1 The classical decision problem for FO

(1)

Algorithmic Model Theory SS 2010

Prof. Dr. Erich Grädel

Mathematische Grundlagen der Informatik RWTH Aachen

(2)

c b n d

This work is licensed under:

http://creativecommons.org/licenses/by-nc-nd/3.0/de/

Dieses Werk ist lizenziert unter:

http://creativecommons.org/licenses/by-nc-nd/3.0/de/

http://www.logic.rwth-aachen.de

1 The classical decision problem for FO

The classical decision problem for first-order logic was considered the main problem of mathematical logic by Hilbert and Ackermann and its undecidability was shown by Church and Turing.

The Entscheidungsproblem is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability. [. . . ] The Entscheidungsproblem must be considered the main problem of mathematical logic.

(D. Hilbert and W. Ackermann, 1928) We introduce the classical decision problem for first-order logic, for which we present three equivalent formulations. The importance of the decision problem for first-order logic results from the fact that first-order logic provides a framework to express almost all aspects of mathematics.

Satisfiability: Construct an algorithm that decides for any given formula of FO whether it has a model.

Validity: Construct an algorithm that decides for any given formula of FO whether it is valid, i.e. whether it holds in all models where it is defined.

Provability: Construct an algorithm that decides for any given formula ψof FO whether ⊢ψ, meaningψis provable from the empty set of axioms in some formal system, e.g. sequential calculus.

Sinceψis satisfiable if and only if¬ψis not valid, satisfiability and validity are equivalent problems with respect to computability. The equivalence with provability is a much more intricate result and in fact a consequence of the following

(4)

1 The classical decision problem forFO

Theorem 1.1(Completeness Theorem (Gödel)). For any given set of sentencesΦ⊆FO(τ)and any sentenceψ∈FO(τ)it holds that

Φ|=ψ ⇐⇒ Φ⊢ψ; in particular∅|=ψ⇔∅⊢ψ.

As a direct consequence we get the following

Theorem 1.2. The set of valid first-order formulae is recursively enumerable.

1.1 Basic notions on decidability

In our formulation of the decision problem it was not precisely specified what an algorithm is. It was not until the 1930s that Church and Kleene , Gödel and Turing provided a precise definition of an abstract algorithm.

Their approaches are today known to be equivalent. We introduce the concept of a Turing machine.

Definition 1.3. ATuring machine(TM)Mis a 6-tuple M= (Q,Σ,Γ,q0,F,δ), where

• Qdenotes a finite set of states,

• Σ,Γdenote finite alphabets, whereΣis the working alphabet with a special blank symbol∈Σ,

• Γ⊆Σ\ {}is the input alphabet,

• q₀∈Qdenotes the initial state,

• F⊆Qis the set of final states and

• δ:(Q\F)×Σ→Q×Σ× {−1, 0, 1}is the transition function.

Aconfigurationis an elementC= (q,p,w=w0w₁. . .w_k)∈Q×^N×Σ^∗. The transition functionδ induces a partial function on the set of all configurations

C7→Next(C),

where forδ(q,wp) = (q^′,a,m), the successor configuration of C is defined as Next(C) = (q^′,p+m,w0. . .w_p−1aw_p+1· · ·w_k). Acom- putation of the TM M on an input word x ∈ Γ^∗ is a configuration

sequence C0,C₁, . . .

whereC0=C0(x) := (q0, 0,x)is the input configuration andC_i+1 = Next(Ci)for alli.

M haltsonxif the computation ofMonxis finite, i.e. ends in a final configurationCf = (q,p,w)withq∈F.

The language accepted byMis L(M):={x∈Γ^∗ :Mhalts onx}.

Mcomputes a partial function f_M:Γ^∗ →Σ^∗ with domainL(M) such thatfM(x) =yif and only if the computation ofMonxends in (q,p,y)for someq∈F,y∈Σ^∗andp∈N.

Definition 1.4. A Turing acceptor is a Turing machine M with F = F⁺ ·∪F⁻whereM accepts xif the computation ofMonxends in a state inF⁺. M rejects xif the computation ofMonxends in a state inF⁻. Definition 1.5.

• L⊆Γ^∗ isrecursively enumerable (r.e.) if there exists a TMMwith L(M) =L.

• L⊆Γ^∗ isco-recursively enumerable (co-r.e.)ifL:=Γ^∗\Lis r.e..

• A (partial) function f:Γ^∗→Σ^∗is(Turing) computableif there is a TMMwith f_M= f.

• L⊆Γ^∗ isdecidableif there is a Turing acceptorMsuch that for all x∈Γ^∗

x∈L⇒Macceptsx x∈/L⇒Mrejectsx

or, equivalently,Lis decidable if its characteristic function χL:Γ^∗→ {0, 1}is Turing computable.

Theorem 1.6. A language L ⊆Γ^∗ is decidable if and only if Lis r.e.

and co-r.e.

(5)

Definition 1.7. Let A ⊆ Γ^∗,B ⊆ Σ^∗. We say that Ais(many-to-one) reducibletoB,A≤B, if there is a total computable functionf :Γ^∗→Σ^∗ such that for allx∈Γ^∗we havex∈A⇔ f(x)∈B.

