to construct a new machine M ′ “simulating” themachine M such that Acc(D M ′ ) = Acc( M ′ ). More specifically, for any spinelessequation [D] we can construct a 3-ACM M K = ( R , Q K , P K , q F ) for which it willbe register-admissible, for some K > M in the following way: q r n r n ∈ Acc( M ) if and only if q r K n r K n ∈ Acc( M K ) . So, the content n of a register r is not represented by r n but by r K n . Com-putationally, this will be achieved by replacing each increment- r (decrement- r )instruction by a distinct set of instructions (i.e., a program ) that will carry out theprocess of multiplying (dividing) the contents in register r by K . The auxiliaryregister r will be necessary to carry out such computations, and the faithfulnessof this encoding will be guaranteed by the insistence that accepted configurationsare those which have a computation resulting in a finite join of configurationslabeled by state q F where all the registers are empty.We will first show how we can achieve multiplying the contents of a register r ∈ R by a fixed constant K >
1. Let Q + K = { a , ..., a K } be a set of ( K + 1)-many fresh state-variables. We can add K tokens to the contents of the auxiliaryregister r by starting from the state a and applying the instructions P + K = { + i : i = 1 , ..., K } , where + i : a i − ≤ a i r , and reaching state a K . Then foreach 0 ≤ i < K , we have a ≤ + K a i r N ⇐⇒ N = i, where ≤ + K is the computation relation defined in the usual way from P + K .Hence a x ≤ + K a K y iff y = x r K , for each x, y ∈ R ∗ .Now, if we have N tokens in register r and we are at state a K , then by removingone r -token, moving to state a by the instruction × loop : a K r ≤ a , and using P + K to add K r -tokens and repeating this process, we can essentially exchange N r -tokens for N K r -tokens. E.g., a K r N ≤ × loop a r N − ≤ + K a K r N − r K ≤ × loop a r N − r K ≤ + K · · · a K r NK . If we set ≤ × r to be the computation relation defined from P r × K = P + K ∪{× loop } ,then it is easily verified by induction that for each N, M, n, m ∈ N and 0 ≤ i ≤ K :2 NIKOLAOS GALATOS AND GAVIN ST. JOHN a K r N ≤ × r a i r n r m ⇐⇒ KN = Kn + m + ( K − i ) a i r n r m ≤ × r a K r M ⇐⇒ M = Kn + m + ( K − i )(8)Observe that multiplying the contents of register r is achieved by an iterativeprocess of adding K -many tokens to r and then looping the process by removingone from r . We can define a division program analogously. However, in bothcases, we start with tokens in r and compute the product (or quotient) by K inthe register r by emptying r . We would then like our machine to transfer thosecontents back to the original register r to complete this program.Let T r be the program with fresh states Q T r = { t , t } and instructions P T r = { T − , T + } given by T − : t r ≤ t and T + : t ≤ t r . Defining its computationrelation to be ≤ T r , we easily obtain for all N, M, n, m ∈ N : t r N ≤ T r t δ r n r m ⇐⇒ N = n + m + δt δ r n r m ≤ T r t r M ⇐⇒ n + m + δ = M (9)If we wish to implement the transfer program after, say, executing the programfor multiplying by K , then we need an instruction to switch from the state a to the state t . This can naively be achieved by a forking instruction, say p : a K ≤ t , which would allow for the computation a K r N ≤ × r a K r NK ≤ p t r NK ≤ T r t r NK . However, since the instruction p can be applied to any configuration labeledby a K , even those for which the register r is nonempty, we also get unwantedinstances of the form a K r N ≤ t r M where M = Km + N − m for each 0 ≤ m ≤ N .Since we want our simulation to be faithful, we need a way of switching to thetransfer program only when the register r is empty. What we are asking for is similar to (whatis commonly called) a zero-test instruction of a standard counter machine, i.e.,an instruction which is applicable only when a specified register is empty. Sincewe require our computation relations to be compatible with multiplication, suchan insistence is impossible. I.e., we insist q ≤ q ′ to entail qx ≤ q ′ x for any term x . However, following the ideas in [11], we construct a program that has a simi-lar behavior utilizing the insistence that accepted configurations are those thatcompute final ID’s, i.e., finite joins of the configuration labeled by a final state q F with all registers empty.We define the (sub-)machine ø = ( R , Q ø , P ø , q F ), with a fresh set of variables Q ø = { z , z , z , q F } and instructions P ø are given by:ø ij : z i r j ≤ z i ø iF : z i ≤ q F ∨ q F , for each i, j ∈ { , , } with i = j , resulting in a total of 6 + 3 = 9 instructions.The addition of the auxiliary states z , z , z is explained by the fact that therole of z i is to empty the contents of all registers other than r i and transitionto the final state q F . Thus it detects situations where r i is not already empty; Technically, p : a ≤ t ∨ t since ∨ is not idempotent, but this technicality is unnecessaryfor the example. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE q in to q out only when the register r i is empty, we can start a parallel computation,the main branch of which moves to state q out (even if r i is non-empty) but theauxiliary branch involving the z i terminates successfully only if r i is empty. This z i -branch ensures/safeguards that the combined computation acts as intended.We call the above machine the zero-test program , and we denote its compu-tation relation by ≤ ø . The zero-test program for a register r i is implementedby a zero-test r i instruction p , where p is of the form q in ≤ q out ∨ z i . Since thedesired final ID’s of M K will consist of only joins of the configuration q F , i.e.all registers are empty, the above instruction copies the contents of the registersand creates two paths; one path with the state q out where r i is intended to beempty, and the second with a state z i where the program ø is intended to emptyregisters r j and r k and then output to the final state. Below is an example ofimplementing the zero-test on register r via the instruction p : q in ≤ q out ∨ z on the configuration q in r r r : q in r r r ≤ p q out r r r ∨ z r r r ≤ ø q out r r r ∨ z r r ≤ ø q out r r r ∨ z r ≤ ø F q out r r r ∨ q F r ∨ q F r . As we see, the above (maximal in ø) computation detected that register r is not empty in the configuration q r r r since the final ID contains the con-figuration q F r , and there are no q F -instructions. In fact, z r r r Acc(ø)since there is no instruction applicable to the state z which alters the contentsof register r . By a similar analysis, we obtain the following, Lemma . z i r n r n r n ∈ Acc(ø) if and only if n i = 0 . Let P be a program (i.e. a sub-machine) and ≤ P be its corresponding compu-tation relation. We define the relation ⊑ P on Conf ( P ) via C ⊑ P C ′ iff C ≤ P C ′ ∨ u ,where either u = ⊥ or u ∈ ID(ø) with u ∈ Acc(ø). If P contains no Q ø -instructions (i.e., no instruction z i x ≤ · · · ), then C ⊑ P C ′ iff there is a com-putation from C to C ′ ∨ u with instructions from P such that every zero-testwas properly applied. Note that ⊑ P is transitive on configurations and C ≤ P C ′ implies C ⊑ P C ′ . We obtain the following lemma. Lemma . Let p be the instruction q in ≤ q out ∨ z i with distinct q in , q out Q ø .For x, x ′ ∈ R ∗ , q in x ⊑ { p } q out x ′ if and only if x = x ′ = r n r n r n and n i = 0 . Proof.
Let x = r n r n r n . The only instruction applicable to q in x is p , sofrom q in ≤ q out ∨ z i we obtain q in x ≤ p q out x ∨ z i x . Since the only instructionsapplicable are those from { p } and q in = q out , the computation cannot proceedfrom this configuration. Hence by Lemma 7.1, q in x ⊑ { p } q out x ′ ⇐⇒ x = x ′ and z i x ≤ ø q F ⇐⇒ x = x ′ and n i = 0 . ⊣ If p is an instruction, by C ⊑ { p } C ′ we mean C ≤ p C ′ ∨ u. NIKOLAOS GALATOS AND GAVIN ST. JOHN
We are now ready to faithfully simulatean increment- r instruction by a program that multiplies the contents of register r ∈ { r , r } by the fixed constant K . For p : q in ≤ q out r , we define the program × ( p ) to have states Q × ( p ) = Q + K ∪ Q T r and instructions P × ( p ) = P + K ∪ P T r ∪{× in , × T , × out } , where × in : q in ≤ a K ∨ z × T : a K ≤ t ∨ z i × out : t ≤ q out ∨ z . The instruction × in is intended to verify that the auxiliary register r is in factempty, and initiate the process of storing in r K -times the contents in theactive register r . The instruction × T is meant to check that all the contents ofthe active register r have been emptied (and thus K -times that amount is in r ),and initiate the transfer program. The instruction × out is intended to end theprogram by transitioning to the state q out only when the transfer is complete,i.e., when r has been emptied. Below is an example of × ( q in ≤ q out r ) runningon the configuration q in r r : q in r r ⊑ {× in } a K r r ≤ × r a K r r K ⊑ {× T } t r r K ≤ T r t r K r ⊑ {× out } q out r K r In this way, we obtain the following technical lemma by induction as a conse-quence of Lemma 7.2 and Equations 8 and 9. We state the lemma for r = r ,but the same holds when swapping the roles of r and r . Lemma . Let p : q in ≤ q out r be an increment- r instruction, where q in , q out Q × ( p ) . Then q in r N r N r N ⊑ × ( p ) q out r M r M r M if and only if M = KN , M = N , and M = N = 0 . In fact, for each n , n , n ∈ N and state q ∈ Q × ( p ) , q in r N r N r N ⊑ × ( p ) q r n r n r n iff N = 0 , N = n , and KN = (cid:26) n + n + δ if q = t δ where δ ∈ { , } ,Kn + n + ( K − δ ) if q = a δ where ≤ δ ≤ K . q r n r n r n ⊑ × ( p ) q out r M r M r M iff M = 0 , M = n , and M = (cid:26) n + n + δ if q = t δ where δ ∈ { , } ,Kn + n + ( K − δ ) if q = a δ where ≤ δ ≤ K .
