A primitive positive definition (pp-definition) over a relational structure is a first-order formula with free variables where is a conjunctive formula. Primitive positive definitions have been extremely influential in the last decades due to their one-to-one correspondence with term algebras in universal algebra, making them a cornerstone in the algebraic approach for studying computational complexity [1, 10]. In short, pp-definitions can be used to obtain classical “gadget reductions” between problems by replacing constraints by their pp-definitions, which in the process might introduce fresh variables viewed as being existentially quantified. This approach has successfully been used to study the complexity of e.g. the constraint satisfaction problem (CSP) which recently led to a dichotomy between tractable and NP-complete CSPs [6, 31]. However, these reductions are typically not sufficient for optimisation problems and other variants of satisfiability, where one needs reductions preserving the number of models, so-called parsimonious reductions. Despite the tremendous advances in the algebraic approach there is currently a lack of methods for studying problems requiring parsimonious reductions, and in this paper we take the first step in developing such a framework. The requirement of parsimonious reductions can be realised by restricting existential quantification to unique quantification (), where we explicitly require that the variable in question can be expressed as a unique combination of other variables. That is, if and only if there exists a function such that , for all where . This notion of unique quantification is not the only one possible and we discuss an alternative viewpoint in Section 5. As a first step in understanding the applicability of uniqueness quantification in complexity classifications we are interested in studying the expressive power of unique existential quantification when used in place of existential quantification in pp-definitions, which we call upp-definitions. Any variables introduced by the resulting gadget reductions are then uniquely determined and do not affect the number of models.
Our main question is then: for which relational structures is it the case that for every pp-formula there exists a upp-formula such that for all ? If this holds over then uniqueness quantification has the same expressive power as existential quantification. The practical motivation for studying this is that if upp-definitions are as powerful as pp-definitions in , then, intuitively, any gadget reduction between two problems can be replaced with a parsimonious reduction. Given the generality of this question a complete answer for arbitrary relational structures is well out of reach, and we begin by introducing simplifying concepts. First, pp-definitions can be viewed as a closure operator over relations, and the resulting closed sets of relations are known as relational clones, or co-clones . For each universe the set of co-clones over then forms a lattice when ordered by set inclusion, and given a set of relations we write for the smallest co-clone over containing . Similarly, closure under upp-definitions can also be viewed as a closure operator, and we write for the smallest set of relations over containing and which is closed under upp-definitions. Using these notions the question of the expressive strength of upp-definitions can be stated as: for which sets of relations is it the case that ? The main advantage behind this viewpoint is that a co-clone can be described as the set of relations invariant under a set of operations , , such that the operations in describe all permissible combinations of tuples in relations from . An operation is also said to be a polymorphism of and if we let be the set of polymorphisms of then is called a clone. This relationship allows us to characterise the cases that need to be considered by using known properties of , which is sometimes simpler than working only on the relational side. This strategy will prove to be particularly useful for Boolean sets of relations since all Boolean clones and co-clones have been determined .