Lemma 1.8.

• A≤B,Bdecidable⇒Adecidable

• A≤B,Br.e.⇒Ar.e.

• A≤B,Aundecidable⇒Bundecidable.

There surely are undecidable languages since there are only count- ably many Turing machines but uncountably many languages. Un- fortunately, among these languages there are quite relevant classes of languages. For example we cannot even decide whether a TM halts on a given input.

Definition 1.9(Halting Problems). Thegeneral halting problemis defined as

H:={ρ(M)#ρ(x):MTuring machine,x∈L(M)}

whereρ(M)andρ(x)are encodings of the TMMand the inputxover a fixed alphabet{0, 1}such that the computation of Mon xcan be reconstructed from the encodingsρ(M)andρ(x)in an effective way.

There is a universal TMUwhich, givenρ(M)andρ(x), simulates the computation ofMonxand halts if and only ifMhalts onx. Thus, L(U) =Hfrom which we conclude thatHis r.e..

We introduce two special variants of the halting problem

• Self-application problem

H₀:={ρ(M):ρ(M)∈L(M)}

• Halting on the empty word Hε:={ρ(M):ε∈L(M)} Theorem 1.10. H,H0,Hεare undecidable.

Proof.

• H₀is not co-r.e. and thus undecidable. OtherwiseH₀=L(M₀)for some TMM0. Then

ρ(M0)∈H0⇔M0halts onρ(M0)⇔ρ(M0)∈H0.

• H₀is a special case ofH,H₀≤H, and thusHis undecidable.

• We can reduceHtoHε, thusHεis undecidable. q.e.d.

As a consequence of the next theorem we cannot algorithmically prove whether a program computes a given function, i.e. we cannot algorithmically prove the correctness of a program. Note that this does not mean that we cannot prove the correctness of a single given program. Instead the statement is that we cannot do so algorithmically for all programs.

Theorem 1.11(Rice). LetRbe the set of all computable functions and letS⊆ Rbe a set of computable functions such thatS̸=∅andS̸=R. Then code(S):={ρ(M): fM∈S}is undecidable.

Proof. Let⇑be the everywhere undefined function, i.e. Def(⇑) =∅.

Obviously,⇑is computable. Assume that⇑̸∈S (otherwise consider R \Sinstead of S. Clearly if code(R \S) is undecidable then so is code(S).)

AsS ̸= ∅, there exists a function f ∈ S. Let M_f be a TM that computes f, i.e. f_M_f = f. We define a reduction Hε ≤ code(S) by describing a total computable functionρ(M)7→ρ(M^′)such that

Mhalts onε⇔ fM^′ ∈S.

Specifically, givenρ(M), we construct the encoding of a TMM^′which, given an inputx, proceeds as follows:

• first simulateMonε(i.e. apply the universal TMUtoρ(M)#ε);

• then simulate M_f on x (i.e. apply the universal TM U to ρ(M_f)#ρ(x)).

(6)

It is clear that the reduction function is computable. Furthermore, ifMhalts onεthen fM^′(x) = f(x)for all inputs x, i.e. fM^′ = f, so f_M′ ∈S. IfMdoes not halt onεthenM^′does not halt onxfor anyx, i.e. f_M^′ =⇑, so f_M^′ ̸∈S. q.e.d.

Definition 1.12(Recursive inseparability). LetA,B⊆Γ^∗be two disjoint sets. We say that AandBarerecursively inseparableif there exists no recursive setC⊆Γ^∗such thatA⊆CandB∩C=∅.

Example. (A,A)are recursively inseparable if and only ifAis undecidable.

Lemma 1.13. LetA,B⊆Γ^∗,A∩B=∅be recursively inseparable. Let X,Y⊆Σ^∗,X∩Y=∅, and let f be a total computable function such thatf(A)⊆Xand f(B)⊆Y. ThenXandYare recursively inseparable.

Proof. Assume there exists a decidable set Z ⊆Σ^∗ such thatX ⊆ Z andY∩Z =∅. ConsiderC ={x∈ Γ^∗ : f(x) ∈ Z}. Cis decidable, A⊆C,B∩C=∅, thusCseparatesA,B. q.e.d.

Notation:We write(A,B)≤(X,Y)if such a function f exists.

Example. (A,A)≤(B,B)⇔A≤B.

As a preparation to prove Trakhtenbrot’s theorem, we consider a refinement ofHε

H_ε⁺:={ρ(M):Macceptsε} H_ε⁻:={ρ(M):Mrejectsε}

H_ε^∞:={ρ(M): the computation ofMonεis infinite and does not cycle.}

H₀⁺, H₀⁻, H₀^∞ are defined analogously, with respect to self- application.

Theorem 1.14. H⁺_ε ,H_ε⁻andH_ε^∞are pairwise recursively inseparable.

Proof.

