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8Логические исследования 2021. Т. 27. № 2. С. 93-120 УДК 510.643
Logical Investigations 2021, Vol. 27, No. 2, pp. 93-120 DOI: 10.21146/2074-1472-2021-27-2-93-120
Mikhail Rybakoy, Dmitry Shkatoy
Undecidability of QLTL and QCTL with two variables and one monadic predicate letter
Mikhail Rybakov
Tver State University,
Zhelyabova Street, 33, Tver, 170100, Russia. E-mail: m_rybakov@mail.ru
Dmitry Shkatov
University of the Witwatersrand, Johannesburg, Private Bag 3, WITS2050, South Africa. Tver State University,
Zhelyabova Street, 33, Tver, 170100, Russia. E-mail: shkatov@gmail.com
Abstract: We study the algorithmic properties of the quantified (first-order) temporal logics QLTL and QCTL in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. We prove that satisfiability problems for QLTL and QCTL in languages with two individual variables and one monadic predicate letter are £ 1 -hard. Thus, QLTL and QCTL are n 1 -hard, and so not recursively enumerable, in such languages. The results, which hold both for the expanding domain and for the constant domain semantics, are obtained by reduction from a £l-hard N x N recurrent tiling problem. The results presented in this paper strengthen those by I. Hodkinson, F. Wolter, and M. Zakharyaschev, who have shown that the satisfiability problem for QLTL (and, by a straightforward modification of the proof, for QCTL) is £1 -hard in languages with two individual variables and an unlimited supply of monadic predicate letters.
Keywords: quantified temporal logic, first-order temporal logic, linear-time temporal logic, branching-time temporal logic, restricted languages, undecidability, recursive enumerabil-ity, arithmetic complexity, algorithmic classification problem, satisfiability problem, validity problem
For citation: Rybakov M., Shkatov D. "Undecidability of QLTL and QCTL with two variables and one monadic predicate letter", Logicheskie Issledovaniya / Logical Investigations, 2021, Vol. 27, No. 2, pp. 93-120. DOI: 10.21146/2074-1472-2021-27-2-93-120
1. Introduction
Temporal logics [Emerson, 1990; Goldblatt, 1992; Gabbay et al., 1994; De-mri et al., 2016] (we use the term "temporal logics" for formalisms studied
© Rybakov M., Shkatov D.
primarily in the context of computer science; by contrast, the term "tense logics" is commonly used for formalisms studied primarily by philosophers and linguists), having appeared in the 1980s [Pnueli, 1986] as a tool for the verification of concurrent systems, have since been extensively used as specification languages [Manna, Pnueli, 1992; Manna, Pnueli, 1995; Clarke et al., 2000], knowledge representation formalisms [Baader et al., 2015; Borgwardt et al., 2015; Bourgaux et al., 2019; Artale et al., 2004], and query languages [Chomicki, Niwinski, 1993; Chomicki, 1994; Abiteboul, 1996], often in combination with other formalisms, such as epistemic [Halpern, Vardi, 1998; Fa-gin et al., 1995; van der Hoek, Wooldridge, 2003] and description [Wolter, Za-kharyaschev, 2000; Artale, Franconi, 2000; Baader et al., 2012] logics.
Most applications of temporal logics in computer science have been those of propositional temporal logics, mostly because the quantified temporal logics of relevance to computer science are computationally badly behaved — most of them are not even recursively enumerable1 and, hence, deductively incomplete: no effective proof system can capture all the valid inferences in quantified temporal languages [Andreka et al., 1979; Szalas, 1986; Szalas, Holenderski, 1988; Abadi, 1989; Merz, 1992; Garson, 2001]. By comparison, propositional temporal logics, which are typically PSPACE-, EXPTIME-, or 2EXPTIME-complete [Fischer, Ladner, 1979; Sistla, Clarke, 1985; Vardi, Stockmeyer, 1985; Emerson, Halpern, 1985; Goranko, van Drimmelen, 2006; Walther et al., 2006; Schewe, 2008] and usually have sound and complete proof systems [Emerson, Halpern, 1985; Goldblatt, 1992; Goranko, van Drimmelen, 2006; Goranko, Shkatov, 2009d]2, look quite attractive.
In some applications, however, — temporal databases being an example — propositional logics do not suffice; hence, a need for quantified temporal logics. The bad computational behaviour of quantified temporal logics does not have to spell end to their applicability: the classical first-order logic, despite being undecidable [Church, 1936; Turing, 1936], has had a very long and highly successful history of applications in computer science and artificial intelligence. What is important from the point of view of applications is not whether a logic as a whole is decidable, but whether it has sufficiently expressive decidable fragments: no application requires the full expressive power of a logic. Therefore, the study of the algorithmic properties of fragments of quantified temporal
1For a brief discussion of how this relates to their propositional bases, see the authors' earlier article [Rybakov, Shkatov, 2020a].
2These can be used to obtain, by being combined with proof systems for other formalisms such as epistemic logics [Goranko, Shkatov, 2008; Goranko, Shkatov, 2009a; Ajspur et al., 2013], proof systems for combined systems such as temporal-epistemic logics [Goranko, Shkatov, 2009b; Goranko, Shkatov, 2009c].
logics is crucial for grasping the range of their applicability within computer science and artificial intelligence.