Our main research question is to identify such that for each such that . If this holds we say that is -covered. The main difficulty for proving this is that it might not be possible to directly transform a pp-definition into an equivalent upp-definition. To mitigate this we analyse relations in co-clones using partial polymorphisms, which allows us to analyse their expressibility in a very nuanced way. In Section 3.1 we show how partial polymorphisms can be leveraged to prove that a given co-clone is -covered. Most notably, we prove that is -covered if consists only of projections of the form , or of projections and constant operations. As a consequence, pp-defines all relations over if and only if upp-defines all relations over . One way of interpreting this result is that if is “sufficiently expressive” then pp-definitions can always be turned into upp-definitions. However, there also exists -covered co-clones where the reason is rather that is “sufficiently weak”. For example, if is invariant under the affine operation , then existential quantification does not add any expressive power over unique existential quantification, since any existentially quantified variable occurring in a pp-definition can be expressed via a linear equation, and is therefore uniquely determined by other arguments. In Section 3.2 we then turn to the Boolean domain, and obtain a full classification of the -covered co-clones. Based on the results in Section 3.1 it is reasonable to expect that the covering property holds for sufficiently expressive languages and sufficiently weak languages, but that there may exist cases in between where unique quantification differs from existential quantification. This is indeed true, and we prove that the Boolean co-clones corresponding to non-positive Horn clauses, implicative and positive clauses, and their dual cases, are not -covered. Last, in Section 4 we demonstrate how the results from Section 3 can be used for obtaining complexity classifications of computational problems. One example of a problem requiring parsimonious reductions is the unique satisfiability problem over a Boolean set of relations () and its multi-valued generalization the unique constraint satisfaction problem (), where the goal is to determine if there exists a unique model of a given conjunctive -formula. The complexity of was settled by Juban  for finite sets of relations , essentially using a large case analysis. Using the results from Section 3.2 this complexity classification can instead be proved in a succinct manner, and we are also able to extend the classification to infinite and large classes of non-Boolean . This systematic approach is also advantageous for proving lower bounds, and we relate the complexity of to the highly influential exponential-time hypothesis (ETH) , by showing that none of the intractable cases of admit subexponential algorithms without violating the ETH.
Primitive positive definitions with uniqueness quantification appeared in Creignou & Hermann  in the context of “quasi-equivalent” logical formulas, and in the textbook by Creignou et al.  under the name of faithful implementations. Similarly, upp-definitions were utilised by Kavvadias & Sideri  to study the complexity of the inverse satisfiability problem. A related topic is frozen quantification, which can be viewed as uniqueness quantification restricted to variables that are constant in any model .
2.1 Operations and Relations
In the sequel, let be a finite domain of values. A -ary function is sometimes referred to as an operation over and we write to denote the arity . Similarly, a partial operation over is a map where is called the domain of , and we let be the arity of . If and are -ary partial operations such that and for each then is said to be a suboperation of . For and we let be the th projection, i.e., for all . We write for the set of all operations over and for the set of all partial operations over . As a notational shorthand we for write for the set . For we by denote the constant -ary tuple . Say that a -ary is essentially unary if there exists unary and such that for all .
Given an -ary relation we write to denote its arity . If is an -ary tuple we write to denote the th element , and to denote the projection on the coordinates . Similarly, if is an -ary relation we let . The th argument of a relation is said to be redundant if there exists such that for each , and is said to be fictitious if for all and have where and .
We write for the equality relation over . We will often represent relations by their defining first-order formulas, and if is a first-order formula with free variables we write to define the relation is a model of . We let be the set of all -ary relations over , , and . A set will sometimes be called a constraint language. Each -ary operation can be associated with a -ary relation , called the graph of .
2.2 Primitive Positive Definitions and Determined Variables
We say that an -ary relation has a primitive positive definition (pp-definition) over a set of relations over a domain if where each is a tuple of variables over of length and each . Hence, can be defined as a (potentially) existentially quantified conjunctive formula over . We will occasionally be interested in pp-definitions not making use of existential quantification, and call pp-definitions of this restricted type quantifier-free primitive positive definitions (qfpp-definitions).
Let be an -ary relation over a domain . We say that is uniquely determined, or just determined, if there exists and a function such that for each .
When defining relations in terms of logical formulas we will occasionally also say that the th variable is uniquely determined, rather than the th index.
An -ary relation has a unique primitive positive definition (upp-definition) over a set of relations if there exists a pp-definition
of over where each is uniquely determined by .
We typically write for the existentially quantified variables in a upp-definition. Following Nordh & Zanuttini  we refer to unique existential quantification over constant arguments as frozen existential quantification ( is constant if there exists such that for each ). If is upp-definable over via a upp-definition only making use of frozen existential quantification then we say that is freezingly pp-definable (fpp-definable) over . Let us define the following closure operators over relations.