• (H_ε⁺,H_ε^∞): We show that every setCwithH_ε⁺⊆CandH_ε^∞∩C=

∅is undecidable by reducing the halting problemHεtoC. Define the functionρ(M)7→ρ(M^′)as follows. From a given codeρ(M) construct the code of a TMM^′that simulatesMand simultaneously counts the number of computation steps since the start. IfMhalts (accepting or rejecting),M^′accepts.

It is clear that the reduction function is computable. IfMhalts onε thenM^′halts onεas well and accepts, soρ(M^′)∈H_ε⁺⊆C. IfM does not halt on εthenM^′ does not halt either, soρ(M^′)∈ H_ε^∞ and asH^∞_ε ∩C=∅, we haveρ(M^′)̸∈C.

• The statement forH_ε⁻andH_ε^∞is proven analogously.

• (H_ε⁻,H_ε⁺): Show that(H₀⁻,H⁺₀)≤(H_ε⁻,H_ε⁺)and that(H₀⁻,H₀⁺) are recursively inseparable.

– (H₀⁻,H⁺₀)≤(H_ε⁻,H_ε⁺):

For a given input TMMconstruct a TM M^′that ignores its own input and simulatesMonρ(M). Obviously,M^′can be constructed effectively, say by a computable functionh. Now h(M)acceptsεiffMacceptsρ(M) andh(M) rejectsεiff M rejectsρ(M).

– (H₀⁻,H⁺₀)recursively inseparable:

Assume there exists a decidableCwithH₀⁻⊆CandH₀⁺⊆C.

Consider a machineM₀that decidesC. There are two cases:

(1) M0acceptsρ(M0). Thenρ(M0)∈Cby definition ofM0. Thenρ(M₀)̸∈H₀⁺by definition ofC. On the other hand, ifM0acceptsρ(M0)thenρ(M0)∈ H₀⁺ (by definition of H₀⁺), a contradiction.

(2) M0 rejectsρ(M0). Thenρ(M0)̸∈Cby definition ofM0. Thenρ(M₀)̸∈H₀⁻by definition ofC. On the other hand, ifM0rejectsρ(M0)thenρ(M0) ∈ H⁻₀ (by definition of H₀⁻), a contradiction.

q.e.d.

(7)

1.2 Trakhtenbrot’s Theorem

In the following, we consider FO, more precisely first-order logic with equality. We restrict ourselves to a countable signature

τ∞:={Rⁱ_j:i,j∈^N} ∪ {f_jⁱ:i,j∈^N}

whereRⁱ_j stands for a relation symbol of arityiand f_jⁱ stands for a function symbol of arityi.

We encode formulae over a fixed alphabet Γ:={R,f,x, 0, 1,[,]} ∪ {=,¬,∧,∨,→,↔,∃,∀.(,)}, and uniquely encode the relational and functional symbols

relation symbols: Rⁱ_j 7−→ R[bini][binj] functional symbols: f_jⁱ 7−→ f[bini][binj]

variables: x_j 7−→ x[binj].

Thus, every formulaϕ∈FO is a word inΓ^∗.

Let X ⊆ FO be a class of formulae. We analyse the following decision problems:

Sat(X):={ψ∈X:ψhas a model} Fin-sat(X):={ψ∈X:ψhas a finite model}

Val(X):={ψ∈X:ψis valid} Non-sat(X):=X\Sat(X)

Inf-axioms(X):=Sat(X)\Fin-sat(X)

={ψ∈X: ψis an infinity axiom, i.e.ψhas a model but no finite model}. Theorem 1.15. LetX⊆FO be decidable. Then

(1) Val(X)is r.e.

(2) Non-sat(X)is r.e.

(3) Sat(X)is co-r.e.

1.2 Trakhtenbrot’s Theorem (4)Fin-sat(X)is r.e.

(5)Inf-axioms(X)is co-r.e.

Proof. (1) ϕis valid⇔ ⊢ϕ(Completeness Theorem). Thus we can systematically enumerate all proofs and halt if a proof for ϕ is listed.

(2) ϕvalid⇔ ¬ϕis not satisfiable.

(3) Follows from Item (2).

(4) Systematically generate all finite models and halt if a model ofϕis found.

(5) FO\Inf-axioms(X) =Non-sat(X)∪Fin-sat(X)is r.e. q.e.d.

Definition 1.16. A classX⊆FO has thefinite model property(FMP) if every satisfiableϕ∈Xhas a finite model, i.e. ifSat(X) =Fin-sat(X). Theorem 1.17. Suppose thatX⊆FO is decidable and thatXhas the FMP. ThenSat(X)is decidable.

Proof. Sat(X)is co-r.e. and sinceSat(X) =Fin-sat(X)andFin-sat(X)is r.e. alsoSat(X)is r.e. ThusSat(X)is decidable. q.e.d.

In this case alsoFin-sat(X),Non-sat(X),Val(X)are decidable and of courseInf-axioms(X) =∅is decidable.

Theorem 1.18 (Trakhtenbrot). There is a finite vocabulary τ ⊆ τ∞

such that Fin-sat(FO(τ)),Non-sat(FO(τ)) and Inf-axioms(FO(τ)) are pairwise recursively inseparable and therefore undecidable.