For a configuration C = q r n r m in M , by C K we denote the configuration q r K n r K m in M K . Corollary . Let p be an increment instruction from some -ACM M .Then C ≤ p C ′ if and only if C K ⊑ × ( p ) C ′ K for any configurations C , C ′ in Conf ( M ) . OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE r instruction p : q in r ≤ q out ,we define the division by K program ÷ ( p ) as follows. For its set of states Q ÷ ( p ) ,we define a fresh set of states Q r − K = { s , ..., s K } and set Q ÷ ( p ) = Q r − K ∪ Q T r .We take as its instructions P ÷ ( p ) = P r − K ∪ P T r ∪ {÷ in , ÷ T , ÷ out } , where P r − K contains the instruction ÷ loop : s K ≤ s r and K -many instructions of the form − i : s i − r ≤ s i , for 1 ≤ i ≤ K , and finally ÷ in : q in ≤ s ∨ z ÷ T : s ≤ t ∨ z i ÷ out : t ≤ q out ∨ z . The instruction ÷ in is intended to verify that the auxiliary register r is infact empty, and initiate the process of storing in r the quotient by K of thecontents in the active register r . The instruction × T is meant to check that allthe contents of the active register r have been emptied (and thus K -divided bythat amount is in r ), and initiate the transfer program. The instruction ÷ out is intended to end the program transitioning to the state q out only when thetransfer is complete, i.e., when r has been emptied. Below is an example of ÷ ( q in r ≤ q out ) running on the configuration q in r K r : q in r K r ⊑ {÷ in } s r K r ≤ r − K s K r K r ≤ ÷ loop s r K r r ≤ r − K s K r r ≤ ÷ loop s r r ⊑ {÷ T } t r r ≤ T r t r r . ⊑ {÷ out } q out r r The following technical lemma is easily verified by induction. We state the lemmafor r = r , but the same holds by swapping the roles of r and r . Lemma . Let p : q in r ≤ q out be a decrement- r instruction. Then q in r N r N r N ⊑ ÷ ( p ) q out r M r M r M if and only if KM = N > , M = N , and M = N = 0 . In fact, for each n , n , n ∈ N and state q ∈ Q ÷ ( p ) , q in r N r N r N ⊑ ÷ ( p ) q r n r n r n iff N = 0 , N = n , and N = (cid:26) n + n + δ if q = t δ where δ ∈ { , } ,Kn + n + ( K − δ ) if q = s δ where ≤ δ ≤ K . q r n r n r n ⊑ × ( p ) q out r M r M r M iff M = 0 , M = n , and KM = (cid:26) n + n + δ if q = t δ where δ ∈ { , } ,Kn + n + ( K − δ ) if q = s δ where ≤ δ ≤ K .
Corollary . Let p be a decrement instruction from some -ACM M and C , C ′ be configurations in M . Then C ≤ p C ′ if and only if C K ⊑ ÷ ( p ) C ′ K . Corollary . Assume that p : q in r ≤ q out is a decrement instruction, C in is a q in -configuration, C is a Q ÷ ( p ) -configuration and C out is q out -configuration.If C in ⊑ ÷ ( p ) C and C in ⊑ ÷ ( p ) C out , then C ⊑ ÷ ( p ) C out . NIKOLAOS GALATOS AND GAVIN ST. JOHN
Proof.
Since C in ⊑ ÷ ( p ) C and C in ⊑ ÷ ( p ) C out , by Lemma 7.5 the values of C in and C , as well as the values of C in and C out are linked. Therefore, the values of C out and C are also linked and hence by Lemma 7.5 we obtain C ⊑ ÷ ( p ) C out . ⊣ M K . Let M = ( R , Q , P , q f ) be a 2-ACM and let K > q f is accepted in M by definition, we willneed ( q f ) K = q f r r to be accepted in M K . To accommodate this, we definethe end program as follows. For a fresh variable c F , we define the set of states Q F = { c F } and the set of instructions P F = { F , F } by: F : q f r ≤ c F F : c F r ≤ q F . By ≤ F we denote the computation relation for the end program. Lemma . For q ∈ Q F ∪ { q f } , q r n r n r n ≤ F q F if and only if n = 0 and ( n , n ) = (1 , if q = q f (0 , if q = c F (0 , if q = q F . We write the instructions P of M as the disjoint union P + ∪ P − ∪ P ∨ of itsincrement, decrement, and forking instructions, respectively. We can now for-mally define the 3-ACM simulation of M to be the machine M K = ( R , Q K , P K , q F ),where • Q K is the (disjoint) union of Q , Q ø , Q F , Q × ( p ) for each p ∈ P + , and Q ÷ ( p ) foreach p ∈ P − . • P K is the (disjoint) union of P ∨ , P ø , P F , P × ( p ) for each p ∈ P + , and P ÷ ( p ) for each p ∈ P − . • q F is the final state of M K .Formally, we view all states and instructions in some multiply/divide program P ( p ) (where p is an increment/decrement instruction from M ) as being labeledby the instruction p , e.g., a state from Q P ( p ) is of the form q p , and an instruction P P ( p ) is of the form ρ p . In other words, we make states and instructions in eachsubprogram disjoint. In fact, since there are no instructions in P ( p ) of the form · · · ≤ q F · · · , we obtain the following useful observation. Lemma . Let p be an increment or decrement instruction from M and P ( p ) its corresponding program in M K . If C is a configuration in M K labeled by astate from Q P ( p ) then the only instructions applicable to C are those from P P ( p ) .Furthermore, if q out is the output state of p , then C being accepted in M K implies C ⊑ P ( p ) C ′ ∈ Acc( M K ) , where C ′ is labeled by q out . Recall, for a configuration C = q r n r m in M , by C K we denote the configuration q r K n r K m in M K . Lemma . The following hold for any 2-ACM M = ( R , Q , P , q f ) and K > . A configuration C is accepted in M iff C K is accepted in M K . Furthermore,any accepted configuration in M K labeled by a state from Q must be of theform C K where C is accepted in M . OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE
Let p be an increment or decrement instruction of M and C a configurationof the corresponding program P ( p ) ( P ∈ {× , ÷} ) . Then C is accepted in M K iff there are accepted configurations C ′ , C ′′ in M such that C ′ ≤ p C ′′ and C ′ K ⊑ P ( p ) C ⊑ P ( p ) C ′′ K . Proof.
For (1), let C be a configuration in M . Since there are no q f -instructionsin M by definition, if C is labeled by state q f then it is accepted in M iff C = q f ,i.e., both registers r and r are empty. By definition, the only q f -instructionsin M K are those found in the end program. By Lemma 7.8, the only acceptedconfiguration in M K labeled by q f is C K . Now, suppose p is a q -instruction from M . Clearly, if p is a forking instruction, then C ≤ p C ′ ∨ C ′′ in M iff C K ≤ p C ′ K ∨ C ′′ K .Otherwise, p is an increment or decrement instruction, and by Corollaries 7.4and 7.6, C ≤ p C ′ in M iff C K ⊑ P ( p ) C ′ K in M K . The claim therefore follows byinduction on the computation lengths.For (2), consider a configuration C in M K labeled by a state from some program P ( p ), where p is an increment or decrement instruction from M . Let q in and q out be the input and output states of p , respectively. By Lemma 7.9, we concludethat if a computation witnesses C being accepted in M K it must implement theoutput instruction of P ( p ). That is C ⊑ P ( p ) q out r n r n r n . By (1), n and n are powers of K while n = 0. By Lemmas 7.3 and 7.5, the result follows. ⊣ Let ˜ M be the 2-ACM given by Theorem 3.2. Since membership of Acc(˜ M ) isundecidable, we obtain the following consequence of Lemma 7.10(1): Corollary . Membership in the set
Acc(˜ M K ) is undecidable for K > . M K . Consider an n -variable simple equa-tion [D], a 2-ACM M , and an integer K >
1. To show the register-admissibilityof [D] in M K , we need only show that for each configuration C in M K , if the ID W d ∈ D C d is obtained by an instance of ≤ D R from C and W d ∈ D C d is accepted in M K , then C is accepted in M K . By Lemma 3.4(2), this implication is equivalentlystated as C ≤ D R _ d ∈ D C d & ( ∀ d ∈ D)( C d ∈ Acc( M K )) = ⇒ C ∈ Acc( M K ) . Since we are only considering applications of [D] to the register contents, wecan split our analysis into cases depending upon the state q ∈ Q K that labels theconfigurations. The following useful observation follows from the fact that everyvariable that appears on the right-hand side of a simple equation appears alsoon the left-hand side. Lemma . If a substitution sends all the joinands of a simple equation to , then it sends all variables of the equation to . In the following for two tuples σ and d of the same length, σd denotes theirdot product. In the next section we will actually view σ as a row-matrix and d as a column-matrix, so σd will be their matrix product. In this way, focusingon the list/column vector d of exponents of the variables in [D] and also onthe list/row vector σ of the exponents of the images of the variables via a one-variable substitution, the above lemma can be stated as: for an n -variable simple8 NIKOLAOS GALATOS AND GAVIN ST. JOHN equation [D], if σd = 0 for each d ∈ D, then σ must be the constantly zero vector ∈ N n .As observed in Section 5.1, if C ≤ D R W d ∈ D C d is an instance of ≤ D R , we maywrite C = qx x n1 and C d = qx x n d for each d ∈ D, where x ∈ R ∗ and x n =( x , . . . , x n ) ∈ ( R ∗ ) n . Let x = r C r C r C , where C , C , C ≥
0, and for each j ∈ { , , } , define σ j ∈ N n via x i = r σ ( i )1 r σ ( i )2 r σ ( i )3 , for each i ∈ { , ..., n } .Then, C = q r C + σ r C + σ r C + σ , and for each d ∈ D, C d = q r C + σ d r C + σ d r C + σ d . Lemma . The zero-test program is register-admissible for any simple equa-tion, and the end program is register-admissible for any non-mingly simple equa-tion.