The study of the algorithmic properties of fragments of a logic, which typically involves identification of it maximal decidable and minimal undecidable fragments — the decidable fragments that become undecidable when slightly extended and the undecidable fragments that become decidable when slightly restricted, — is known as an algorithmic classification problem for a logic. Such a study pursues at least two objectives: to understand the root causes of the bad computational behaviour of a logic and to obtain expressive, yet computationally feasible, fragments. The first objective is pursued mainly through identification of undecidable, the second through identification of decidable, fragments.
The main success story of algorithmic classification research has been the classical first-order logic QCl [Borger et al., 1997]. The algorithmic classification problem for non-classical — modal, temporal, and superintuitionistic — logics has been much less studied: despite extensive literature [Kripke, 1962; Arte-mov, Dzhaparidze, 1990; Gabbay, Shehtman, 1993; Wolter, Zakharyaschev, 2001; Kontchakov et al., 2005; Rybakov, Shkatov, 2019c; Shehtman, Shkatov, 2019; Rybakov, Shkatov, 2020b; Rybakov, Shkatov, 2020c; Shehtman, Shkatov, 2020; Rybakov, Shkatov, 2021b; Rybakov, Shkatov, 2021c; Rybakov, Shkatov, 2021d; Rybakov, Shkatov, 2021e], much less is known about the algorithmic properties of fragments of non-classical logics than about the algorithmic properties of fragments of QCl.
The present paper aims to contribute to the algorithmic classification problem for quantified temporal logics. We originally focus on the quantified lineartime temporal logic QLTL, both the simplest and the most widely used quantified temporal logic, and then show how our treatment of QLTL can be extended to the quantified branching-time temporal logic QCTL, and hence to the quantified alternating-time temporal logic QATL. Clearly, our results also apply to QCTL* and QATL*, since QCTL can be viewed as a fragment of QCTL* and QATL as a fragment of QATL*.
The largest known decidable fragments of QLTL are monodic fragments [Hodkinson et al., 2000, Theorem 15] (see also [Gabbay et al., 2003, Ch. 11]): a monodic fragment is obtained by adding to a decidable fragment of QCl valid temporal formulas in which scopes of temporal operators do not contain more than one free variable (many temporal formulas of interest, including the Barcan formula corresponding to the semantic condition of constant domains, have this form). Following up on the discovery of monodic fragments, the complexity of a number of decidable fragments of QLTL has been iden-
tified [Hodkinson et al., 2003], and effective axiomatizations of monodic fragments have been obtained [Wolter, Zakharyaschev, 2002].
The smallest known undecidable fragment of QLTL [Hodkinson et al., 2000, Theorem 2] contains formulas with two individual variables and an unlimited supply of monadic predicate letters.
Thus, the gap between the known decidable and the known undecidable fragments of QLTL is quite narrow: the fragment with two individual variables, an unlimited supply of monadic predicate letters, and unrestricted use of temporal operators is undecidable, but the fragment with two variables, only monadic predicate letters, and the monodic use of temporal operators — i.e., using the temporal operators so that their scopes contain formulas with at most one free variable — is decidable (the corresponding fragment of QCl is decid-able [Mortimer, 1975; Gradel et al., 1997]; hence, it forms a suitable classical basis for a monodic fragment).
Nevertheless, in light of recent results [Rybakov, Shkatov, 2019c; Rybakov, Shkatov, 2020b; Rybakov, Shkatov, 2020c; Rybakov, Shkatov, 2021b; Rybakov, Shkatov, 2021c] on undecidability of quantified modal and superintuitionistic logics, close relations of quantified temporal logics, — which show that fragments with a few variables (two [Rybakov, Shkatov, 2019c; Rybakov, Shkatov, 2021c] or three [Rybakov, Shkatov, 2020c; Rybakov, Shkatov, 2021b], depending on the logic) and a single monadic predicate letter often turn out to be undecidable when no restrictions are placed on the use of non-classical operators — a very natural question arises of whether the gap between the known decidable and the known undecidable fragments can be tightened still further: can we prove undecidability of QLTL in languages with two variables and a single monadic predicate letter?
In the present paper we do just that: we narrow the gap between the known decidable and the known undecidable fragments of QLTL by proving that the fragment containing formulas with two variables and a single monadic predicate letter is already undecidable — more precisely, this fragment is n}-hard, i.e. has the same computational complexity as the full logic (we are not aware of upper-bound results for QLTL; whatever the upper bound is, it is surely inherited by all the fragments).