The proof of Trakhtenbrot’s theorem introduces a proof strategy that can be applied in many other undecidability proofs. (Do not focus on the technicalities but on the general idea to construct the reduction formulae.)

Proof. LetMbe a deterministic Turing acceptor. We show that there is an effective reductionρ(M)7→ψ_Msuch that

(1) Macceptsε =⇒ ψ_Mhas a finite model.

(2) Mrejectsε =⇒ ψMis unsatisfiable.

(8)

(3) The computation ofMonεis infinite and non-periodic =⇒ ψ_M is an infinity axiom.

Then the theorem follows by Lemma 1.13.

LetMbe a Turing acceptor with statesQ={q0, . . . ,qr}, initial state q0, alphabetΣ={a0, . . . ,as}(wherea0=), final statesF=F⁺∪F⁻ and transition functionδ.

ψ_Mis defined over the vocabularyτ={0,f,q,p,w}where 0 is a constant, f,q,pare unary functions andwis a binary function. Define the termkas f^k0.

By constructing a formula we intend to have a model A_M = (A, 0,f,q,p,w)describing a run ofMon the inputεwhere

• universeA={0, 1, 2, . . . ,n}orA=N;

• f(t) =t+1 ift+1∈Aandf(t) =t, iftis the last element ofA;

• q(t) =iiffMis at timetin stateqi;

• p(t)is the head position ofMat timet;

• w(s,t) =iiff symbola_iis at timeton tape-cells.

Note that we cannot enforce this model, but ifψMis satisfiable this one will be among its models.

ψ_M:= START ∧ COMPUTE ∧ END START := (q0=0∧p0=0∧ ∀x w(x, 0) =0).

[Enforces input configuration onεat time 0]

COMPUTE :=NOCHANGE∧ CHANGE

NOCHANGE :=∀x∀y(py̸=x→w(x,f y) =w(x,y))

[content of currently not visited tape cells does not change]

CHANGE := ^{^}

δ:(qi,aj)7→(qk,aℓ,m)

∀y(α_i,j→β_k,ℓ,m) where

α_ij:= (qy=i∧w(py,y) =j)

[Mis at timeyin stateqiand reads the symbolaj] β_k,ℓ,m:= (q f y=k∧w(py,f y) =ℓ∧MOVEm)

1.2 Trakhtenbrot’s Theorem

and

MOVEm:=









p f y=py ifm=0 p f y= f py ifm=1

∃z(f z=py∧p f y=z) ifm=−1.

END := ^{^}

δ(qi,aj)undef.

qi̸∈F⁺

∀y¬α_ij

[The only way the computation ends is in an accepting state]

Remark1.19.

• ρ(M)7→ψ_Mis an effective construction.

• IfMacceptsε, the intended model is finite and is indeed a model A_M|=ψM, thusψM∈Fin-sat(FO(τ)).

• If the computation of Mon ε is infinite, the intended model is infinite andA_M|=ψ_M.

It remains to show that ifMrejectsε, thenψ_Mis unsatisfiable, and if the computation ofMonεis infinite and aperiodic, thenψM is an infinity axiom.

SupposeB= (B, 0,f,q,p,w)|=ψM.

Definition 1.20. Benforces at timet the configuration (q_i,j,w) with w=a_i₀. . .a_i_m ∈Σ^∗if

(1)B|=qt=i, (2)B|=pt=j,

(3) for allk≤m,B|=w(k,t) =ikand for allk>m,B|=w(k,t) =0.

SinceB|=ψM, the following holds:

• BenforcesC₀= (q₀, 0,ε)at time 0 (sinceB|= START.)

• IfBenforces at timeta non-final configurationCt, thenBenforces the configurationC_t+1=Next(Ct)at timet+1.

• Especially, the computation ofMcannot reach a rejecting configuration. It follows that ifMrejectsε, thenψMis unsatisfiable.

(9)

Consider an infinite and aperiodic computation ofM, and assume B|=ψMis finite. SinceBis finite, it enforces a periodic computation in contradiction to the assumption that the computation ofM is aperiodic.

C0⊢. . .⊢ Cr ⊢. . .⊢ ^Ct−1

We have shown:

• IfMacceptsε, thenψ_Mhas a finite model.

• IfMrejectsε, thenψ_Mis unsatisfiable.

• If the computation ofMis infinite and aperiodic, thenψMis an

infinity axiom. q.e.d.

We now know that the sets of all finitely satisfiable, all unsatisfiable and all only infinitely satisfiable formulae are undecidable for FO(τ) whereτconsists of only three unary functions and one binary function.

This raises a number of questions.

(1) For which other vocabulariesσdo we have similar undecidability results for FO(σ)?

(2) For whichσis satisfiability of FO(σ)decidable?

(3) Is there a complete classification? In this case, we want to find min- imal vocabulariesσsuch that the above problems are undecidable, i.e. vocabularies such that any further restriction yields a class of formulae for which satisfiability is decidable.