Proof.
Let [D] be a simple equation. If q is the final state q F , then C d isaccepted iff all registers are empty, i.e. C d = q F for each d ∈ D. Hence x = 1and σ j d = 0 for each d ∈ D and j ∈ { , , } . For each j ∈ { , , } , this impliesthat σ j = , by Lemma 7.12. Therefore C = q F ∈ Acc( M K ).Suppose q = z i , and without loss of generality, let i = 3. By Lemma 7.1, C d is accepted iff register r is empty, i.e., C d ∈ Acc(ø) iff C + σ d = 0, for each d ∈ D. This implies C = 0 and σ d = 0, for each d ∈ D. So by Lemma 7.12, σ = . Hence C + σ = 0 and C ∈ Acc(ø) ⊆ Acc( M K ).Lastly, suppose q = c F . By Lemma 7.8, C d ∈ Acc( F ) iff C d = c F r . Hence C = C = 0, σ d = σ d = 0 for each d ∈ D, and C + σ d = 1. Again, byLemma 7.12, σ = σ = . Let λ = σ . Then λ is positive since [D] is a simpleequation. If λ = 1, then C = c F r and we are done. If λ = 1 then σ is asubstitution witnessing that [D] is mingly. ⊣ Now, suppose C is labeled by a state q ∈ Q from M . By Lemma 7.10, C d isaccepted in M K only if the contents of the registers r , r are each powers of K and the register r is empty. That is, C + σ d and C + σ d are powers of K and C + σ d = 0. On the one hand, Lemma 7.12 ensures that σ is the zerovector and C = 0, and so r is empty in C .By the motivation in Section 5.1, a natural condition to consider would beto stipulate that [D] satisfies ( ⋆K ). In such a case, if C + σ d is a power of K for each d ∈ D then there exists ¯ d ∈ D such that σ = σ ¯ d . Similarly, if C + σ d is a power of K for each d ∈ D, then there exists ¯ d ′ ∈ D such that σ = σ ¯ d ′ . However, there is no reason a priori that entails ¯ d = ¯ d ′ and thus C ∈ { C d : d ∈ D } , which would be sufficient to ensure that C would be acceptedif W d ∈ D C d were accepted.Since the most naive and obvious way to ensure acceptance is to ask that theleft-hand side C appears as one of the joinands C d on the right-hand side, it issufficient to stipulate that [D] satisfies the following condition:If the exponents of each variable in the right-hand side of [D] produced by a2-variable substitution are translated powers of K , then the substitutioninstance is trivial. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE σ, σ ′ ∈ N n and for all C, C ′ ∈ N , if C + σd and C ′ + σ ′ d are powers of K for each d ∈ D , then there exists ¯ d ∈ D such that σ ¯ d = σ and σ ′ ¯ d = σ ′ . ( ⋆⋆K )In this case, we say [D] satisfies ( ⋆⋆K ). We also consider the condition ( ⋆⋆ ):there exists K > ⋆⋆K ) holds. Note that, by setting σ = σ ′ , we seethat if [D] satisfies ( ⋆⋆K ) then it satisfies ( ⋆K ). So, we obtain the followinglemma.
Lemma . If a simple equation satisfies ( ⋆⋆ ) then it satisfies ( ⋆ ) . It is clear then that when q ∈ Q , if [D] satisfies ( ⋆⋆K ) then the acceptanceof W d ∈ D C d in M K implies the acceptance of C in M K by Lemma 7.10(1) and theobservations above.As it turns out, the remaining cases can be reduced to the above, and sosatisfying the condition ( ⋆⋆K ) alone is sufficient to ensure register-admissibility.The only remaining cases to verify are when the state q is internal to a multiply ordivide by K program. Let p ∈ P be some increment or decrement instruction for M . The idea is that if an instance of [D], which leads to acceptance in M K , occursinternal to a program P ( p ) then, by using Lemma 7.10(2) such an instance couldhave equivalently occurred at the end (or beginning) of executing the program P ( p ). Without loss of generality, suppose the instruction p acts on register r with input and output states q in and q out , respectively.For instance, if q is a transfer state q = t δ , where δ ∈ { , } , then by Lem-mas 7.3(2) and 7.5(2), C ⊑ P ( p ) C ′ := q out r ( C + σ )+( C + σ )+ δ r C + σ , and by Lemmas 7.3(2), 7.5(2), and 7.10(2), for each d ∈ D, C d ⊑ P ( p ) C ′ d := q out r ( C + σ d )+( C + σ d )+ δ r C + σ d ∈ Acc( M K ) . We see that by setting C = C + C + δ , C ′ = C , σ = σ + σ , and σ ′ = σ , weobtain the instance C ′ ≤ D W d ∈ D C ′ d ∈ Acc( M K ). Since C ≤ M K C ′ and C ′ is accepted( C ′ is labeled by state q out ∈ Q , which was handled above), it follows that C isaccepted.Similarly, if q is a multiply state q = a δ , for some δ ≤ K , then by Lemma 7.3(2), C ⊑ × ( p ) C ′ := q out r K ( C + σ + K − δ )+( C + σ )1 r C + σ and by Lemmas 7.3(2) and 7.10(2), for each d ∈ D, C d ⊑ × ( p ) C ′ d := q out r K ( C + σ d + K − δ )+( C + σ d )1 r C + σ d ∈ Acc( M K ) . So by setting C = KC + C + K − δ , C ′ = C , σ = Kσ + σ , and σ ′ = σ , weobtain the instance C ′ ≤ D W d ∈ D C ′ d ∈ Acc( M K ). Since C ≤ M K C ′ and C ′ is accepted( C ′ is labeled by state q out ∈ Q , which was handled above), it follows that C isaccepted. Surprisingly, we prove in Theorem 8.10 that the converse holds for all K sufficiently large. NIKOLAOS GALATOS AND GAVIN ST. JOHN
Lastly, we consider when q is a division state q = s δ , for some δ ≤ K . ByLemma 7.5(2), C ′ := q in r ( C + σ + δ )+ K ( C + σ )1 r C + σ ⊑ ÷ ( p ) C , and by Lemmas 7.5(2) and 7.10(2), for each d ∈ D, C ′ d := q in r ( C + σ d + δ )+ K ( C + σ d )1 r C + σ d ⊑ ÷ ( p ) C d ⊑ ÷ ( p ) C ′′ d ∈ Acc( M K ) , where C ′′ d is the unique output configuration of ÷ ( p ) labeled by q out .Now, it is clear that by setting C = C + KC + δ , C ′ = C , σ = σ + Kσ ,and σ ′ = σ , we have that C ′ ≤ D W d ∈ D C ′ d . Hence C ′ = C ′ ¯ d for some ¯ d ∈ D by( ⋆⋆K ), and so C ′ ¯ d ⊑ ÷ ( p ) C . Since C ′ ¯ d ⊑ ÷ ( p ) C ′′ ¯ d , by Corollary 7.7, it follows that C ⊑ ÷ ( p ) C ′′ ¯ d . Therefore C is accepted in M K if W d ∈ D C d is accepted in M K .By the arguments above the following lemma is established: Lemma . Let M be a -ACM and K > . If a non-mingly simple equationsatisfies ( ⋆⋆K ) then it is register-admissible in M K . ( ⋆⋆ ) and undecidability. Assume that [D] is a non-minglysimple equation that satisfies ( ⋆⋆ ). Since it is non-mingly, by Lemma 6.5 weget that [D] is state-admissible in any machine. Since it also satisfies ( ⋆⋆ ), byLemma 7.15 we have that [D] is register-admissible in M K for some integer K > M is any machine. In particular, [D] is admissible in ˜ M K , where ˜ M is themachine with undecidable halting problem. By Corollary 7.11, the machine ˜ M K has an undecidable set of accepted configurations for any K >
1. By Lemma 6.3we obtain W +˜ M K ∈ CRL + [D]. Therefore,
CRL + [D] has an undecidable wordproblem by Theorem 4.5. This proves the following result.
Corollary . For any finite set Γ of non-mingly equations that satisfy ( ⋆⋆ ) , every subvariety of RL containing CRL +Γ has an undecidable word problem. As motivation for the general case, we show that the 1-variable basic equations[ n, P ] : x n ≤ W p ∈ P x p , where P contains at least two distinct positive integers,considered in Lemma 5.4, define varieties with undecidable word problem. Theresults of the next section will show that this holds for many more equations, allspineless equations. Theorem . Let [ n, P ] be a -variable basic equation where P containsat least two distinct positive integers. Then the variety CRL + [ n, P ] has anundecidable word problem. If additionally P only contains integers strictly greaterthan n , then the variety CRL + [ n, P ] has an undecidable equational theory. Proof.
