One-dimensional fragment of first-order logic
We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and non-uniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest.
Decidability questions constitute one of the core themes in computer science logic. Decidability properties of several fragments of first-order logic have been investigated after the completion of the program concerning the classical decision problem. Currently perhaps the most important two frameworks studied in this context are those based on the guarded fragment  and two-variable logics.
Two-variable logic was introduced by Henkin in  and showed decidable in  by Mortimer. The satisfiability and finite satisfiability problems of two-variable logic were proved to be -complete in . The extension of two-variable logic with counting quantifiers, , was shown decidable in . It was subsequently proved to be -complete in .
Research concerning decidability of variants of two-variable logic has been very active in recent years. Recent articles in the field include for example  , , , and several others. The recent research efforts have mainly concerned decidability and complexity issues in restriction to particular classes of structures, and also questions related to different built-in features and operators that increase the expressivity of the base language.
Guarded fragment was originally conceived in . It is a restriction of first-order logic that only allows quantification of “guarded” new variables—a restriction that makes the logic rather similar to modal logic.
The guarded fragment has generated a vast literature, and several related decidability questions have been studied. The fragment has recently been significantly generalized in . The article introduces the guarded negation first-order logic . This logic only allows negations of formulae that are guarded in the sense of the guarded fragment. The guarded negation fragment has been shown complete for in .
Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not based on restricting the quantification patterns of formulae, unlike the prefix classes studied in the context of the classical decision problem. Surprisingly, not many such frameworks have been investigated in the literature.
In this paper we introduce a novel decidable fragment that allows arbitrary quantifier alternation patterns. The uniform one-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential (universal) quantifiers that leave at most one free variable in the resulting formula. Additionally, a uniformity condition applies to the use of atomic formulae: if , then a Boolean combination of atoms and is allowed only if . Boolean combinations of formulae with at most one free variable can be formed freely.
We establish decidability of the satisfiability and finite satisfiability problems of . We also show that if the uniformity condition is lifted, we obtain an undecidable logic. Furthermore, if we keep uniformity but go two-dimensional by allowing existential (universal) quantifier blocks that leave two variables free, we again obtain an undecidable formalism. Therefore, if we lift either of the two restrictions that our fragment is based on, we obtain an undecidable logic.
In addition to studying decidability, we also show that is incomparable in expressive power with both and .
In , Vardi initiated an intriguing research effort that aims to understand phenomena behind the robust decidability of different variants of modal logic. In addition to , see also for example  and the introduction of . Modal logic indeed has several features related to what is known about decidability. In particular, modal logic embeds into both and .
However, there exist several important and widely applied decidable extensions of modal logic that do not embed into both and . Such extensions include Boolean modal logic (see , ) and basic polyadic modal logic, i.e, modal logic containing accessibility relations of arities higher than two (see ). Boolean modal logic allows Boolean combinations of accessibility relations and therefore can express for example the formula . Polyadic modal logic can express the formula . Boolean modal logic and polyadic modal logic are both inherently one-dimensional, and furthermore, satisfy the uniformity condition of . Both logics embed into . The notions of one-dimensionality and uniformity can be seen as novel features that can help, in part, explain decidability phenomena concerning modal logics.
Importantly, also the equality-free fragment of embeds into . In fact, when attention is restricted to vocabularies with relations of arities at most two, then the expressivities of and the equality-free fragment of coincide. Instead of seeing this as a weakness of , we in fact regard as a canonical generalisation of (equality-free) into contexts with arbitrary relational vocabularies. The fragment can be regarded as a vectorisation of that offers new possibilities for extending research efforts concerning two-variable logics. It is worth noting that for example in database theory contexts, two-variable logics as such are not always directly applicable due to the arity-related limitations. Thus we believe that the one-dimensional fragment is indeed a worthy discovery that extends the scope of research on two-variable logics to the realm involving relations of arbitrary arities.
Instead of extending basic techniques from the field of two-variable logic, our decidability proof is based on a direct satisfiability preserving translation of into monadic first-order logic. The novel proof technique is mathematically interesting in its own right, and is in fact a central contribution of this article; the proof technique is clearly robust and can be modified and extended to give other decidability and complexity results. Furthermore, as a by-product of our proof, we identify a natural polyadic modal logic , which is expressively equivalent to the one-dimensional fragment. This modal normal form for the one-dimensional fragments is also—we believe—a nice contribution.