We first define what it means that a fragment of FO is as hard for satisfiability as the whole FO.

Definition 1.21. X⊆FO is areduction classif there exists a computable function f : FO→Xsuch thatψ∈Sat(FO)⇔ f(ψ)∈Sat(X).

LetX,Y⊆FO. Aconservative reduction of X to Yis a computable function f :X→Ywith

• ψ∈Sat(X)⇔ f(ψ)∈Sat(Y), and

• ψ∈Fin-sat(X)⇔ f(ψ)∈Fin-sat(Y).

1.2 Trakhtenbrot’s Theorem Xis aconservative reduction classif there exists a conservative reduction of FO toX.

Corollary 1.22. Let X be a conservative reduction class. Then Fin-sat(X),Inf-axioms(X)andNon-sat(X)are pairwise recursively inseparable, and thus Fin-sat(X), Sat(X),Val(X),Non-sat(X),Inf-axioms(X) are undecidable.

Proof. A conservative reduction from FO toXyields a uniform reduction fromFin-sat(FO), Inf-axioms(FO) and Non-sat(FO) toFin-sat(X), Inf-axioms(X)andNon-sat(X), respectively. q.e.d.

We now observe that we can indeed give a complete classification of signaturesσsuch that FO(σ)is decidable.

Theorem 1.23. If σ ⊆ {P0,P₁, . . .} ∪ {f} consists of at most one unary function f and an arbitrary number of monadic relations P₀,P₁, . . ., thenSat(FO(σ))is decidable. In all other cases,Sat(FO(σ)), Inf-axioms(FO(σ))andNon-sat(FO(σ))are pairwise recursively inseparable, and FO(σ)is a conservative reduction class.

A full proof of this classification theorem is rather difficult. In particular, the decidability of the monadic theory of one unary function, which implies the decidability part, is a difficult theorem due to Rabin.

On the other side, one has to show that Trakhtenbrot’s theorem applies to the vocabularies

τ₁={E}whereEis a binary relation, τ₂={f,g}where f,gare unary functions, τ3={F}whereFis a binary function, and hence to all extensions ofτ₁,τ₂,τ₃.

Of course, we may also look at other syntactic restrictions besides restricting the vocabulary. One possibility is to restrict the number of variables. This is only interesting for relational formulae. If we have functions, satisfiability is undecidable even for formulae with only one variable as we shall see.

Define FO^kas first-order logic with relational symbols only and a fixed amount ofkvariables, sayx1, . . . ,x_k.

(10)

Theorem 1.24.

• FO²has the finite model property and is decidable (see Chapter 2).

• FO³is a conservative reduction class.

Another possibility is to restrict the structure of quantifier prefixes.

Definition 1.25(Prefix-Vocabulary Classes). A string in{∀,∃}^∗is called prefix, and anarity sequenceis a sequence assigning all positive integers values inN∪ {ω}.

For any set of prefixesΠand any arity sequencespand f,[Π,p,f] and[Π,p,f]=denote the collection of all formulae ϕ∈FO in prenex normal form without equality and with equality, respectively, such that

• the prefix ofϕbelongs toΠ,

• the number ofn-ary predicate symbols inϕis at mostp(n)and

• the number ofn-ary function symbols inϕis at most f(n).

• Except for the logical constantstrue and false, ϕ has no nullary predicate symbols, no nullary function symbols and no free variables.

The prefix set containing all prefixes and the arity sequence that assigns ωto eachnwill be denotedall.

We write arity sequences as tuples, e.g.,(2, 1,ω),(0)to express that two predicate symbols of arity 1, one of arity 2, unboundedly many of arity 3 and no other predicate or function symbols are allowed.

Theorem 1.26 (Gurevich). Let X be a prefix class, p,q two arity sequences andX= [Π,p,q]₌.

• Xis a conservative reduction class if it contains any of (1) [∀,(0),(2)]₌

(2) [∀,(0),(0, 1)]=

(3) [∀²∃,(ω, 1),(0)]=

(4) [∃^∗∀²∃,(0, 1),(0)]=

(5) [∀²∃^∗,(0, 1),(0)]=.

• IfXis contained in one of the following classes, thenSat(X)and Inf-axioms(X)are decidable

1.3 Domino problems (6) [∃^∗∀^∗,all,(0)]₌

(7) [∃^∗,all,all]₌ (8) [all,(ω),(1)]₌ (9) [∃^∗∀∃^∗,all,(1)]₌.

This gives a complete classification.

1.3 Domino problems

Domino problems are a simple and yet general tool for proving undecidability without talking about Turing machines.

The informal idea is the following: a domino (type) is an oriented square with unit length and coloured edges. We consider the following decision problem.

Given: a finite set of domino types (infinite supply of each).

Question: does there exist a tiling ofN×^Nsuch that adjacent edges have the same colour?

The undecidability of the stated problem is established by encoding computations of Turing machines in an appropriate way. A row of the tiling represents a configuration of a Turing machine.

Definition 1.27. Adomino systemis a structureD= (D,H,V)with

• a finite setD,

• horizontal and vertical compatibility relationsH,V⊆D×D.