The main technical novelty of the present paper is modelling of an arbitrary number of monadic predicate letters with a single one for logics of linear structures. The techniques used in the proofs of analogous results for most modal and superintuitionistic logics [Rybakov, Shkatov, 2019c; Rybakov, Shkatov, 2020c; Rybakov, Shkatov, 2021b] are inapplicable to QLTL: being ultimately based on the propositional level techniques [Halpern, 1995; Chagrov, Rybakov, 2003; Rybakov, Shkatov, 2018a; Rybakov, Shkatov, 2018b; Rybakov,
Shkatov, 2019a; Rybakov, Shkatov, 2021a] developed for modal logics of frames without restrictions on the branching factor, they cannot be used for logics of linear frames, such as QLTL, since they rely on transformations of models that arbitrarily increase their branching factor. We adopt here to QLTL an approach recently [Rybakov, Shkatov, 2020b; Rybakov, Shkatov, 2021c] used for monomodal logics of linear frames: in that setting, however, it is not clear how undecidability can be obtained for languages with a single monadic predicate letter. In the present paper, we adapt those techniques to QLTL to obtain undecidability for languages with two variables and a single monadic predicate letter.3
The proof proceeds by reduction from the X}-hard [Harel, 1986] N x N recurrent tiling problem. The initial reduction to satisfiability for QLTL in languages with two individual variables, one binary, and an unlimited supply of monadic letters, carried out in Lemma 1, resembles the reduction from a known proof by I. Hodkinson, F. Wolter, and M. Zakharyaschev [Hodkinson et al., 2000, Theorem 2]; it is not, however, enough for our purposes to reuse their formulas — we, therefore, define our own reduction from the recurrent tiling problem preparing the ground for subsequent reductions to less expressive subfragments, carried out in Lemmas 2 and 3.
The paper is structured as follows. In Section 2, we introduce the syntax and semantics of QLTL. In Section 3, we present our undecidability result for QLTL. In Section 4, we show how to obtain an analogous result for QCTL. We conclude in Section 5 by discussing directions for future study.
We assume the reader's familiarity with the basic notions of computability theory; the reader wishing a reminder may consult [Rogers, 1967] and [Enderton, 2011].
Throughout the paper, we denote by N the set of natural numbers, which includes zero, and by N+ the set of positive integers, which excludes zero.
2. Preliminaries
In this section, we present the syntax and semantics of first-order quantified linear time temporal logic QLTL.
2.1. Syntax of QLTL
The full, unrestricted, language of QLTL contains:
• countably many individual variables, denoted by x,y,xi,yi,...;
• for every n e N, countably many n-ary predicate letters, denoted by Pn,Qn,Rn,...; when the arity of a predicate letter is clear from the
3The result itself was announced, without proof, in the Discussion section of an earlier paper [Rybakov, Shkatov, 2021c].
context, the superscript is usually omitted; nullary predicate letters are propositional variables;
• the Boolean constant ± (falsity);
• the binary Boolean connective ^ (implication);
• the binary temporal connective U (until);
• the quantifier symbol V (for every);
• technical symbols: parentheses (, ), and the comma.
Formulas are defined by the BNF expression
f := P (xi,...,xn) (f ^ f) | (f U f) |Vxf,
where P ranges over n-ary predicate letters. Atomic formulas are those of the form P(xi,..., xn). Closed formulas are those without free occurrences of variables. We use the standard abbreviations
= (f ^
T = -±;
f A 0 = -f ^ -0;
f V 0 = ^ 0;
( o 0 = (f ^ 0) A (0 ^ ();
Of = ^U (;
Of = f V (TUf);
□f = -O-f;
3x f = -Vx -f,
as well as
o0f = f; On+1f = OOnf, for every n e N.
When parentheses are omitted, -, o, O, □, V, and 3 are assumed to bind tighter than A, V, and U, which are assumed to bind tighter than ^ and o. To enhance readability of formulas, we occasionally use square brackets in place of parentheses.
2.2. Semantics of QLTL
The semantics of QLTL-formulas is defined in terms of Kripke frames, also known — particularly, in computer science — as transition systems.
A Kripke frame is a structure F = (S, where S is a non-empty set of states and ^ is a binary transition relation on S satisfying the seriality condition: for every s e S, there exists t e S such that s ^ t. If s ^ t, we say that t is accessible from s. The relation ^ is not required to be functional: more than one state can be accessible from any state.
A path in a frame (S, is an infinite sequence n of elements of S such that n[i] ^ n[i + 1], for every i e N; thus, n[i] is the ith state of n. For every i e N, we denote by n[i, to] the infinite suffix of n beginning with n[i], i.e., the sequence n[i],n[i + 1],... of states.
A predicate Kripke frame with a system of expanding domains is a tuple 3d = (S, ^,D}, where (S, is a Kripke frame and D is a function from S into the set of non-empty subsets of some set, the domain of 3D; the function D is required to satisfy the monotonicity condition: s ^ s' implies D(s) C D(s'). The set D(s) is the domain of state s. We often write Ds for D(s). We shall also consider predicate frames satisfying the stronger constancy condition: s ^ s' implies D(s) = D(s'); such predicate frames are called predicate frames with constant domains. By predicate frame simpliciter we mean predicate frames with expanding domains. If needed, consult [Gabbay et al., 1994; Hughes, Cresswell, 1996; Fitting, Mendelsohn, 1998; Braiiner, Ghilardi, 2007; Goldblatt, 2011] and [Gabbay et al., 2009, §3.1] for fuller description of Kripke semantics for predicate temporal — and closely related modal — logics.
A Kripke model is a tuple M = (S, D, I}, where (S, D} is a predicate Kripke frame and I, called the interpretation of predicate letters with respect to states of S, is a function assigning to a state s e S and an n-ary predicate letter P an n-ary relation I(s,P) on Ds; thus, I(s,P) C Dn. In particular, if P is nullary, I(s,P) C D° = {(}}; thus, we can identify truth with {(}} and falsity with 0. We often write P1,s for I(s,P). We say that (S, ^,D,/} is a model over the frame (S, and over the predicate frame (S, D}. We also say that (S, is the underlying frame, and (S, D} the underlying predicate frame, of a model (S, ^,D,/}. If n is a path in a frame underlying a model M, we also say that n is a path in M.