Let denote the set of positive integers. Let denote a complete relational vocabulary, i.e., , where denotes a countably infinite set of -ary relation symbols. Each vocabulary we consider below is assumed to be a subset of . A -formula of first-order logic is a formula whose set of non-logical symbols is a subset of . A -model is a model whose set of interpreted non-logical symbols is .
Let denote the countably infinite set of variable symbols. We define the set of -formulae of first-order logic in the usual way, assuming that all variable symbols are from . Below we use meta-variables in order to denote variables in . Also symbols of the type and , where , will be used as meta-variables. In addition to meta-variables, we also need to directly use the variables below. Note that for example the meta-variables and may denote the same variable in , while the variables of course simply are different variables.
Let be a -ary relation symbol, . An atomic formula is called -ary if there are exactly distinct variables in the set . For example, if are distinct variables, then and are binary, and is -ary. An -ary -atom is an atomic formula that is -ary, and the relation symbol of the formula is in .
Let . Let a -model with the domain . A function that maps some subset of into is an assignment. Let be a -formula with the free variables . Let be an assignment that interprets the free variables of in . We write if satisfies when the free variables of are interpreted according to . Let . Let be a -formula whose free variables are among . We write if for some assignment such that for each .
By a non-empty conjunction we mean a finite conjunction with at least one conjunct; for example and are non-empty conjunctions.
By monadic first-order logic, or , we mean the fragment of first-order logic without equality, where formulae contain only unary relation symbols.
Let . A -permutation is a bijection . When is irrelevant or clear from the context, we simply talk about permutations.
Let . We let and denote the -tuple containing copies of the object . When , this tuple is identified with the object .
Let and be positive integers. Let be a set, and let be a tuple. We let denote the tuple . Let be an -ary relation. We let denote the -ary relation defined such that for each , we have iff for some tuple .
Recall that is assumed to be always true, while is always false.
3 The one-dimensional fragment
We shall next define the uniform one-dimensional fragment of first-order logic. Let be a set of variable symbols, and let be a -ary relation symbol. An atomic formula is called a -atom if . A finite set of -atoms is called a -uniform set. When is irrelevant or known from the context, we may simply talk about a uniform set. For example, assuming that are distinct variables, and are uniform sets, while is not. The empty set is a -uniform set.
Let . The set , or the set of -formulae of the one-dimensional fragment, is the smallest set satisfying the following conditions.
Every unary -atom is in , and .
If , then . If , then .
Let be a set of variable symbols. Let be a finite set of formulae whose free variables are in . Let . Let be a -uniform set of -atoms. Let be any Boolean combination of formulae in . Then .
If , then .
Notice that there is no equality symbol in the language. Notice also that the formation rule (iv) is strictly speaking not needed since the rule (iii) covers it. Concerning the rule (i), notice that also atoms of the type , where , are legitimate formulae. Let denote the set .
3.1 Intuitions underlying the decidability proof
We show decidability of the satisfiability and finite satisfiability problems of by translating -formulae into equisatisfiable -formulae. We first translate into a logic . This logic is a normal form for such that all literals of arities higher than one appear in simple conjunctions, as for example in the formula . The logic is then translated into a modal logic , which is an essentially variable-free formalism for . In Section 4 we show how formulae of the logic are translated into equisatisfiable formulae of , which is well-known to have the finite model property.
The semantics of is defined (see Section 3.4) with respect to pointed models , where . If is a formula of , we let denote the set . In Section 4 we fix a - formula and translate it to an -formula . We prove that if , then is satisfied in a model , whose domain is , where is the domain of an -dimensional hypertorus of arity . Such a hypertorus is a structure , where the different relations are all -ary. Intuitively, the domain of consists of several copies of , one copy for each point of the hypertorus. Let denote the set of subformulae of . The vocabulary of consists of monadic predicates and , where and . The predicates are interpreted such that and .
We will give a rigorous and self-contained proof of the decidability of , but to get an (admittedly very rough) initial idea of some of the related background intuitions, consider the following construction. (It may also help to refer back to this section while internalizing the proof.)