The meaning ofHandVis that

• (d,d^′)∈Hif the right colour ofdis equal to the left colour ofd^′,

• (d,d^′)∈Vif the top colour ofdis equal to the bottom colour ofd^′ (see Figure 1.1).

A tiling ofN×^NbyDis a functionσ:N×^N→Dsuch that for all x,y∈^N

• (σ(x,y),σ(x+1,y))∈Hand

• (σ(x,y),σ(x,y+1))∈V.

(11)

A periodic tiling ofN×^NbyDis a tilingσfor which two integers h,v∈^Nexist such that for allx,y∈^Nit holdsσ(x,y) =σ(x+h,y) = σ(x,y+v). The decision problem DOMINO is described as

DOMINO :={D: there exists a tiling ofN×^NbyD}

a b

c d

c

•

• b

Figure 1.1.Domino adjacency condition

An important variant is the origin constrained tiling.

Definition 1.28. Anorigin constrained domino systemis a system(D,D0) withD₀⊆D. A tiling with origin constraintD₀is a tilingσsuch that σ(0, 0)∈D0. The corresponding decision problem is

CORNER-DOMINO :={(D,D0): there exists a tiling ofN×^N with origin constraintD0}. Theorem 1.29(Wang, Büchi). CORNER-DOMINO is undecidable.

Proof. We reduceH_ε^ω={ρ(M): the computation ofMonεis infinite}, which is co-r.e., to CORNER-DOMINO.

Consider a 1-tape TMM= (Q,Σ,q₀,δ,F), and construct(D,D₀) such that the computation ofMonεis infinite if and only if there exists a tiling ofN×^NbyDwith origin constraintD0.

Assume w.l.o.g. that M never moves off-tape to the left, i.e. in configurations(q, 0,w)it is never the case thatδ(q,w0) = (q^′,a,−1).

1.3 Domino problems Dconsists of the following domino types.

For eacha∈Σ ↔

a

↔ a

For each(q,a)∈Q×Σwith

δ(q,a) = (q^′,a^′, 0) ↔

(q^′,a^′)

↔ (q,a)

For each(q,a)∈Q×Σwith δ(q,a) = (q^′,a^′, 1), for each b∈Σ

↔ a^′

(q,a) (q,a)

(q,a) (q^′,b)

↔ b

For each(q,a)∈Q×Σwith δ(q,a) = (q^′,a^′,−1)for each b∈Σ

↔ (q^′,b)

(q,a) b

(q,a) a^′

↔ (q,a)

Additionally there exist

dominoes ⊢

(q0,)

↔0

⊥

↔0

⊥

The origin constraintD0con-

sists of ⊢

(q0,)

↔0

⊥ Note that(D,D₀)can be constructed effectively fromM.

There is precisely one way of tiling the first row:

(12)

⊢ (q0,)

↔0

⊥ σ(0, 0)

↔0

⊥

σ(0,i) for alli>0

· · ·

Assume the firstjrows have been tiled correctly. Then the top edge of rowjreads

w0. . .wi−1(q,wi)wi+1. . .

forCj= (q,i,w0,w1, . . .), thejth configuration ofMonε.

This tiling can be extended to a tiling of rowj+1 if and only if there existsCj+1=Next(Cj).

Conclusion: The computation ofMonεis infinite if and only if there exists a tiling ofN×^Nby(D,D0). q.e.d.

Stronger forms of this result are the following

Theorem 1.30(Berger, Robinson). DOMINO (without origin constraint) is co-r.e. and undecidable.

Theorem 1.31. The problem of tilingZ×^Zis reducible to the problem of tilingN×^N. (Proof via König’s Lemma).

Theorem 1.32. The set of domino systems admitting a periodic tiling ofN×^N, those that admit no tiling ofN×^Nand those that admit a tiling but not a periodic one are pairwise recursively inseparable.

Definition 1.33. A computable function f is a reduction from domino systemstoXif, for all domino systemsD, f(D) =ϕ_D is inXand the following holds:

• Dadmits a periodic tiling ofN×^N⇒ψ_Dhas a finite model

• Dadmits no tiling ofN×^N⇒ψ_D is unsatisfiable

• Dadmits a tiling ofN×^Nbut no periodic one⇒ψ_Dis an infinity axiom.

1.4 Applications of the domino method Remark1.34. Let X ∈ FO. If there exists a reduction from domino systems toXthenXis a conservative reduction class.

Proof. SinceFin-sat(FO) andNon-sat(FO)are recursively enumerable andInf-axioms(FO)is co-recursively enumerable, we can associate with every first-order formulaψa Turing machineMsuch that

• ψ∈Fin-sat(FO)⇒ρ(M)∈H⁺_ε ,

• ψ∈Non-sat(FO)⇒ρ(M)∈H_ε⁻,

• ψ∈Inf-axioms(FO)⇒ρ(M)∈H_ε^∞.