Let [D] be the n -variable simple equation that is the linearization of[ n, P ] over CRL (given by Equation 4 in Lemma 5.4) and let p, q ∈ P be such that p > q >
0. Note that by Lemma 5.4, [D] is spineless and hence it is not mingly byLemma 6.4. By Corollary 7.16, to establish the first claim it is enough to show[D] satisfies ( ⋆⋆ ). We will show that [D] satisfies ( ⋆⋆K ), for every K > p − q ;since p > q , this implies that K >
C, C ′ ∈ N and 1-variable substitutions σ, σ ′ such that C + σd and C + σd ′ are powers of K for each d ∈ D. We will show that σ and OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE σ ′ are trivial substitutions (i.e., all entries are 0), and hence σ = 0 = σ ¯ d and σ ′ = 0 = σ ′ ¯ d for every ¯ d ∈ D.Arguing towards contradiction, suppose that σ is nontrivial with σ ( i ) > i ≤ n . Now (by Equation 4) the terms x pi and x qi appear as joinands on theright-hand side of [D], i.e., D contains d and d ′ such that d ( i ) = p , d ′ ( i ) = q , and d ( j ) = d ′ ( j ) = 0 for each j = i . By the assumption on σ and C , C + σd = K a + b and C + σd ′ = K a , for some a, b ∈ N , with b > p > q . We have that, K a ( K − ≤ K a ( K b −
1) = K a + b − K a = σd − σd ′ . Also, σd = σ ( i ) p and σd ′ = σ ( i ) q (by definition of d, d ′ ), so we obtain σd − σd ′ = σ ( i ) p − σ ( i ) q = σ ( i )( p − q ) ≤ K a ( p − q ) , where the last inequality follows from σ ( i ) ≤ σ ( i ) q ≤ K a ; note that q ≥ K − ≤ p − q and K ≤ p − q , acontradiction. Hence [D] satisfies ( ⋆⋆K ). Furthermore, if all elements from P are larger then n , then [ n, P ] is an expansive equation. Therefore the secondclaim follows by Corollary 5.8. ⊣ §
8. Characterization of spineless equations.
In this section we provethat a simple equation is spineless if and only if it satisfies ( ⋆⋆K ) for every K sufficiently large. CRL as sets of tuples.
Given thenatural ordering of the variable set { x i : i ∈ Z + } , note that using the above-mentioned vector notation, every commutative monoid term can be written inthe form x n f , for some n ∈ Z + and some n -tuple f of natural numbers; recallthat x n = ( x , . . . , x n ). If we actually extend our notation to the case where x ∞ = ( x i ) i ∈ Z + = ( x , x , . . . ) and f is a sequence of natural numbers that iseventually constantly zero, then every commutative monoid term is of the form x ∞ f , and thus it is fully specified by such an f . In the following we will workinterchangeably in the free monoid over the variable set { x i : i ∈ Z + } and also inthe isomorphic monoid F of eventually-zero sequences of natural numbers. Moreformally, N Z + denotes the set of all functions from Z + to N and for f ∈ N Z + ,we define supp( f ) := { i ∈ Z + : f ( i ) = 0 } to be the support of f . Then theset F := { f ∈ N Z + : | supp( f ) | < ∞} of all functions of finite support formsa commutative monoid ( F , + , ), under addition and with unit the constantly-zero function . Clearly, this monoid is simply an additive rendering of the freecommutative monoid on countably many generators and is isomorphic to theabove multiplicative rendering by exactly the map f x ∞ f . Up to now we havefavored the multiplicative representation due to its connection with machines,but from now on we will use the additive one as it connects better with the linearalgebra arguments of this section. Under this isomorphism the variable x i mapsto the generator e i , which has 1 in the i -th entry and 0 everywhere else.For reasons that will be clear soon, we view the elements of F as columnvectors and we also consider the bijective set F ⊤ of the row vectors, which arethe transposes of the elements of F . In particular, for f ∈ F and σ ∈ F ⊤ , thematrix product σf yields a 1 × f and σ are each of infinite2 NIKOLAOS GALATOS AND GAVIN ST. JOHN dimension, they both have finite support, so their product is well defined. Fora subset S of Z + we define F S to be the set of eventually zero functions from S to N , so F = F Z + ; we identify F S with the corresponding subset of F in thenatural way, as every function in F S is the restriction to S of the function in F that is defined to be zero outside S . We write F n for F { ,... ,n } . So, if f ∈ F withsupport included in { , ..., n } , we will identify f with the corresponding elementof F n . We define sets F ⊤ S and F ⊤ n in a similar way. Therefore, F n is the set of all n × F ⊤ n is the set of all 1 × n matrices.For a set X ⊆ F , we write σX := { σf ∈ N : f ∈ X } and supp( X ) := S f ∈ X supp( f ). For each n ∈ Z + , we define the column vector n ∈ F to contain1 in its first n entries and 0 everywhere else. A substitution σ on F is fullydetermined by its application on the generators e i f i ∈ F for each i ∈ Z + ,and as it is a homomorphism, namely an additive/linear map, its application isgiven by multiplication of an associated matrix M σ ; so σ ( f ) = M σ f . Since weonly consider finite subsets A of F in basic equations [ f, A ], we may view A as asubset of F n , where n is the largest index in supp( A ∪ { f } ) and, in this way, willonly consider substitutions σ : F n → F k , in which case the associated M σ is a k × n matrix; in this case, we say that σ is a k -variable substitution . We will write σ i ∈ F ⊤ n for the i -th row of M σ for each i ≤ k and also M σ = [ σ i ] ki =1 . Abusingnotation, we will identify σ with M σ and also we use σ [ f, A ] for the resultingbasic inequality. As we have seen in the statement of condition ( ⋆⋆ ), 1-variablesubstitutions play an important role. Actually, every substitution σ is renderedas the product of 1-variable substitutions σ i (the ones corresponding to the rowsof M σ = [ σ i ] ki =1 ) as for every variable x j , σ ( x j ) = σ ( x j ) · σ ( x j ) · · · σ k ( x j ), whenusing multiplicative notation, and as a sum of 1-variable substitutions σ i as forevery e j , we have σ ( e j ) = σ ( e j ) + σ ( e j ) + · · · + σ k ( e j ), when using additivenotation. Let [ f, V] be a k -variable spinal equation. We de-fine v := and V = V ∪ { v } . Using additive notation, it follows fromDefinition 5.2:1. V contains a subset V + consisting of k ≥ v , ..., v k , where v j ( i ) is positive if i = j and zero if i > j .2. V is exactly either V + or V .3. f is a vector in F k with all entries positive such that f V.We write [ v · · · v k ] for the matrix with columns v , . . . , v k , in that order.Observe that (1) is equivalent to [ v · · · v k ] being a k × k upper-triangularmatrix whose diagonal entries are positive.Using this additive perspective, we will demonstrate why spinal equations failto satisfy the condition ( ⋆⋆ ), and thus the argument for register-admissibility inthe machines M K found in Lemma 7.15 is not applicable to extensions by suchequations. In fact, we prove a much stronger property for spinal equations whichentails such an argument will fail, not just for our exponential encoding, but forany similar sort of encoding in general. Specifically, we mean the following: Let M be a 2-ACM and φ : N → N any (computable)function with infinite range. Let M φ be an ACM constructed so that the register contents h n, m i of a configuration from M are stored as h φ ( n ) , φ ( m ) , , ..., i in M φ , and programs constructed OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE S of natural numbers, we consider the following property:If the exponents in the right-hand side of [D] produced by a 1-variablesubstitution are in a translation of S (by the same constant), then thesubstitution instance is trivial.In symbolic terms this can be written asIf for some σ ∈ F ⊤ n and C ∈ N , every C + σd is in S , for d ∈ D,then there exists ¯ d ∈ D such that σ ¯ d = σ ( ⋆S )In more compact terms, this can be written as( ∃ σ ∈ F ⊤ n , ∃ C ∈ N , C + σ D ⊆ S ) ⇒ σ ∈ σ D . Clearly, what we called ( ⋆K ) is simply ( ⋆S ), where S is the set of all powers of K . In Lemma 8.1, we essentially show that ( ⋆S ) fails for any prespinal equation ε and infinite set S . Lemma . If a simple equation satisfies ( ⋆S ) for an infinite subset S of N ,then it is spineless. Proof.