An assignment in a model is a function g associating with every individual variable x an element g(x) of the domain of the underlying predicate frame. We write g' = g if assignment g' differs from assignment g in at most the value of variable x.
Formulas are evaluated in Kripke models relative to paths (rather than states, as in modal, and some other temporal, logics). The satisfaction relation between models M, paths n, assignments g, and formulas ^ is defined by recursion:
• M,n=g P(xi,...,xn) ^ (g(xi),...,g(xn)} e P/,n[°];
• M, n =g ±;
• M, n |=g ^ ^ M, n |=g or M, n |=g
• M, n == f1 Uf2 ^ M, n[i, to] |= for some i > 0, and
M, n[j, to] == f1, for every j with 0 < j < i;
• M, n ==g Vxf1 ^ M, n ==g' f1, for every g' such that g' = g
and g'(x) e D(n[0]).
This definition implies that
• M,n ==g Of ^^ M,n[1, to] = f;
• M, n O f ^^ M, n[i, to] ==g f, for some i e N;
• M, n ==g □ f ^^ M, n[i, to] ==g f, for every i e N.
We note that, if M = (S, D, I} is a Kripke model, s e S, and Is(P) = I(s, P), then the tuple Ms = (Ds, Is} is a classical predicate model, or structure.
We also note that we take the strict version of "until" as our primitive temporal connective: n == f1 Uf2 implies that f2 is satisfied at a future state of n and f 1 is satisfied at every future state of n preceding the state satisfying f2. The non-strict "until", allowing f2 to be satisfied at the current state of n, can be expressed using the strict "until": f1 U°f2 = f2 V (f1 Uf2). Conversely, the strict until can be expressed using U° and O: f1 Uf2 = 0(f1 U°f2). Thus, it is immaterial whether {o, U°} or {U} is taken as the set of primitive temporal operators.
We shall often use the following notation. Let M = (S, D, I} be a Kripke model, n a path in M, and a1,...,an elements of D(n[0]); let also f(x1,... ,xn) be a formula whose free variables are among x1,... ,xn and g be an assignment with g(x1) = a1, ..., g(xn) = an. Then, we shall often write M, n == f(a1,..., an) instead of M, n ==g f(x1,..., xn). This notation is unambiguous since the languages we consider do not contain constants.
Since the satisfaction of atomic formulas at a path depends only on its initial state, we often write M, s == P(a1,..., an) instead of M, n == P(a1,..., an) whenever n[0] = s. If M, s == P(a1,..., an), we also say that P(a1,..., an) is true at s in M.
A formula f is satisfied at a path n in a model M (symbolically, M, n == f) if M, n ==g f, for every g assigning to the free variables of f elements of D(n[0]). A formula f is satisfied at a state s of a model M (symbolically, M, s == f) if M, n == f, for every path n with n[0] = s. A formula f is true in a model M (symbolically, M == f) if M, s == f, for every state s of M. A formula f is valid on a predicate frame Fd (symbolically, Fd == f) if f is true in every model over FD. A formula f is valid on a frame F (symbolically, F == f) if f is valid on every predicate frame (F, D}.
To simplify the notation, we write n = ^ and s = rather than, respectively, M, n = ^ and M, s == when M is clear from the context.
A formula ^ is satisfiable in a model M = (S, D, I} if there exists a path n in M and an assignment g assigning to the free variables of ^ elements of D(n[0]) such that M,n ==g p.
We note that satisfiability of a closed formula in a model does not depend on the assignment: a closed formula ^ is satisfiable in a model M = (S, D, I} if, and only if, there exists a path n in M such that M, n ==
A formula ^ is QLTL-valid if it is valid on every frame; it is QLTL-satis-fiable if it is satisfiable in some model.
In this paper, we are concerned with the computational complexity of the following computational problems:
• Validity problem for QLTL: given a formula determine whether ^ is QLTL-valid.
• Satisfiability problem for QLTL: given a formula determine whether ^ is QLTL-satisfiable.
We note that ^ is valid if, and only if, is not satisfiable; therefore, validity and satisfiability for QLTL are dual problems.
3. Main results
In this section, we prove that satisfiability for QLTL is I-hard — hence, validity for QLTL is n1-hard — in languages with two individual variables and a single monadic predicate letter. This implies that the set of validities of QLTL in such languages is not recursively enumerable, and hence cannot be captured by an effective sound and complete proof system.
3.1. Reduction from the recurrent tiling problem
We proceed by reduction from a Incomplete [Harel, 1986] recurrent tiling problem for N x N.4 We are given a set of tiles, a tile t being a 1 x 1 square, with a fixed orientation, whose edges are "colored" with left(t), right(t), up(t), and down(t). A tile type is fully determined by the edge colors. Every tile belongs to one of the finitely many types T = {t°,..., ta}, with tiles of each type being in unlimited supply. A tiling in an arrangement of tiles on a rectangular N x N grid so that the edge colors of the adjacent tiles match, both horizontally and vertically. We are to determine whether there exists a tiling in which a tile of
4D. Harel proves ^^-completeness of this problem by one-to-one reduction from a Ei-complete set of nondeterministic Turing machines: each machine is mapped to an instance of the recurrent tiling problem, i.e., to a set of tile types.
type t0 occurs infinitely often in the leftmost column, i.e., whether there exists a function f : N x N ^ T such that, for every n, m e N,
(T1) right(f(n, m)) = left(f(n + 1, m));
(T2) up(f (n, m)) = down(f (n, m + 1));
(T3) the set {m e N : f (0, m) = t0} is infinite.