Let be a set of relations. Then we define (1) has a pp-definition over , (2), has a upp-definition over , (3), has an fpp-definition over , and (4), has a qfpp-definition over .
In all cases is called a base. If is singleton then we write instead of , and similarly for the other operators. Sets of relations of the form are usually called relational clones, or co-clones, sets of the form weak systems, or weak partial co-clones, and sets of the form are known as frozen partial co-clones. Note that for any .
Co-clones and weak systems can be described via algebraic invariants known as polymorphisms and partial polymorphism. More precisely, if and is a -ary operation, then for we let . We then say that a -ary partial operation preserves an -ary relation if or there exists such that , for each sequence of tuples . If preserves then is also said to be invariant under . Note that if is total then the condition is simply that for each sequence . We then let , , , and . Similarly, if is a set of (partial) operations we let be the set of relations invariant under , and write if is singleton. It is then known that is a co-clone if and that is a weak system if . More generally, and , resulting in the following Galois connections.
Let and be two sets of relations. Then if and only if and if and only if .
Last, we remark that sets of the form and are usually called clones, and strong partial clones, respectively, and form lattices when ordered by set inclusion. Boolean clones are particularly well understood and the induced lattice is known as Post’s lattice . If then we write for the intersection of all clones over containing . Hence, is the smallest clone over containing .
2.3 Weak and Plain Bases of Co-Clones
In this section we introduce two special types of bases of a co-clone, that are useful for understanding the expressibility of upp-definitions.
Definition 5 (Schnoor & Schnoor ).
Let be a co-clone. A base of with the property that for every base of is called a weak base of .
For we let where is the sequence of -ary tuples such that is a lexicographic enumeration of .
Given a relation and a set of operations over a domain , we let
We typically write instead of if the domain is clear from the context, and say that a co-clone has core-size if there exist relations such that , , and . Weak bases can then be described via core-sizes as follows (a clone is finitely related if there exists a finite base of ).
Theorem 7 (Schnoor & Schnoor ).
Let be a finitely related clone where has core-size . Then is a weak base of for every .
See Table 2 for a list of weak bases for the Boolean co-clones of interest in this paper [20, 21]. Here, and in the sequel, we use the co-clone terminology developed by Reith & Wagner  and Böhler et al. , where a Boolean co-clone is typically written as . Many relations in Table 2 are provided by their defining logical formulas; for example, is the binary relation . See Table 1 for definitions of the remaining relations. As a convention we use to indicate a variable which is constant 0 in any model, and for a variable which is constant 1. On the functional side we use the bases by Böhler et al.  and let , , , , , , , , and , where and where are shorthands for the two constant Boolean operations. We conclude this section by defining the dual notion of a weak base.
Definition 8 (Creignou et al. ).
Let be a co-clone. A base of with the property that for every base of is called a plain base of .
Clearly, every co-clone is a trivial plain base of itself, but the question remains for which co-clones more succinct plain bases can be found. For arbitrary finite domains little is known but in the Boolean domain succinct plain bases have been described  (see Table 2).
|Weak base of||Plain base of|
Many questions concerning Boolean co-clones can be simplified by only considering parts of Post’s lattice. If is -ary then the dual of , , is the operation , and we let for a set . Each Boolean clone can then be associated with a dual clone . Similarly, for we let and for . It is then known that .
3 The Expressive Power of Unique Existential Quantification
The main goal of this paper is to understand when the expressive power of unique existential quantification coincides with existential quantification in primitive positive formulas. Let us first consider an example where a pp-definition can be rewritten into a upp-definition.
Consider the canonical reduction from -SAT to -SAT via pp-definitions of the form . In this pp-definition the auxiliary variable is not uniquely determined since, for example, and are both consistent with . On the other hand, if we instead take the pp-definition , which can be expressed by -SAT, it is easily verified that is determined by and .
Using the algebraic terminology from Section 2 this property can be phrased as follows.