We argue by contraposition, assuming that a simple equation ε isprespinal. So there is a substitution σ such that [ f, V] := σ ε is a spinal equation,where every column vector of V has k entries/rows. We will construct a 1-variablesubstitution τ = [ t t · · · t k ] ∈ F ⊤ k such that C + τ V ⊆ S , for some C , and τ f τ V (hence also τ f τ V, as V ⊆ V ); let V = { v , v , . . . , v k } . This willimply that ε falsifies ( ⋆S ) by the 1-variable substitution τ σ and constant C .First note that for any τ ∈ F ⊤ k we have τ v = 0, so C + τ V ⊆ S iff C ∈ S and C + τ V + ⊆ S . Observe that V + := [ v · · · v k ] is an upper-triangular k × k matrix whose entries are non-negative integers; note the different font from theset V + . Furthermore, the determinant δ := det V + = v (1) · · · v k ( k ) is positivesince v n ( n ) is positive for each 1 ≤ n ≤ k by definition. Hence V + is invertibleand V − = δ − adj V + , where the adjoint adj V + is an upper-triangular matrixwith integer entries which furthermore has positive entries on its diagonal (eachof the form δ/v n ( n )).Now, if C ∈ S and τ ∈ F ⊤ k then C + τ V + ⊆ S ⇐⇒ τ V + ∈ ( S − C ) k ⇐⇒ τ ∈ ( S − C ) k V − . Observe that( S − C ) k V − = ( S − C ) k δ adj V + = (cid:18) S − Cδ (cid:19) k adj V + . Therefore, C + τ V + ⊆ S ⇐⇒ τ ∈ (cid:18) S − Cδ (cid:19) k adj V + . We claim that there is a C ∈ S such that the set ( S − C ) /δ has an infinitesubset in the positive integers. Indeed, since S is infinite there exists a coset so that increments n n + 1 [decrements n n −
1] of a register in M are simulated by φ ( n ) φ ( n + 1) [ φ ( n ) φ ( n − M φ . Lemma 8.1 ensures that the corresponding argumentfor register-admissibility is not valid for spinal equations without having more informationabout Acc( M ). NIKOLAOS GALATOS AND GAVIN ST. JOHN C + N δ that has infinite intersection with S , where we can take C ∈ S withoutloss of generality; let ¯ N ⊆ N be the infinite set such that C + ¯ N δ is the intersectionof S with C + N δ . Hence ¯ N is such an infinite subset of ( S − C ) /δ , and actually0 ∈ ¯ N since C ∈ S . Consequently, if τ ∈ ¯ N k adj V + , then C + τ V + ⊆ S . Notethat ¯ N k adj V + is an infinite set and all if its entries are integers, while we need τ ∈ F ⊤ k . Therefore, it is enough to be able to find [ x · · · x k ] ∈ ¯ N k such that[ t t · · · t k ] = [ x · · · x k ] adj V + , where the entries t i are nonnegative andfurther τ f τ V .Note that since adj V + is upper-triangular, the value of t n , for each n ≤ k ,is determined only by the values x , . . . , x n ; t n is a linear combination of only x , ..., x n . This allows us to recursively choose the values of the x i ’s, in orderto specify the values t i one-by-one. Furthermore, at the recursive step wherewe have already determined the values of x , ..., x n − , the value of x n can bechosen arbitrarily large from the infinite set ¯ N ; moreover, in the linear combi-nation specifying t n the coefficient of x n is the ( n, n )-entry of adj V + , which ispositive; this allows for the value of t n to be as large as we want (in particularnonnegative). Therefore, the only thing that we have to ensure is that x , ..., x n are chosen in ¯ N so that furthermore τ f τ V , i.e. τ f = τ v i for all 1 ≤ i ≤ k .Below we first prove that τ f = τ v k and then that τ f > τ v i , for all i < k .Since [ f, V] is a spinal equation, we have f V and in particular f = v k . Let m be the largest number in { , . . . , k } such that f ( m ) = v k ( m ) and f ( i ) = v k ( i )for all i > m . We now define x i = 0 for each i < m ; note that since 0 ∈ ¯ N , allof these values are in ¯ N . Since t i is a linear combination of x , . . . , x i , we havethat t i = 0 for i < m . We define x m to be any positive number in ¯ N , resultingin a positive value for t m , since x i = 0 for i < m . Since f ( m ) = v k ( m ), we get t m f ( m ) = t m v k ( m ), and since f ( i ) = v k ( i ) for all i > m we obtain τ f = k X i = m t i f ( i ) = t m f ( m ) + X i>m t i v k ( i ) = t m v k ( m ) + X i>m t i v k ( i ) = τ v k , for all possible values of t i for i > m . So any choice of a positive value for x m in¯ N ensures that τ f = τ v k .For m < i < k , we continue choosing positive values for x i in ¯ N that arelarge enough to ensure that t i is nonnegative, as explained above. Finally, atthe last step, we have chosen x , . . . , x k − and therefore determined the valuesof t , . . . , t k − . Note that furthermore the values of τ v , . . . , τ v k − have alsobeen determined: since V + is an upper triangular matrix, we have v i ( j ) = 0 foreach j > i , so τ v i = t v i (1) + · · · + t i v i ( i ) = 0 for all i < m ; in particular eventhough t k appears in τ , it does not appear in the values of τ v , . . . , τ v k − . Wenow choose x k ∈ ¯ N so that t k > τ v i for all i < k . Since [ f, V] is a basic equationby definition, f is positive in each of its entries and in particular f ( k ) ≥
1, sowe obtain τ f ≥ t k f ( k ) ≥ t k > τ v i for each i < k . ⊣ Corollary . If a simple equation satisfies ( ⋆ ) , then it is spineless. Thus we obtain the following from Lemma 7.14 and Corollary 8.2:
Lemma . If a simple equation satisfies ( ⋆⋆ ) then it is spineless. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE We will begin with a concrete example of a prespinalequation before illustrating the general case.
Example . Consider the 8-variable simple equation ε : stuvwxyz ≤ ∨ sw xz ∨ s ∨ s tx yz ∨ s tz ∨ s twy ∨ s z ∨ s vwx y ∨ s t ∨ s uvx y, where for better readability we use the letters s, ..., z for the formal variables x , ..., x , in that order; we will use both names for each variable below.Since ε is an 8-variable equation with 10 distinct joinands, and all spinalequations in 8-variables have no more than 9 distinct joinands, the equation ε isnot spinal. However, it is easily verified that ε is prespinal, as witnessed by the3-variable substitution σ defined via: s t x , z x , y x x , v x ,and u, w, x x . Indeed, we have σ ε : x x x ≤ ∨ x ∨ x x ∨ x x x . We name the joinands in the right-hand side of ε in order of appearance fromleft to right: d = 1 , d = sw xz , . . . , d = s uvx y ; we do the same for σ ε : v = 1, v = x , v = x x , and v = x x x . Also, we define the setsD = { d , ..., d } and V = { v , ..., v } . Given this particular ordering of variablesand joinands, the set-theoretic equation σ D = V induces the matrix equation σ D = V (note the different font for the sets D , V and the matrices
D, V ), where σ = s t u v w x y z , D = stuvwxyz d d d d d d d d d d , V = v v v v v v v v v v Note that the ( i, j )-entry of D represents the degree of the i -th variable x i in thejoinand d j , and the ( j, i )-entry of σ represents the degree of x j in σ ( x i ).Note that, by omitting v = , the 3 × v v v ] is upper triangularwith a positive diagonal (as demanded in the definition of σ ε being spinal)and this is the reason for the particular naming of v , v , v , in that order. Inturn, given this order, the substitution partitions the set of joinands D, i.e.,the columns of D into D = { d , d } , D = { d , d , d } , D = { d , d , d } andD = { d , d } , so that σ D j = { v j } , for all j . Guided by this ordering, wefurther rearrange the columns of V into a new matrix V ′ and the columns of D into a new matrix D ′ , where the ordering of the columns within each D i is donerandomly. If we represent D ′ symbolically as [D D D D ], then we also have σ D ′ = [ σ D σ D σ D σ D ] and the equation σ D ′ = V ′ .We can improve the presentation of this equation even more by putting σ and D ′ in a triangular form, at least in blocks. More specifically, we now rearrange therows of D ′ and simultaneously the columns of σ (this corresponds to permutingthe variables of ε ) to obtain new matrices D ′′ and σ ′ , yielding the equation σ ′ D ′′ = V ′ :6 NIKOLAOS GALATOS AND GAVIN ST. JOHN s t z w x y u v d d d d d d d d d d = v v v v v v v v v v . Finally, we observe that the rearrangement of the rows results in a partition ofthe set of rows such that the two partitions (of the set of rows and the set ofcolumns) induce a blocking (given by the solid lines above) that has an upper-triangular shape. We denote by D b and σ b the resulting block matrices, andthe equation σ ′ D ′′ = V ′ of matrices yields the equation σ b D b = V b of blockmatrices. We call the elements of D b blocks and they are submatrices of D ′′ ; wedenote by ( D b ) ij the ( i + 1 , j + 1)-block of D b , where 0 ≤ i, j ≤