To make the underlying idea clearer, we initially encode the recurrent tiling problem with formulas of two individual variables, without regard for the number of predicate letters used. This encoding is similar to the one used by I. Hodkinson, F. Wolter, and M. Zakharyaschev [Hodkinson et al., 2000, Theorem 2] (similar techniques have been used elsewhere [Spaan, 1993; Marx, 1999; Wolter, Zakharyaschev, 2001; Kontchakov et al., 2005] and [Gabbay et al., 2003, Theorem 11.1]), but our encoding is different from theirs since we need to prepare the ground for the subsequent reductions, presented in Lemmas 2 and 3, eliminating all the predicate letter bar a single monadic one in the formula obtained in the initial encoding. Thus, the formulas we use in the initial encoding resemble, but are not the same as, those used by I. Hodkinson, F. Wolter, and M. Zakharyaschev [Hodkinson et al., 2000, Theorem 2].
To obtain a grid for the tiling, we shall think of the states of the path satisfying a formula as rows, and of elements of the domain of the state satisfying a formula as columns, of the N x N grid. Indeed, a path can be thought of as the set of natural numbers with the immediate successor relation; we shall also write formulas that capture, within the domain of a state, the structure of the naturals with the immediate successor relation. Then, to describe a tiling on thus obtained grid, we use a family {Pt : t e T} of monadic predicate letters corresponding to the tile types: intuitively, n[m] == Pt(an) shall mean that the square (n, m} is tiled with a tile of type t from T.
Not all the states on the satisfying path shall be part of the tiling: to prepare the ground for subsequent reductions, we create gaps, non-empty equal-sized sequences of states that are not part of the tiling, in the timeline; the states meant to partake in the tiling shall be marked off by a propositional variable p. The size of the gaps, a + 6, is proportional to the number of monadic predicate letters we need to eliminate in subsequent reductions.
Thus, let < be a binary, Pt, for every t e T, monadic, and p nullary predicate letter.
Let (for brevity, in formulas we write l, r, u, and d instead of left, right, up, and down)
A° = p ad (p o oa+6p);
Ai = 3x no(p A Pi0(x));
A2 = Vx3yx<y;
A3 = VxVy [x<iy ^n(p ^ x<y)];
A4 = Vx □ [p ^ (V Pt(x) A A(Pt(x) ^-Pf (x)))];
teT t'=t
A5 = vxv^ ^(x<y a Pt(x) ^ V Pt' (y));
teT
r(t)=i(f)
A6 = Vx □ A (Pt(x) ^ oa+6 V Pt' (x)),
teT u(t)=d(t')
and let A be the conjunction of A° through A6. Notice that the formula A contains only two individual variables.
3(a + 6) • p
p
2(a + 6) i p
f
p
a + 6 • p
a + 5 I. -p
1 fJ 0 I p
Pf (°,3)(0) Pf (1,3) (1) Pf (2,3) (2) Pf (3,3) (3)
Pf (°,2) (0) Pf (1,2) (1) Pf (2,2) (2) Pf (3,2) (3)
Pf (°,1)(0) Pf (1,1) (1) Pf (2,1) (2) Pf (3,1) (3)
Pf (°,°)(0) Pf (1,°)(1) Pf (2,°) (2) Pf (3,°) (3) 0123
Fig. 1. Model Mo
Lemma 1. There exists a recurrent tiling of N x N satisfying (T1) through (T3) if and only if A is satisfiable.
Proof. ("if") Suppose A is satisfiable, i.e., M, n == A, for some model M = (S, D, I} and some path n in M.
By Ao,
n[m] == p ^^ m = 0 mod (a + 6). (1)
By A1, there exists a0 e D(n[0]) such that n[0] == Pt0(a0). By A2, there exists an infinite sequence a0, a1, a2,... of elements of D(n[0]) such that
a0 ^ 01 ^ a2 ^ ... . By A3 and (1), for every m such that m = 0 mod (a + 6),
o0 ^H o1 ^H o2 ^H ....
By A4, for every n e N and every m such that m = 0 mod (a + 6), there exists a unique t e T such that n[m] == Pt(on). Hence, we can define a function f : N x N ^ T by
f(n, m)= t whenever n[m(a + 6)]== Pt(on).
Since n == A1 A A5 A A6, conditions (T1) through (T3) are satisfied for f. Therefore, f is a required function.
("only if") Suppose f is a function satisfying conditions (T1) through (T3). We obtain a model satisfying A. Let
• S = N,
• s ^ s' ^ s' = s + 1,
• D(n) = N, for every n e N,
and let M0 = (N, <, D, I} be a model, depicted in Figure 1, such that, for every n e N,
• n == p ^ n = 0 mod (a + 6);
• n== k < l ^ n = 0 mod (a + 6) and l = k + 1;
• n == Pt(k) ^ 3 m e N (n = m(a + 6) and f (m, k) = t).
Lastly, let n0 be the path defined by n0[n] = n, for every n e N.