Consider a formula of ordinary unimodal logic
and a Kripke model .
We can maximize the accessibility relation of
by defining a new relation such
that iff for all formulae ,
If we replace by in , then each point in the new model will satisfy exactly the same subformulae of as satisfied in the old model. Thus we can encode information concerning by using the (so-called filtration) condition given by Equation 1. The equation talks about the sets and , and thus it turns out that we can encode the information given by the equation by monadic predicates and corresponding to the sets and (cf. the formulae and in Section 4.1). This way we can encode information concerning accessibility relations by using formulae of .
This construction does not work if one tries to maximize both a binary relation and its complement at the same time: the problem is that the maximized relations and will not necessarily be complements of each other. For this reason we need to make enough room for maximizing accessibility relations. Below we will simultaneously maximize several types of accessibility relations that cannot be allowed to intersect. Thus we need to use an -dimensional hypertorus (rather than a usual 2D torus). Each -ary accessibility relation type of the translated -formula will be reserved a sequence of copies of from the domain of . Information concerning will be encoded into this sequence of models.
Let be a finite vocabulary. Let be an integer, and let be a set of distinct variable symbols. A uniform -ary -diagram is a maximal satisfiable set of -atoms and negated -atoms of the vocabulary . (The empty set is not considered to be a uniform -ary -diagram; this case is relevant when contains no relation symbols of the arity or higher.)
For example, let , where the arities of , , are , , , respectively. Now is a uniform binary -diagram. Here we assume that and are distinct variables.
Let be a finite vocabulary. The set is the smallest set satisfying the following conditions.
Every unary -atom is in . Also .
If , then . If , then .
Let be a uniform -ary -diagram in the variables ,…,, where . Let be a non-empty conjunction of a finite set of formulae in whose free variables are among ,…,. Then .
If has at most one free variable, , then .
Let denote the set of exactly all formulae such that for some finite , we have . translates effectively into ; see the appendix for the proof. Here we briefly sketch the principal idea behind the translation. Consider a -formula , where denotes a tuple of variables. Put into disjunctive normal form . Thus translates into the formula , where the formulae are conjunctions. Each conjunction is equivalent to a disjunction , where is of the desired type .
We next define a class of hypertori. It may help to have a look at Lemma 3.1 before internalizing the definition. Let and be integers. Define Let be a tuple. Let . Let . For each , let such that the following conditions hold.
Let us call such a tuple the -th good -ary sequence originating from . Define the relation such that iff is the -th good -ary sequence originating from . We call the structure the -dimensional hypertorus of the arity .
Let be an -dimensional hypertorus of the arity . Let and . Then the following conditions hold.
For each , there exists exactly one tuple such that . We have for all such that .
Let . Let be a -permutation, and let . Then .
Let . Let be any -permutation other than the identity permutation. Then .
In the rest of the article, we let denote the -dimensional hypertorus of the arity . We let and denote, respectively, the domain and the relation of .
3.4 Translation into a modal logic
Let be a finite vocabulary, and let be an integer. Let be a -model with the domain . Let be a uniform -ary -diagram in the variables . Notice that here we use the standard variables from . The diagram is a standard uniform -ary -diagram. We define to be the relation Standard variables are needed in order to uniquely specify the order of elements in tuples of .
Let be a standard uniform -ary -diagram. Let be a positive integer. Let be a surjection. We let denote the set obtained from by replacing each variable by .
Let and be positive integers such that . Let and be standard uniform -ary and -ary -diagrams, respectively. Let be a surjection. Assume that , i.e., the implication holds for each -model and each assignment interpreting the variables in the domain of . Then we write .
We then define a modal logic that provides an essentially variable-free representation of . Define the set to be the smallest set such that the following conditions are satisfied.
If is a relation symbol of any arity, then . Also .
If , then . If , then .
If and is a standard uniform -ary -diagram, then .
If , then . (Here denotes the universal modality; see below for the its semantics.)