3. We observethat1. each ( D b ) ij is the zero matrix when i > j (all blocks below the diagonalare zero matrices) and2. each row and each column of ( D b ) jj contains a nonzero entry for j ≥ D b ) ).We call such a partition of the matrix into a block matrix with these twofeatures a blocking of the matrix. Each blocking specifies an (ordered) partitionof the set of rows and an (ordered) partition of the set of columns of a matrixin the obvious way (with the provision that the first class in this ordered listmay be the empty set), but each of these two partitions is special as we willexplain. Given a column partition, we define below an associated list of sets ofrows. Whenever the original partition comes from a blocking, the resulting listis actually an (ordered) partition (every set in the list, except possibly the firstone, is non-empty). The same holds with the roles of rows and columns swapped.We will explain that blockings correspond bijectively to column-partitions thathappen to induce ordered row-partitions and to row-partitions that happen toinduce ordered column-partitions.Given an I × J matrix D , formally viewed as a function from I × J , as usual D ij denotes its entry in the i -th row and j -th column, for i ∈ I and j ∈ J ;usually I and J are taken to be initial segments of the positive integers as inthe example above. We denote by D i the i -th row and by D j the j -th columnof D . Given an ordered partition ( C , C , . . . , C k ) of J (i.e., C may be empty,but not all of J , and the remaining list forms a partition of J ) we define thelist ( R , R , . . . , R k ) of subsets of I as follows: for m ≥ R m contains those i ∈ I such that the entry D in is zero for n belonging to parts C ℓ with ℓ < m and there is a non-zero entry D in for some n belonging to the part C m . In otherwords, if we group the columns of D according to the ordered partition, then R m corresponds to those rows that are fully zero on all columns before C m and arenot fully zero on C m . We define R as containing the remaining i ’s that are notin R ∪ · · · ∪ R k . (Note that we allow our ordered partitions to have an optional OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE R .) In the example above, the sets R n are all non-empty,thus resulting into a partition of the set of columns; however, this may not be thecase when ( C , C , . . . , C k ) of J is an arbitrary ordered partition of J . Note thatwhenever ( R , R , . . . , R k ) is a partition, we can define a partition of I × J intoblocks of the form R m × C n with the feature that, for n ≥
1, each block R n × C n is such that no column and no row of D in that block is fully zero. Therefore,blockings correspond to column-partitions that happen to induce ordered row-partitions. Often, instead of writing a partition ( C , C , . . . , C k ) of the columnindex set J , we will be writing the partition (D , D , . . . , D k ) of the set D of thecorresponding columns.Conversely, given an ordered partition ( R , R , . . . , R k ) of I we define a list( C , C , . . . , C k ) of subsets of J as follows: for n ≥ C n contains those j ∈ J such that the entry D mj is zero for m belonging to parts R ℓ with ℓ > n , andthere is a non-zero entry D mj for some m belonging to the part C n . In otherwords, if we group the rows of D according to the ordered partition, then R n corresponds to those rows that finish with zeros on all rows after C n and are notfully zero on C n . We define C as containing the remaining i ’s that are not in C ∪ · · · ∪ C k . Again we can see that this yields an ordered partition (i.e., thesets C n , n ≥
1, are non-empty) iff this corresponds to a blocking of the matrix.Given a blocking b of I × J on a matrix D , we obtain the block matrix D b andobserve that it has an upper-triangular form. In the following we will consider ablocking given by either its ordered row-partition or its ordered column-partition.In analogy with our notation for entries, rows and columns of a matrix, we define D b mn := { D ij : i ∈ R m , j ∈ C n } , D b m = { D ij : i ∈ R m } and D b n = { D ij : j ∈ C n } . If D is a set of column vectors, we say b is a blocking of D if it is ablocking of a matrix D whose set of columns form D, and we define D b j to bethe j -th set of column vectors D b j . Observe that if b = ( R , . . . , R k ) is a rowblocking for D, then by how the associated list of columns are defined, we getD b j = { d ∈ D : supp( d ) ∩ R j = ∅} \ (D b j +1 ∪ · · · D b k ), for each j ≥
1, andD b = D \ (D b ∪ · · · D b k ).The blocking in the Example 8.4 is induced by σ in the sense that the corre-sponding partition on the set { , . . . , } of columns of D along the vertical linesin D b into the sets C = { , } , C = { , , } , C = { , , } , and C = { , } is given by the stipulation that C n is exactly the set of the columns that aremapped by σ to the same column vector, v n , of V , for all n ∈ { , , , } . Thepartition of the set of rows { , . . . , } of D (along the horizontal lines of D b )yields the sets R = { , } , R = { , , } , R = { , } , and R = { } .As in the example, every substitution σ that maps an I × J matrix D to aspine V = [ v v · · · v k ] induces a blocking b via a partitioning of the columns: C n = { j ∈ J : σ D j = v n } . The blockings induced by substitutions enjoy furtherproperties (in addition to yielding upper-triangular block matrices with diagonalblocks that have no fully zero row or column). Returning to the example we seethat σ D b j = v j for each 0 ≤ j ≤
3. We say that a substitution is a solution fora set/matrix of column vectors if it sends all of the vectors of the set/matrix tothe same vector. In this terminology, σ is a solution for each of the sets D b ,8 NIKOLAOS GALATOS AND GAVIN ST. JOHN D b and D b . Note that a substitution is a solution for a set/matrix iff each ofits rows is a solution for it.Also, looking at the induced partition on the rows, the R m part of each row σ m of σ is not fully zero and all of its elements are non-negative. For f ∈ F and T ⊆ Z + we say f is T -positive if f T > , i.e., f T = and f ( i ) ≥ i ∈ T ;put differently f F T c , where T c is the complement of T . In this terminology,the row σ m of σ is R m -positive, for each m . Finally, we note that σ is anelement of F ⊤ R , σ is an element of F ⊤ R ∪ R , and σ is an element of F ⊤ R ∪ R ∪ R .In general, for every blocking defined by a substitution σ with respect to a spine, σ m is an element of F ⊤ R + m , where R + m := R m ∪ · · · ∪ R k . For T ⊆ S , we say f is( T, S ) -positive if f is T -positive and f ∈ F S ; put differently f ∈ F S and f F T c .Therefore, the row σ m of σ is ( R m , R + m )-positive, for each m .Given a row blocking b = ( R , . . . , R k ) of a set D, a 1-variable substitution σ ∈ F ⊤ and 1 ≤ i ≤ k , if σ is ( R i , R + i )-positive and σ is a solution for each setof columns D b j , then we say that σ is a ( b , i ) -solution for D. Finally, we say thata k -variable substitution σ is a b -solution for D, where b = ( R , . . . , R k ), if forall i , the i -th row σ i of σ is a ( b , i ) -solution for D. σ b · · · σ b i · · · σ b k ... ... ... ... ... ... · · · σ b ii · · · σ b ik ... ... ... ... ... ... · · · · · · σ b kk D b D b · · · D b i · · · D b k D b · · · D b i · · · D b k ... ... ... ... ... ... · · · D b ii · · · D b ik ... ... ... ... ... ... · · · · · · D b kk = v (1) · · · v i (1) · · · v k (1)... ... ... ... ... ... · · · v i ( i ) · · · v k ( i )... ... ... ... ... ... · · · · · · v k ( k ) , The following lemma and theorem then follow by the definition of b -solutions. Lemma . If a k -variable substitution σ is a b -solution for D , then thematrix [ v v · · · v k ] , where v j = σ D b ∗ j , is a k × k upper-triangular matrixwhose diagonal contains positive entries. Theorem . An n -variable simple equation [D] is prespinal if and only if D has a b -solution σ , for some row blocking b , such that σ n and σ D b k differ. The importance of Theorem 8.6 is that it characterizes the notion of prespinal-ity without a reference to a spine.Using Theorem 8.6 we now characterize which equations are spineless. Forinstance, we can verify that equations (iii)-(v) from Table 1 are spineless. Ineach equation, the set D has only one blocking b , given by (D b , D b ) where D b contains all the nonzero columns from D. We see that there is no nonzero ( b , b , σ = [1 1 1], but σ ∈ σ D. Thereforeequations (iii)-(v) are spineless. ( ⋆⋆ ) . Theorem 8.6 provides the founda-tional link between an equation being spineless and satisfying ( ⋆⋆ ), namely bythe (non-)existence of certain 1-variable substitutions viewed as solutions to par-ticular linear systems.We prove that every spineless equation satisfies ( ⋆⋆K ), for sufficiently large K , contrapositively. So, we assume that a simple equation [D] satisfies the OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE ⋆⋆K ) and in particular there exists a (nontrivial) 1-variable sub-stitution σ for which σ D is contained in some shift of K N . The following lemmaensures that, if K is chosen large enough, σ induces a blocking b on D in thesense that the naturally ordered column-partition (D , . . . , D k ) of D induces b , i.e., D j = D b j for each j . Here we conventionally order the sets so that σ D j < σ D j +1 for j < k , and we always take D to be the (possibly empty) setof all d ∈ D such that σd = 0. Note that if a σ induces a blocking then it mustbe that k ≥ σ is not the zero vector.For a finite set D ⊆ F , we define ∆D = P ni =1 max {| d ( i ) − d ′ ( i ) | : d, d ′ ∈ D } . Lemma . Assume that for a finite D ⊆ F and for some 1-variable substi-tution σ that is nontrivial on D , σ D is contained in some shift of K N , where K > ∆D + 1 . Then σ induces a blocking on D . Proof.