It is straightforward to check that M0,n0 == A, so we leave this to the reader. ■
Thus, in the proof of the "if" part of Lemma 1, we obtained a grid for the required tiling by treating the states of path n as rows and elements o0, o1, o2, . . . of the domain of the state n[0] as columns.
3.2. Elimination of the binary predicate letter
We next eliminate, in a satisfiability-preserving way, all predicate letters of the formula A bar a single monadic one, without increasing the number of individual variables in the resultant formula. We, thus, obtain a reduction from the recurrent tiling problem to satisfiability for QLTL in languages with two variables and a single monadic predicate letter.
The elimination of predicate letters is carried out in two steps: first, we simulate the binary letter < with two monadic predicate letters; then, we simulate all the predicate letters of thus obtained formula with a single monadic one.
From now on, we assume, for ease of notation, that the formula A contains monadic predicate letters P0,..., Pa — rather than a letters Pt, for each t € {to,..., ta} — to refer to the tile types.
First, using a suitable adaptation of the ideas of Kripke's [Kripke, 1962], we eliminate the binary predicate letter < of A.
Recall that Kripke's construction [Ibid.] transforms a model M satisfying a formula containing a binary predicate letter, and no modal operators, at a state s in such a way that a sufficiently large number of states — more precisely, one state for every pair of elements of the domain of s — accessible from s is added to M. This idea cannot be applied to QLTL in a straightforward manner, for two reasons. First, in Kripke's version, the added states do not form a path, which is needed for QLTL. Second, since < occurs in the scope of the temporal operator □ in A, we need to simulate < not just at the initial state of a path satisfying A, but at every state affecting the satisfaction of A.
We resolve these difficulties by working the model M0 defined in the proof of Lemma 1 and by exploiting its properties: the domains of the states of M0 are all equal, and the interpretation of < is the same at every state of M0 affecting the satisfaction of A.
Let Pa+i and Pa+2 be monadic predicate letters distinct from P0,..., Pa and from each other, and let ■' be the function substituting
0(Pa+l(x) л Pa+2(y)), for X < y.
Lemma 2. There exists a recurrent tiling of N x N satisfying (T1) through (T3) if and only if A' is satisfiable.
Proof. ("if") Suppose M, n |= A', for some model M = (S, D, I} and some path n. Define an interpretation I' on the frame (S, ^,D} so that, for every s € S,
I'(s, <) = {(a, b} : M, s |= 0(Pa+i(a) Л Pa+2(b))}
and, for every s e S and every Q e {p, P0,... Pa},
I '(s, Q) = I (s, Q).
and let M' = (S, D, I'}. Then, M', n == A. Hence, by Lemma 1, there exists a recurrent tiling of N x N satisfying (T1) through (T3).
("only if") Suppose f is a function satisfying conditions (T1) through (T3). Let M0 and n0 be, respectively, the model and the path defined in the proof of the "only if" part of Lemma 1. As we have seen, M0,n0 == A. We use M0 to obtain a model satisfying A'. Let a be the infinite sequence
0, 0,1, 0,1, 2, 0,1, 2, 3, 0,1, 2, 3, 4, ...
and let ak, for each k e N, be the kth element of a.
Let M0 = (N, D, I'} be a model such that, for every n, c e N,
M0, n == Pa+1(c) ^ 3 m e N (n = m(a + 6) and c = am); M0, n == Pa+2(c) ^ 3 m e N (n = m(a + 6) and c = am + 1),
and for every n e N and every S e {p, P0,..., Pa},
I'(n, S) = I (n, S).
We show that M0,n0[0] = A'.
Since M0, n0[0] == A, it suffices to prove, for every m, o, b e N,
M0,m(a + 6)= o < b ^^ M0,m(a + 6) = 0(Pa+1(o) A Pa+2(b)).
Assume M0,m(a + 6) == o<b. Then, b = o + 1, by definition of M0. Choose k e N so that m ^ k and ak = o; by definition of a, such a number k certainly exists. Then, M0,k(a + 6) == Pa+1 (o) A Pa+2(b), by definition of M0 Hence, M0, m(a + 6) = 0(Pa+1(o) A Pa+2(b)).
Conversely, assume M0,m(a + 6) == 0(Pa+1(o) A Pa+2(b)), i.e., M0,v == Pa+1 (o) A Pa+2(b), for some v ^ m(a + 6). The definition of M0 of implies that v = k(a + 6), for some k ^ m. Hence, by definition of M0, both o = ak and b = ak + 1. Hence, b = o + 1. Therefore, M0, m(a + 6) == o < b, by definition of M0. ■
(j + 1)(a + 6) (a + 5)+ j(a + 6) (a + 4) + j(a + 6) 5 + j(a + 6) 4 + j(a + 6) 3 + j(a + 6) 2 + j(a + 6) 1 + j(a + 6) j(a + 6) 0
for k such that h(oj) = (k, k + 1)
Z/(0,j+1)(0) C/(1,j+1)(1) C/(2,j+1)(2) C/(3,j+1)(3) P(k), for every k such that f (j, k) = t0 P(k), for every k such that f (j, k) = t1 P(k), for every k such that f (j, k) = ta
P (k) P (k + 1) P(k), for every k e N P(k), for every k e N Z/ (0j)(0) C/ (1,j)(1) C/ (2,j)(2) C/ (3,j)(3) C/(0,0) (0) C/(1,0) (1) C/(2,0) (2) C/(3,0)(3) Fig. 2. Model M0
3.3. Elimination of monadic predicate letters
We lastly simulate the occurrences of predicate letters p, P0,..., Pa+2 in A' with a single monadic letter P, without increasing the number of individual variables in the resultant formula. We, thus, obtain a reduction of the recurrent tiling problem to satisfiability for QLTL with formulas containing two individual variables and a single monadic predicate letter P.