A co-clone is -covered if for every base of .
Thus, we are interested in determining the -covered co-clones, and since every constraint language belongs to a co-clone, namely , Definition 10 precisely captures the aforementioned question concerning the expressive strength of uniqueness quantification in primitive positive formulas. The remainder of this section will be dedicated to proving covering results of this form, with a particular focus on proving a full classification for the Boolean domain. See Figure 1 for a visualisation of this dichotomy. We begin in Section 3.1 by outlining some of the main ideas required to prove that a co-clone is -covered, and consider covering results applicable for arbitrary finite domains. In Section 3.2 we turn to the Boolean domain where we prove the classification in Figure 1.
3.1 General Constructions
Given an arbitrary constraint language it can be difficult to directly reason about the strength of upp-definitions over . Fortunately, there are methods to mitigate this difficulty. Recall from Definition 5 that a weak base of a co-clone is a base which is qfpp-definable by any other base of , and that a plain base is a base with the property that it can qfpp-define every relation in the co-clone. We then have the following useful lemma.
Let be a co-clone with a weak base and a plain base . If then is -covered.
Let be a base of and take an arbitrary -ary relation . First, take a qfpp-definition over . By assumption, can upp-define every relation in , and it follows that
for a -formula since each constraint in can be replaced by its upp-definition over . Last, since can qfpp-define , we can obtain a upp-definition of by replacing each constraint in by its qfpp-definition over . ∎
Although not difficult to prove, Lemma 11 offers the advantage that it is sufficient to prove that for two constraint languages and . Let us now illustrate some additional techniques for proving that is -covered. Theorem 7 in Section 2.3 shows that the relation is a weak base of for larger than or equal to the core-size of . For smaller than the core-size we have the following description of .
Let be a finitely related clone over a finite domain . Then, for every , for every base of .
The intuitive meaning behind the relation is that it may be viewed as a relational representation of the set of all -ary operations of a clone , in the sense that there for each -ary exists such that , where . Moreover, the operation preserves if and only if . In a qfpp-definition of we then associate each variable with an element of , and then for each and add the constraint . For further details, see Theorem 2 in Bodnarchuk et al. , or Theorem 15 in Dalmau . ∎
The applications of Theorem 12 in the context of upp-definitions might not be immediate. However, observe that each argument of is determined by at most other arguments, and if is sufficiently simple, this property can be proved to hold also for . This intuition can then be formalised into the following general theorem.
Let be a clone over a finite domain such that each is a constant operation or a projection. Then is -covered.
Let be a set of operations such that . We may without loss of generality assume that for unary operations such that for some . Take an arbitrary -ary relation . Let and consider the relation from Definition 6. Our aim is to prove that can upp-define , which is sufficient since via Theorem 12. Let denote the indices satisfying .
If , and consists only of projections, then , and each argument in is already determined by , and by the preceding remark . Therefore, assume that . For each then observe that and that . Choose such that for if and only if , for a such that . Thus, we choose a pair of indices differing in if and only if the projection on is constant. Such a choice is always possible since the arguments of enumerate all -ary tuples over . Then construct the relation . It follows that , and that every argument is determined by . Hence, . ∎
Theorem 13 implies that is -covered if is sufficiently powerful, and in particular implies that is -covered for every finite . Hence, pp-defines every relation if and only if upp-defines every relation. However, as we will now illustrate, this is not the only possible case when a co-clone is -covered.
Let be a set of operations over a finite domain . If each argument is either fictitious or determined for every , then is -covered.
Let be a set of relations such that , and let be an -ary relation. Let denote a pp-definition of over . First consider the relation which is upp-definable (indeed, even qfpp-definable) over . Hence, is preserved by , implying that the th argument is either fictitious or determined. In the first case we construct the relation
In the second case, we can directly upp-define the -ary relation as
Since , it is clear that this procedure can be repeated until the relation is upp-defined. ∎
Let be a finite domain