Suppose D ⊆ F n and let (D , ..., D k ) be the column partition ofD such that 0 = σ D < · · · < σ D k . We will show that the associated list( R , R , . . . , R k ) of sets of rows is an ordered row-partition, i.e., each of the R , . . . , R k is not empty. Recall that R j is the set of all indices i ∈ { , ..., n } such that D ij = but D il = for each l < j , where D := [D · · · D k ] (the orderof the columns within each D i plays no role). We first prove that R k inducesD k , i.e., each d ∈ D k must have some nonzero entry/row that is zero in every d ′ ∈ D \ D k . If not, then there exists some d ∈ D k such that for each i ∈ supp( d )there exists j < k and vector d ′ ∈ D j with i ∈ supp( d ′ ). By assumption, σd and σd ′ are in the same shift of K N , say K N − C for some C ∈ N , and because of ourordering convention, σd ′ < σd . So, σd = K a +1 − C and σd ′ ≤ K a − C for some a ≥
0. Since d ′ ( i ) ≥
1, we have σ ( i ) ≤ σ ( i ) d ′ ( i ) ≤ σd ′ ≤ K a , so K a ( K −
1) = K a +1 − K a ≤ σd − σd ′ ≤ σ | d − d ′ | ≤ K a ∆ { d, d ′ } ≤ K a ∆D , where the entries of | d − d ′ | are the absolute values of the corresponding entriesof d − d ′ . Therefore, K ≤ ∆D + 1, which contradicts the assumption on the sizeof K . So, R k is nonempty and we obtain D k = { d ∈ D : supp( d ) ∩ R k = ∅} .Continuing in this way for 1 ≤ j < k , set D ′ = D ∪ · · · ∪ D j . Since D ′ ⊆ Dimplies ∆D ′ ≤ ∆D, the same argument shows each d ∈ D j contains a nonzeroentry that is zero for each d ′ ∈ D ′ \ D j . Hence R j induces D j , i.e.,D j = { d ∈ D : supp( d ) ∩ R j = ∅} \ (D j +1 ∪ · · · ∪ D k ) . Given R = supp(D) \ R +1 by definition, we conclude that b = ( R , ..., R k ) is ablocking on D such that D b j = D j for each j ≤ k , i.e., σ induces the blocking b on D. ⊣ Suppose σ induces a row blocking b = ( R , R , . . . , R k ) of D . Since σ D > σ is R -positive, and since σ D = 0 by definition,it must be that the support σ is contained in R +1 . That is, σ must be ( R , R +1 )-positive. Since it is also a solution for each D b j by definition, this proves thefollowing lemma. Lemma . If a -variable substitution σ induces a blocking b on D , thenit is a ( b , -solution for D . In particular, if σ is any b -solution for D , thesubstitution obtained by replacing the first row of σ by σ is also a b -solution for D . NIKOLAOS GALATOS AND GAVIN ST. JOHN
We now demonstrate the converse of Corollary 8.3 by proving the followingstronger statement. The proof relies on Lemma 8.13, which we prove in the nextsection.
Lemma . A spineless equation satisfies ( ⋆⋆K ) for all sufficiently large K . Proof.
Toward establishing the contrapositive, we assume that the n -variablesimple equation [D] fails ( ⋆⋆K ) for infinitely many K ∈ N . Then there areinfinitely many pairs of 1-variable substitutions σ, σ ′ witnessing such failuresthat furthermore induce pairs of blockings for D by Lemma 8.7. Since there areonly finitely many blockings, and thus finitely many pairs of them, there mustexist blockings b = ( R , ..., R k ) and c = ( R ′ , ..., R ′ l ) that are witnessed infinitelyoften as a pair. By Lemma 8.13, [D] has a b -solution σ and a c -solution σ ′ . ByLemma 8.8, we can replace the first rows of σ and σ ′ with 1-variable substitutions σ and σ ′ (from the above-mentioned infinitely many) witnessing the failure of( ⋆⋆K ) and inducing b and c , respectively.If either σ n = σ D b k or σ ′ n = σ ′ D c l , then [D] is prespinal by Theorem 8.6,and we are done. If not, we have σ n = σ D b k and likewise σ ′ n = σ ′ D c l . Inparticular, σ n = σ d for every d ∈ D b k and σ ′ n = σ ′ d for every d ∈ D c l , andhence for every d ∈ D b k ∩ D c l . If D b k and D c l were not disjoint, there would be a d ∈ D such that σ n = σ d and σ ′ n = σ ′ d , which would imply that σ , σ ′ satisfy ( ⋆⋆K ), contradicting their choice above. Hence D b k and D c l are disjointand by the definition of blockings, R k and R ′ l are also disjoint.As a result the row partition a := ( B , B ), where B = R k ∪ R ′ l and B = { , ..., n } \ B , is actually a row blocking on D which furthermore induces thepartition of columns D a = D b k ∪ D c l and D a = D \ D a . We will construct a1-variable substitution α that will serve as an a -solution for D witnessing theprespinality of [D].Let σ k and σ ′ l be the bottom rows of σ and σ ′ , respectively. Since σ k is a( R k , R + k )-positive solution for D b k , the value t = σ k D b k is positive, and since R k and R ′ l are disjoint, σ k D c l is zero; in detail R ′ l contains columns that appearin earlier blocks than R k . Similarly, t ′ = σ ′ l D c l is positive and σ ′ l D b k is zero.Note that σ k n = t and σ ′ l n = t ′ follow by the fact that σ n = σ D b k and σ ′ n = σ ′ D c l .We now define the 1-variable substitution α = t ′ σ k + tσ ′ l . Since R k and R ′ l are disjoint, it follows that α D b k = t ′ t + t tt ′ and α D c l = t ′ tt ′ = tt ′ , so α D a = tt ′ ; also α n = t ′ t + tt ′ = 2 tt ′ . In the case when D a is nonempty, we have α D a = 0, because σ k D b j = 0, for j < k , and σ ′ l D c i = 0, for i < l . Since t, t ′ > tt ′ > tt ′ >
0, and therefore it follows that α n α D and so [D] isprespinal by Theorem 8.6. ⊣ Lemmas 8.3 and 8.9 establish the equivalences between a simple equation beingspineless and satisfying ( ⋆⋆ ), and as well as satisfying ( ⋆ ). Theorem . A simple equation is spineless iff it satisfies ( ⋆⋆K ) for everysufficiently large K . Therefore, to establish Theorem 8.10 we must prove Lemma 8.13.
OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE R n . The goal of this section is to prove Lemma 8.13. Toaddress this, we recall a theorem of alternatives for positive solutions to linearsystems accompanied by conventional terminology (see [15]).Let v ∈ R n (viewed as a row vector) and M ⊆ R n . We say that v is orthogonal to M if vM = 0. We say that v is strictly positive if v = and v ( i ) ≥ i ∈ { , ..., n } . The set X n + denotes the set of all strictly positive vectors in X n ,called the strictly positive orthant in X n , where X ∈ { Z , Q , R } . The following(folklore) theorem is equivalent to Farkas’s Lemma (for instance, see Theorem 27[14]). Theorem . Let M ⊆ R n be nonempty set of vectors and i ∈ { , ..., n } afixed index. Then exactly one of the following holds: there exists a strictly positive vector v orthogonal to M where v ( i ) > , or there exists a strictly positive vector w ∈ span( M ) where w ( i ) > . Note that span( M ) S = span( M S ) for any M ⊆ R n and S ⊆ { , ..., n } ; herethe subscript S denotes restriction to S . Corollary . Let M ⊆ R n and T ⊆ S ⊆ { , ..., n } be non-empty. If thereis no T -positive vector in R | S | + orthogonal to M S then there exists L ∈ N suchthat, for any v ∈ R n + orthogonal to M , v ( i ) ≤ L · max { v ( j ) : j ∈ { , ..., n } \ S } for each i ∈ T . Proof.
By Theorem 8.11, for each i ∈ T there exists a strictly positive (in R | S | ) vector w i ∈ span( M S ) where w i ( i ) >
0. Then ¯ w := P i ∈ T w i ∈ span( M S )is strictly positive with T ⊆ supp( ¯ w ). Let w ∈ span( M ) be such that w S = ¯ w .Note that since T is nonempty and T ⊆ supp( w S ), it follows that t := min { w ( i ) : i ∈ T } is positive. Set S c := { , ..., n } \ S and m := P j ∈ S c | w ( j ) | , where we takethe empty sum to be zero. Define L to be the smallest positive integer greaterthan m/t .Now, suppose v ∈ R n + is orthogonal to M . Then vw ⊤ = 0 and hence X j ∈ S w ( j ) v ( j ) = X j ∈ S c − w ( j ) v ( j ) . Let N denote this common value. Considering the right-hand side of the equa-tion, we obtain N ≤ m · max { v ( j ) : j ∈ S c } . Considering the left-hand side ofthe equation, tv ( i ) ≤ w ( i ) v ( i ) ≤ N , for all i ∈ T , since T ⊆ S and w ( i ) ≥ t > i ∈ T , we deduce v ( i ) ≤ N/t ≤ L · max { v ( j ) : j ∈ S c } . ⊣ Lemma . Let D be a finite subset of F n and b be a blocking on D . If thereare infinitely many K ∈ N for which there exists a -variable substitution σ thatinduces b and σ D is contained in some shift of K N , then D has a b -solution. Proof.