Let P be a monadic predicate letter distinct from the letters P0,..., Pa+2.
Define
d = Vx -P (x) AOVxP (x) AO2 VxP (x), and, for every k e {0,..., a + 2},
Cfc (x) = e AO a+5-k P (x);
Ofe(y) = e ^ a+5-k p (y).
Let •* be the function replacing
• p with 0;
• Pfc(x) with (x), for each k € {0,..., a + 2};
• Pk(y) with (y), for each k € {0,..., a + 2},
and let A* be the result of applying the function •* to A'.
To obtain a model satisfying A*, provided a recurrent tiling exists, we use the model M° defined in the proof of Lemma 2. Call a state s of M° a tiling state if M°, s == p (recall that these are the states corresponding to the tiling) and a gap state otherwise. We use gap states to simulate the interpretation of letters p, P°,..., Pa+2 at a tiling state of M°. The truth of Pk(a), for every k € {0,..., a + 2} and every a € N, at a tiling state s is simulated by making P(a) true at the gap state k + 1 steps away from the next tiling state. The interpretation of the nullary letter p at a tiling state s is simulated by making P universally true at the two gap states immediately succeeding s. As we shall show, this pattern of two consecutive states making P universally true does not occur anywhere else — hence, it marks off exactly the tiling states.
Lemma 3. There exists a recurrent tiling of N x N satisfying (T1) through (T3) if and only if A* is satisfiable in QLTL.
Proof. ("if") Suppose M,n == A*, for some model M = (S, D, I} and some path n. Define an interpretation I' on the frame (S, ^,D} so that, for every s € S and every k € {0,..., a + 2},
and let M' = (S,D, I'}. Then, M',n == A'. Hence, by Lemma 2, there exists a recurrent tiling of N x N satisfying (T1) through (T3).
("only if") Suppose f is a function satisfying conditions (T1) through (T3). Let M° and n° be, respectively, the model and the path defined in the "only if" part of the proof of Lemma 2. As we have seen, M°, n[0] == A'. We use M° to obtain a model satisfying A*.
Let (N, D} be the frame underlying M°, i.e. let D(s) = N, for every s € N. Define a model M* = (N, D, I*} so that, for every s € N,
I'(s, Pk) = {(a} : M,s = Cfc(a)}
and, for every s € S,
I '(s,p)
• if s = 0 mod (a + 6), then I*(s, P) = 0;
• if s = 1 mod (a + 6) or s = 2 mod (a + 6), then I*(s, P) = N;
• if s = j + n(a + 6), for some j € {3,..., a + 5} and some n € N, then for every a € N,
M°, s = P (a) ^ M°, n(a + 6) = P«+5- (a).
Let be a path with n*[0] = s°. It is straightforward to check that M*,n* == A*, so we leave this to the reader.
Thus, A* is satisfiable. ■
We, therefore, obtain the following:
Theorem 1. Satisfiability for QLTL is £^hard in languages with two individual variables and a single monadic predicate letter.
Proof. Immediate from Lemma 3. ■
As we have observed in Section 2, satisfiability and validity for QLTL are dual problems. Hence, we have also proven the following:
Corollary 1. Validity for QLTL is n 1 -hard in languages with two individual variables and a single monadic predicate letter.
Thus, QLTL is not recursively enumerable even in languages with only two variables and only a single monadic predicate letter. Therefore, even for such a fragment of QLTL, no effective sound and complete proof system exists.
In the proofs of this section, we have never relied on the domains of the predicate frames we have worked with being properly increasing. We have, therefore, also established the following:
Theorem 2. Satisfiability for QLTL over predicate frames with constant domains is £ 1-hard in languages with two individual variables and a single monadic predicate letter. Validity for QLTL over predicate frames with constant domains is n 1-hard in languages with two individual variables and a single monadic predicate letter.
4. QCTL and QATL
Since QCTL is neither more nor less expressive than QLTL, an undecid-ability result for either of these logics does not automatically imply an analogous result for the other logic. The formulas used in Section 3 were, however, chosen so that a simple translation, presented in this section, effectively embeds the undecidable fragment of QLTL obtained in Section 3 into a fragment of QCTL. Since the translation does not affect either the individual variables or the predicate letters of a formula, we obtain a result analogous to Theorem 2 for QCTL.
Recall that the main difference between QCTL and QLTL (just as between their propositional bases CTL and LTL [Huth, Ryan, 2004]) is that in QCTL formulas are evaluated at states rather than at paths, as in QLTL, of Kripke models. The language of QCTL contains, in place of formulas (^>i U^>2), formulas EU^>2) and U<^2) ("existential until" and "universal until", respectively; the quantifiers E and A refer to the paths beginning with the state at which the formula is evaluated). The satisfaction conditions for these formulas are (here, s is a state of a Kripke model)
• M, s == EU^>2) ^ there exists a path n with n[0] = s such that
M, n[i] == <^2, for some i > 0, and M, n[j] == for every j with 0 < j < i;
• M, s == A(^1 U^>2) ^ for every n with n[0] = s, both M, n[i] == <^2, for
some i > 0, and M, n[j] == for every j with 0 < j < i.