The semantics of is defined with respect to pointed -models , where is an ordinary -model of predicate logic for some vocabulary , and is an element of the domain of . Obviously we define that always holds, and that never holds. Let be an -ary relation symbol. We define , where is the interpretation of the relation symbol in the model . The Boolean connectives and have their usual meaning. For formulae of the type , we define that if and only if there exists a tuple such that and for each . For formulae , we define if and only if there exists some such that .
When is a -formula and a -model with the domain , we let denote the set We let denote the union of all sets , where is a finite subset of .
It is very easy to show that there is an effective translation that turns any formula into a formula such that for all -models , where is the set of non-logical symbols in . (The set of non-logical symbols in is contained in , and the formula can either be a sentence or have the free variable .)
4 is decidable
Let us fix a formula of . We will first define a translation of to an -formula in Section 4.1. We will then show in Sections 4.2 and 4.3 that the translation indeed preserves equivalence of satisfiability over finite models as well as over all models. Due to the above effective translations from to and from to , this implies that the satisfiability and finite satisfiability problems of are decidable.
4.1 Translating into monadic first-order logic
We assume, w.l.o.g.,, that contains at least one subformula of the type . If not, we redefine . The vocabulary of may of course grow. We also assume, w.l.o.g., that does not contain occurrences of the symbols , . Furthermore, we assume, w.l.o.g., that if is a relation symbol occurring in some diagram of , then also occurs in as a subformula: we can of course always add the conjunct to .
Let be the set of all relation symbols in , whether they occur in diagrams or as atomic subformulae; in fact, due to our assumptions above, the set of atomic formulae in is equal to . Let be the set of relation symbols occurring in the diagrams of . Let denote the set of -ary relation symbols in . Define analogously. Due to the assumption that contains a subformula , each relation symbol of some arity that occurs as an atom in , also occurs in the diagram . (This is due to the definition of .) Thus for all .
Let denote the maximum arity of all diagrams in . For each , let denote the set of exactly all standard uniform -ary -diagrams. Let denote the union of the sets , where . Let . Recall that denotes the domain of the -dimensional hypertorus of the arity . For each , define an injection For a -ary diagram , let denote the -ary relation .
Let denote the set of subformulae of the formula . Fix fresh unary relation symbols and for each formula and torus point . The vocabulary of the translation of will be the set We let denote this set.
We shall next define a collection of auxiliary formulae needed in order to define . If a pointed model satisfies , then will be satisfied in a larger model; the related model construction is defined in the beginning of Section 4.2. The predicates of the type will be used to encode information about sets , while the predicates encode information about the diagrams of . The predicates are crucial when defining a -model that satisfies based on a -model of in Section 4.3.
Let . Define to be the formula
Let be the set of pairs , where is a -ary diagram for some , and is a surjection such that we have . The set is the set of inverse projections of in . Define
The following formula is the principal formula that encodes information about diagrams of (cf. Lemma 4.1).
Let denote the set of relation symbols that occur positively in , i.e., there exists some atom , where is the arity of . Let be the relation symbols that occur negatively in , i.e., for some atom . The following three formulae encode information about atomic formulae in . Define
The next formula is essential in the construction of a -model of from a -model of in Section 4.3. The two models have the same domain. The formula states that each tuple can be interpreted to satisfy some diagram such that information concerning the unary predicates in is consistent with . See the way is defined based on in Section 4.3 for further details. Define
Also the following formula is crucial for the definition of .
Let , , , and be formulae in . The following formulae recursively encode information concerning subformulae of . Define
Let Finally, we define
4.2 Satisfiability of implies satisfiability of
Fix an arbitrary model -model with the domain . Fix a point . Assume . We shall next construct a model with the domain . We then show that , where is a torus point. If is a finite model, then so is .
The domain of the -model consists of copies of , one copy for each torus point . Let us define interpretations of the symbols in . Consider a symbol , where . If , then . Consider then a symbol , where . If , then .
Let and . Then iff .
Define and . Assume . Thus for some tuple such that for each . Hence for each and each torus point . To conclude the first direction of the proof, it suffices to prove that for some torus points .
Let be the torus points such that . In order to establish that assume that , where and . Assume that , and that We must show that .
For each , as , we have by the definition of . As and , we have . Therefore we have . Thus by the definition of .
We then deal with the converse implication of the lemma. Define and . Assume Hence for some tuple