Working contrapositively, we assume that D has no b -solution. Wewill show that if K > σ ∈ F ⊤ n that induces a blocking b on D and that σ D is contained in some shift of K N ,then K can be no larger than a certain multiple of ∆D. Let b = ( R , ..., R k ),where k ≥ A ⊆ F n , fix ¯ a ∈ A and define ¯ A := { a − ¯ a : a ∈ A } ; notethat the entries of the column vectors are in Z . For a set of rows S ⊆ { , ..., n } ,2 NIKOLAOS GALATOS AND GAVIN ST. JOHN σ is a solution for A S iff σA S is a singleton iff σA S = σ ¯ a iff σ ¯ A S = { } iff σ is orthogonal to ¯ A S in R n (regardless of the choice ¯ a ∈ A ). Hence, if T ⊆ S ,then there exists a T -positive solution for A in F ⊤ S iff there exists a T -positivesolution for A S iff there exists a T -positive vector of R | S | + orthogonal to ¯ A S . Now, by definition of being a blocking, for each i ≥ k the set D b i is nonempty,so we can define ¯D b i = { d − ¯ d i : d ∈ D b i } for some fixed ¯ d i ∈ D b i . Since, if ∈ Dthen ∈ D b by definition, we may define the (possibly empty) set ¯D b := D b . Wenote that if σ is a ( b , i )-solution for D then σ ¯D b = 0, where ¯D b := ¯D b ∪ · · · ∪ ¯D b k .Since D has no b -solution, there must be some 1 ≤ i ≤ n for which there isno ( b , i )-solution. However, since σ induces b , Lemma 8.8 implies that σ is a( b , i >
1. Therefore, for some i >
1, there is no R i -positive v ∈ R | S | + orthogonal to M S , where M = ¯D b and S = R + i . Since σ inducesa blocking, σ is strictly positive, and since σ is a ( b , σ isorthogonal to ¯D b ; since further R i ⊆ R + i , by Corollary 8.12 we have that thereexists L ∈ N such that σ ( t ) ≤ L · max { σ ( x ) : x ∈ X } for all t ∈ R i , where X := { , ..., n } \ R + i = R ∪ . . . ∪ R i − .Since σ induces b and since i >
1, we have 0 < σ D b i − < σ D b i . As σ D iscontained in a shift of K N , say K N − C for some C ∈ N , there must be a ≥ b > σ D b i − = K a − C and σ D b i = K a + b − C . Observe that σ ( x ) ≤ K a for all x ∈ X since σ D b j ≤ σ D b i − ≤ K a for each j ≤ i − σ inducing b . Hence σ ( t ) ≤ LK a for all t ∈ X ∪ R i .For d ∈ D b i and d ′ ∈ D b i − , we have supp( { d, d ′ } ) ⊆ X ∪ R i , so K a ( K − ≤ K a ( K b −
1) = σd − σd ′ ≤ σ | d − d ′ | ≤ LK a ∆ { d, d ′ } ≤ LK a ∆D . It follows that K ≤ L ∆D + 1. ⊣ The lemma above completes the proof of Lemma 8.9 and hence Theorem 8.10.Now, if Γ is a finite set of spineless simple equations then each equation in Γmust be non-mingly by Lemma 6.4, and furthermore there must exist a smallest K for which each equation in Γ satisfies ( ⋆⋆K ) as a consequence of Theorem 8.10.Therefore, by Corollary 7.16, we obtain: Corollary . For any finite set of spineless simple equations Γ , everysubvariety of RL containing CRL + Γ has undecidable word problem.
This completes the proof of Theorem 5.5, and therefore also of Theorem 5.9. §
9. Concluding Remarks.
First, we note that the quasiequations used toestablish Theorem 4.5 are in the signature {∨ , · , } , so all complexity lower-bound/undecidability results hold even when restricting the word problem tothe {∨ , · , } -fragments of such varieties. On the other hand, the equations usedto establish Theorem 5.9 make use of the full signature. Corollary . Let ε be a spineless equation that is simple over RL and V a variety of residuated lattices containing CRL ε as a subvariety. Then the word For the reverse direction, if v ∈ R | S | + is orthogonal to ¯ A S , then since ¯ A S has integer entries,by Gaussian Elimination we may assume that v ∈ Q | S | + , and so t · v ∈ Z | S | + for some t ∈ N and t · v is orthogonal to ¯ A S . OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE problem (and hence quasiequational theory) for the {∨ , · , } -fragment of V isundecidable. Furthermore, if ε is expansive then the equational theory for V isundecidable. Given a simple equation ε , by ( ε ) we denote its corresponding sequent-styleinference rule, e.g., if ε : xy ≤ x yx ∨ x ∨
1, thenΓ , ∆ , ∆ , ∆ , ∆ Σ ⊢ Π Γ , ∆ , Σ ⊢ Π Γ , Σ ⊢ ΠΓ , ∆ , ∆ , Σ ⊢ Π ( ε ) . Corollary . Let ε be a spineless simple equation and L any logic con-tained in the interval from FL e + ( ε ) to FL . Then deducibility in the {∨ , · , } -fragment of L is undecidable. Furthermore, if ε is expansive then provability inthe ( -free fragment of ) L is undecidable. In the fol-lowing table we display decidability results for subvarieties of
CRL axiomatizedby 1-variable equations using Lemma 5.4. The numbers n, p, q, ... are distinctand positive, m ≥
0, and furthermore are all given so that ε is not trivial. By(1 ∨ ) we mean 1 may or may not be included in the expression. ε Eq. Th. of
CRL ε Quasi-Eq Th. of
CRL ε x n ≤ x m FMP [17] FEP [17] x n ≤ x m ∨ x n ≤ (1 ∨ ) x p ∨ x q ∨ · · · ? Und. (Thm 5.5) x n ≤ x n + p ∨ x n + q ∨ · · · Und. (Thm 5.9) Und. (Thm 5.5)We note that subvarieties axiomatized by equations of the form x n ≤ x ∨ RL axiomatized bya simple equation in which each term on the right-hand side is linear (i.e., anyvariable occurs at most once in any joinand) has the finite model property (seeTheorem 3.15 in [5]). Similarly, while the subvariety of CRL axiomatized by thesimple equation ε : xyz ≤ xy ∨ xz ∨ yz ∨ x ∨ y ∨ z has an undecidable wordproblem (it is easily verified that ε is spineless), CRL + ε has the FMP.Subvarieties of CRL axiomatized by equations of the form x n ≤ x m +1 ∨
1, thesimplest of which is d : x ≤ x ∨
1, have no known decidability results for their(quasi-)equational theories. Focusing on the equation d , we make the followingobservations: • CRL d does not have the finite embeddability property. In fact, extensionsof CRL by equations of the form x n y m ≤ x n ∨ y m do not have the FEPfor any choice n, m . This follows from the fact that such equations hold inchains and the FEP fails in such extensions of CRL (see [7]). • The quasiequational theory of
CRL d does not have a primitive recursivedecision procedure. This can be shown using Theorem 4.5 and the machineconstructed in [16] (which shows that provability in FL ec , while decidable, isnot primitive recursive). In fact, the same construction can be used to showthat there is no primitive recursive decision procedure for the quasiequa-tional theory of the subvariety of CRL axiomatized by x m ≤ x m + n ( ∨ m, n >
1. Furthermore, by Corollary 5.8 the same holds for the equa-tional theory of the subvariety of
CRL axiomatized by x m ≤ x m + n , as4 NIKOLAOS GALATOS AND GAVIN ST. JOHN this equation is expansive. A more general treatment will be given in aforthcoming paper.
As mentioned above, although the mainreason for our study has been to establish undecidability for commutative vari-eties, we also get results about non-commutative ones. Here we compare thatportion of our results with existing ones. In [8], it is shown that any variety ofresiduated lattices containing (as a subvariety) H = RL + ( x ≤ x ) + ( x ≤ x )has an undecidable word problem witnessed in its {≤ , · , } -fragment; of courseno subvariety of CRL contains H , so this result has no implication about commu-tative varieties. In fact, the algebra W + ∈ H constructed in [8] satisfies everyequation for which the deletion of any collection of its variables results in eithera trivial equation or one with the right-hand side containing a square subterm(i.e., a joinand of the from uv w for u, v, w ∈ X ∗ with v = 1.) As a result,even though Theorem 5.5 covers a lot of commutative varieties, it does not offerany new results for non-commutative varieties axiomatized by 1-variable equa-tions. In fact, the results in [8] entail undecidability of the word problem formany non-commutative extensions of RL by prespinal equations; e.g., RL d hasan undecidable word problem since W + | = x ≤ x ∨ x y ≤ xyx ) is spineless and hencesubvarieties of RL by such equations have an undecidable word problem by The-orem 5.5.More interesting 3-variable basic equations can be obtained by using square-free joinands. Let X = { x, y, z } and let h : X ∗ → X ∗ be the homomorphismextending the assignment h (1) = 1, h ( x ) = xyz , h ( y ) = xz , and h ( z ) = y . Onecan produce square-free words of arbitrary length by considering h ( x ) = x and h k +1 ( x ) = h ( h k ( x )) (see [12]). For any nontrivial 1-variable equation ε : x n ≤ (1 ∨ ) x p ∨ x q ∨ · · · , we denote by h ( ε ) the basic equation: h n ( x ) ≤ (1 ∨ ) h p ( x ) ∨ h q ( x ) ∨ · · · . Since ε is nontrivial, h ( ε ) is nontrivial and furthermore has square-free joinands.Consequently, H 6| = h ( ε ) and so [8] is not applicable to equations of this form.However, if ε is a spineless 1-variable basic equation then it can easily be shownthat h ( ε ) C is also spineless and therefore RL + h ( ε ) has an undecidable wordproblem by Theorem 5.5. E.g., consider the equation ε : x ≤ x ∨ x , then h ( ε ) : xyz ≤ xyzxzy ∨ xyzxzyxyzyxz and h ( ε ) C : xyz ≤ x y z ∨ x y z . It is easily checked using Theorem 8.6 that h ( ε ) C is spineless and hence h ( ε ) isspineless in view of Definition 5.3. Therefore RL + h ( ε ) has an undecidable wordproblem by Theorem 5.5, but H 6| = h ( ε ).While our undecidability results for the word problem in these varieties takesplace in the {∨ , · , } -fragment, we can strengthen such results to the {≤ , · , } -fragment for many non-commutative extensions of residuated lattices (even bysome prespinal equations) using a different encoding not relying on the ∨ oper-ation. Such ideas will be explored in a forthcoming paper. A word w ∈ X ∗ is square-free if w = ux v for any words u, v, x ∈ X ∗ and x = 1. OST SIMPLE EXTENSIONS OF FL e ARE UNDECIDABLE REFERENCES [1]
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DEPARTMENT OF MATHEMATICSUNIVERSITY OF DENVERDENVER, CO, 80210, USA
E-mail : [email protected]
DEPARTMENT OF MATHEMATICSUNIVERSITY OF DENVERDENVER, CO, 80210, USA