The satisfaction clauses for the other formulas are similar to those for QLTL (with paths replaced by states). By analogy with QLTL, the other temporal operators can be defined as
Eo<£ = E (±U^); AO<£ = A(±U^);
EO= VE(TU^); AO^ = ^ VA(TU^);
En = -EO-^>; An = -AO-^>.
Now, if we replace in the formulas defined in Section 3 O with EO, n with A , and with A , then we readily obtain results for QCTL analogous to those obtained in Section 3 for QLTL. This observation yields the following result:
Theorem 3. Satisfiability for QCTL over predicate frames with both expanding and constant domains is hard in languages with two individual variables and a single monadic predicate letter. Validity for QCTL over predicate frames with both expanding and constant domains is n^-hard in languages with two individual variables and a single monadic predicate letter.
The language of quantified alternating-time temporal logic QATL contains — in place of the two path quantifiers, E and A, of QCTL — path quantifiers parametrised by groups of agents; the so obtained formulas are evaluated over concurrent game structures [Alur et al., 2002]. The QCTL path quantifiers E and A can, however, be expressed [Ibid.] as QATL path quantifiers parametrised by, respectively, the empty set of agents and the set of all the agents in the language. Such a replacement of path quantifiers gives us a recursive embedding of QCTL into QATL that affects neither the variables nor the predicate letters of formulas. Hence, from Theorem 3, we obtain the following result:
Theorem 4. Satisfiability for QATL over predicate frames with both expanding and constant domains is £ -hard in languages with two individual variables and a single monadic predicate letter. Validity for QATL over predicate frames with both expanding and constant domains is n 1[-hard in languages with two individual variables and a single monadic predicate letter.
Obviously, Theorems 3 and 4 extend to the more expressive logics QCTL* and QATL*, since QCTL and QATL are fragments of, respectively, QCTL* and QATL* obtained by disallowing formulas of certain form.
5. Conclusion
We conclude with mentioning directions for future research.
First, an important open problem is identification of the minimal undecid-able fragments of QLTL over finite domains (it is known that QLTL over finite domains behaves differently from QLTL over arbitrary domains [Hodkinson et al., 2000, Theorem 25]). The reduction used in the present paper essentially relied on the domains of the states to be infinite; therefore, the construction we used is not applicable to QLTL over finite domains. First-order logics over finite domains are usually computationally harder than logics over arbitrary domains [Trakhtenbrot, 1953; Libkin, 2004], even in languages with a small number of individual variables [Rybakov, Shkatov, 2019b] — is this true for QLTL? Monodic fragments of QLTL over finite domains are known to be decidable [Hodkinson et al., 2000, Theorem 26] — what are the minimal unde-cidable fragments?
Second, similar question arise for QLTL over structures with a finite set of states. Some undecidability results for this logic are known [Cerrito et al., 1999], but the gap between the decidable and the undecidable remains quite wide. We do know that, in structures with finite sets of states, we can model finite domains [Rybakov, Shkatov, 2020c; Rybakov, Shkatov, 2018c, Lemma 3.3]; while we would, in such a setting, be able to model all the predicate letters in
the language with a single monadic one, the number of variables needed would be quite large.
The last, and perhaps most technically challenging, direction for future research is the study of similar questions in the context of QLTL based on intuitionistic [Mints, 2000], rather than classical logic. Intuitionistic modal and temporal logics [Fischer Servi, 1977; Fischer Servi, 1980; Fischer Servi, 1984; Davoren, 2009], which formalise constructive modal and temporal reasoning, have been found of particular interest in computer science [Simpson, 1994; Davies, Pfenning, 2001; de Paiva et al., 2004; Yuse, Igarashi, 2006; Brewka et al., 2011; Davies, 2017]. Recently, motivated by applications in computer science, propositional linear-time temporal logics with an intuitionistic base have been proposed and studied [Maier, 2004; Boudou et al., 2017; Fernandez-Duque, 2018; Dieguez et al., 2018; Boudou et al., 2019; Balbiani et al., 2019]. While a number of techniques for studying decidability and complexity of intuition-istic modal and temporal logics are known [Ono, 1977; Simpson, 1994; Grefe, 1998; Wolter, Zakharyaschev, 1997; Wolter, Zakharyaschev, 1999; Alechina, Shkatov, 2006; Gabbay et al., 2003, Chapter 10], no research, as far as we know, has been done on first-order temporal intuitionistic logic. If this direction is pursued, however, we expect to be able to draw on extensive research into computational properties of various fragments of both propositional [Nishimura, 1960; Rybakov, 2006; Rybakov, 2008] and first-order [Maslov et al., 1965; Mints, 1968; Gabbay, 1981; Kontchakov et al., 2005; Rybakov, Shkatov, 2019c; Rybakov, Shkatov, 2021b] intuitionistic logics.
Acknowledgements. The work on this paper, carried out at Tver State University during the second author's visit, has been supported by Russian Science Foundation, project 21-18-00195.
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