The proof of 1.32 reduces the halting problemsH_ε⁺,H⁻_ε ,H_ε^∞, to the domino problems. There exists a recursive function that associates with every TMMa domino systemDsatisfying

• IfM∈H_ε⁺thenDadmits a periodic tiling ofN×^N.

• IfM∈H_ε⁻thenDadmits no tiling ofN×^N.

• IfM∈H_ε^∞thenDadmits a tiling ofN×^Nbut no periodic one.

Finally, according to to the assumption, there is a reduction D 7→

ϕ_D from domino systems to X Thus, the domino method yields a conservative reduction from FO toX.

q.e.d.

1.4 Applications of the domino method

We now apply the domino method to obtain several reduction classes.

Theorem 1.35. [∀∃∀,(0,ω),(0)]is a conservative reduction class.

Proof. Due to Remark 1.34 it suffices to give a reduction from domino systems toX, i.e. find a mappingD 7→ψ_Dover a vocabulary consisting of binary relation symbols(P_d)_d∈Dsuch that

(1) Dadmits a periodic tiling ofN×^N⇒ψ_D has a finite model (2) Dadmits no tiling ofN×^N⇒ψ_Dis unsatisfiable

(3) Dadmits a tiling ofN×^Nbut no periodic one⇒ψ_Dis an infinity axiom

(13)

The intended model isNwith intended interpretation of P_d = {(i,j)∈^N×^N:τ(i,j) =d}for alld∈D. We defineψ_Dby

ψ_D:=∀x∃y∀z _^

d̸=d^′

P_dxz→ ¬P_d′xz

∧ ^_

(d,d^′)∈H

(Pdxz∧Pd^′yz)∧ ^_

(d,d^′)∈V

(Pdzx∧Pd^′zy).

Obviouslyψ_Dis of the desired format, i.e.ψ_D∈[∀∃∀,(0,ω),(0)]. (1) IfDadmits a periodic tiling ofN×^N, thenψ_D has a finite model.

Letτ:N×^N→Dbe a periodic tiling such that for someh,v∈^N τ(x,y) =τ(x+h,y) =τ(x,y+v)for allx,y. Lett:=lcm(h,v)be the least common multiple ofhandv. Thenτinduces a tiling

τ:Z/tZ×^Z/tZ→D

withτ^′(x,y) =τ(x( mod t),y( mod t)).

Thus, A= (Z/tZ,(P_d)_d∈D)with P_d ={(i,j) : τ^′(i,j) = d}is a finite model (forxinψ_Dchoosey:=x+1 (modt)inψ_D.) (2) Ifψ_Dhas a model, thenDadmits a tiling.

(3) We want to show: ifψ_Dhas a finite model, thenDadmits a periodic tiling. (In the case thatψ_D is unsatisfiable, we show with the same arguments as in(1)that ifDadmits a tiling ofN×^N, thenψ_D has a modelA= (N,(P_d)_d_∈_D).)

Let nowψ_Dhave a finite model. To show that ifψ_Dhas a (finite) model, thenDadmits a (periodic) tiling we consider the Skolem normal formϕ_D ofψ_D:

ϕ_D:=∀x∀z(^{^}

d̸=d^′

Pdxz→ ¬Pd^′xz

∧ ^_

(d,d^′)∈H

(P_dxz∧P_d′f xz)∧ ^_

(d,d^′)∈V

(P_dzx∧P_d′z f x).

• SupposeB= (B,f,(P_d)_d∈D)|=ϕ_D. Define a tilingτ:N×^N→ D as follows: chooseb ∈ B, and setτ(i,j) := dfor the unique

1.4 Applications of the domino method d∈Dsuch thatB|=P_d(fⁱb,f^jb)for alli,j∈^N. SinceB|=ϕ_D,τ is a correct tiling.

• Suppose thatB|=ϕ_Dis finite:

• _f • f b· · · • f

Chooseb∈Bsuch that, for somet≥1, f^tb=b. Then the defined tilingτis periodic.

q.e.d.

Corollary 1.36. FO³is a conservative reduction class.

Later we show thatFO²has the FMP.

Consider sets of formulaeX ⊆FO over functional vocabularies.

FO(τ)is a conservative reduction class ifτcontains

• two unary functions or

• one binary function.

This is even true for sentences of the form∀xϕwhereϕis quantifier- free.

Theorem 1.37. [∀,(0),(2)]₌and[∀,(0),(0, 1)]₌ are conservative reduction classes.

Proof. We apply the domino method for formulae ∀xϕ where ϕ is quantifier-free with any number of unary functions, and then apply a reduction/interpretation to reduce this to two unary/one binary function/s.

Define a mappingD= (D,H,V) 7→ ψ_D whereψ_D is a formula over the vocabulary {f,g,(h_d)_d∈D} where all function symbols are unary. The intended model isN×^Nwith successor functions f andg.

The subformula∀x(f gx=g f x)ensures that the models ofψ_D contain a two-dimensional grid. The fact that a positionxis tiled byd∈Dis

(14)

expressed by requiring thath_dx=x, i.e. thatxis a fixed point ofh_d. Now define

ψ_D:=∀x f gx=g f x∧ ^{^}

d̸=d^′

(h_dx=x→h_d^′x̸=x)

∧ ^_

(d,d^′)∈H

(h_dx=x∧h_d′f x= f x)

∧ ^_

(d,d^′)∈V

(h_dx=x∧h_d^′gx=gx).

We claim that there exists a tilingσ:N×^N→ Dif and only if ψ_D is satisfiable.

”⇒” Assumeσis a correct tiling. Construct the (intended) model A= (N×^N,f,g,(h_d)_d∈D)with

– f(i,j) = (i+1,j), – g(i,j) = (i,j+1), – hd(i,j)





= (i,j) ifσ(i,j) =d

̸= (i,j) otherwise.

ClearlyA|=ψ_D.

”⇐” ConsiderB= (B,f,g,(hd)_d_∈D)|=ψ_D. Choose an arbitraryb∈Band define

σ:N×^N→ D:σ(i,j):=diffB|=h_dfⁱg^jb= fⁱg^jb.

Note that every position is in exactly one of the h_d. Thenσis a correct tiling. IfB is finite, then σis periodic, and thus the reduction is conservative.

We now show that we can sharpen the results, i.e. show that two unary function symbols are sufficient

Consider∀xϕ∈[∀,(0),(ω)]=with monadic function symbols f₁, . . . ,fm. Transformϕinto ˜ϕ:=ϕ[x/hx,f_i/hgⁱ]whereh,gare fresh unary function symbols. This procedure transforms formulae over the vocabulary{f1, . . . ,fm}into formulae over the vocabulary{h,g}. The idea is to replace an application of f_ibyiapplications ofg. The second functionhtakes care of unwanted equalities.

1.4 Applications of the domino method x

• • · · · •

f1 f2 fm

x hx

• • · · · • •_g_m (hx) h

g

g g

Claim: ∀xϕis (finitely) satisfiable⇔ ∀xϕ˜is (finitely) satisfiable.

”⇐” LetB= (B,h,g)|=∀xϕ. Construct˜ A= (A,f1, . . . ,fm)with – A={hb:b∈B}

– fi(a) = (hgⁱ)(a) ThenA|=∀xϕ.

”⇒” LetA= (A,f1, . . . ,fm)|=∀xϕ. ConstructB= (B,g,h)with – B=A× ^Z/(m+1)Z

, – g(a,i) = (a,i+1), – h(a, 0) = (a, 0), – h(a,i) = (fia, 0).

This transformation preserves the meaning of terms: Lett(x) = fi1. . .fikxbe a term inϕ. Then ˜t(x) =hgⁱ¹. . .hgⁱ^khx, and it holds that ˜t^B[a, 0] = (t^A[a], 0). Now the claim follows via induction over the structure ofϕ.

We now show that we need at most one binary function. The idea is to find an interpretation ofg,h: A→ Ain a structureA= (A,F) withF:A×A→Avia

• g(a) =F(a,F(a,a)),

• h(a) =F(F(a,a),a)

(15)

1 The classical decision problem forFO whereF(a,a)̸=a.

Formally, consider a formula∀xϕwith unary function symbols f,g. Introduce a new binary function symbolFand translate

ϕ7→ϕg∧ϕ_h where

ϕg:=ϕ[x/g^∗x,g/g^∗,h/h^∗], ϕ_h:=ϕ[x/h^∗x,g/g^∗,h/h^∗] with

g^∗t=F(t,Ftt), h^∗t=F(Ftt,t).

Claim: ∀xϕ(finitely) satisfiable ⇔ ∀x(ϕg∧ϕ_h) (finitely) satisfiable.

”⇒” LetA= (A,g,h)|=∀xϕbe a model. SetB= (B,F)with – B:=A×^Z/3Z

– F((a,i),(a,i)):= (a,i+1) – F((a,i),(a,i+1)):= (ga, 0) – F((a,i+1),(a,i)):= (ha, 0). Now, for all(a,i)∈B

g^∗(a,i) =F (a,i),F(a,i)(a,i)=F (a,i),(a,i+1)= (ga, 0) and

h^∗(a,i) = (ha, 0).

ThusAis isomorphic to a copy ofAdefined inB. A∼=^A^∗:= ({(a, 0):a∈A},g^∗,h^∗).

Therefore, for all(a,i)

B|=ϕg(a,i)⇔^A^∗|=ϕ(ga, 0)

1.4 Applications of the domino method

⇔^A |=ϕ(ga) and B|=ϕ_h(a,i)⇔^A^∗|=ϕ(ha, 0)

⇔^A |=ϕ(ha).

Thus,A|=∀xϕimpliesB|=∀x(ϕg∧ϕ_k).

”⇐” ForB= (B,F)|=∀x(ϕg∧ϕ_h)letA= (A,g,h)with – A:=g^∗(B)∪h^∗(B)

– g:=g^∗ – h:=h^∗

ThenA|=∀xϕ. q.e.d.

1 The classical decision problem for FO

Algorithmic Model Theory SS 2010

c b n d

Contents

1 The classical decision problem for FO