Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes can be described by a strong height-one Maltsev condition. For infinite-domain CSPs, the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (Q; <); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the Boolean rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parameterized by a number k and a set Q of primes. The intuition is that two equivalent graphs G equiv^IM_{k, Q} H cannot be distinguished by means of partitioning the set of k-tuples in both graphs with respect to any linear-algebraic operator acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences have first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define {LA^{k}}(Q), an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that equiv^IM_{k, Q} is the natural notion of elementary equivalence for this logic. The logic LA^{omega}(Q) = Cup_{k in omega} LA^{k}(Q) is then a natural upper bound on the expressive power of any extension of fixed-point logics by means of Q-linear-algebraic operators. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Fürer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that equiv^IM_{k, Q} is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in LA^{omega}(Q), which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke’s Theorem, an important result from the representation theory of finite groups.

Abstract Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. WhileFPRcan express most of the known queries that separateFPCfromPtime, almost nothing was known about the limitations of its expressive power. In our first main result we show that the extensions ofFPCby rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic ${\text{FPR}}^{\text{*}}$ with an operator that uniformly expresses the matrix rank over finite fields is more expressive thanFPR. One important step in our proof is to consider solvability logicFPSwhich is the analogous extension ofFPCby quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the monomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. We also study the bounded-degree polynomial calculus. Over the rationals, it captures fixed-point logic with counting if we restrict the bit-complexity of the coefficients. For unrestricted coefficients, we can only say that the bounded-degree polynomial calculus is at most as powerful as bounded variable infinitary counting logic, but a precise logical characterisation of its power remains an open problem. These connections between logics and proof systems allow us to establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) published originally on January 23, 2019.

Choiceless Polynomial Time (CPT) is one of the most promising candidates in the search for a logic capturing P time . The question whether there is a logic that expresses exactly the polynomial-time computable properties of finite structures, which has been open for more than 30 years, is one of the most important and challenging problems in finite model theory. The strength of Choiceless Polynomial Time is its ability to perform isomorphism-invariant computations over structures, using hereditarily finite sets as data structures. But, because of isomorphism-invariance, it is choiceless in the sense that it cannot select an arbitrary element of a set—an operation that is crucial for many classical algorithms. CPT can define many interesting P time queries, including (a certain version of) the Cai-Fürer-Immerman (CFI) query. The CFI-query is particularly interesting, because it separates fixed-point logic with counting from P time and has since remained the main benchmark for the expressibility of logics within P time . The CFI-construction associates with each connected graph a set of CFI-graphs that can be partitioned into exactly two isomorphism classes called odd and even CFI-graphs. The problem is to decide, given a CFI-graph, whether it is odd or even. For the case where the CFI-graphs arise from ordered graphs, Dawar, Richerby, and Rossman proved that the CFI-query is CPT-definable. However, definability of the CFI-query over general graphs remains open. Our first contribution generalises the result by Dawar, Richerby, and Rossman to the variant of the CFI-query derived from graphs with colour classes of logarithmic size, instead of colour class size one. Second, we consider the CFI-query over graph classes where the maximal degree is linear in the size of the graphs. For the latter, we establish CPT-definability using only sets of small, constant rank, which is known to be impossible for the general case. In our CFI-recognising procedures we strongly make use of the ability of CPT to create sets, rather than tuples only, and we further prove that, if CPT worked over tuples instead, then no such procedure would be definable. We introduce a notion of “sequencelike objects” based on the structure of the graphs’ symmetry groups, and we show that no CPT-program that only uses sequencelike objects can decide the CFI-query over complete graphs, which have linear maximal degree. From a broader perspective, this generalises a result by Blass, Gurevich, and van den Bussche about the power of isomorphism-invariant machine models (for polynomial time) to a setting with counting.

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.

We prove that the solvability of systems of linear equations and related linear algebraic properties are definable in a fragment of fixed-point logic with counting that only allows polylogarithmically many iterations of the fixed-point operators. This enables us to separate the descriptive complexity of solving linear equations from full fixed-point logic with counting by logical means. As an application of these results, we separate an extension of first-order logic with a rank operator from fixed-point logic with counting, solving an open problem due to Holm [21]. We then draw a connection from this work in descriptive complexity theory to graph isomorphism testing and propositional proof complexity. Answering an open question from [7], we separate the strength of certain algebraic graph-isomorphism tests. This result can also be phrased as a separation of the algebraic propositional proof systems “Nullstellensatz” and “monomial PC”.

We study the descriptive complexity of summation problems in Abelian groups and semigroups. In general, an input to the summation problem consists of an Abelian semigroup G, explicitly represented by its multiplication table, and a subset X of G. The task is to determine the sum over all elements of X. Algorithmically this is a very simple problem. If the elements of X come in some order, then we can process these elements along that order and calculate the sum in a trivial way. However, what makes this fundamental problem so interesting for us is that from the viewpoint of logical definability its tractability is much more delicate. If we consider the semigroup G as an abstract structure and X as an abstract set, without a linear order and hence without a canonical way to process the elements one by one, then it is unclear how to define the sum in any logic that does not have the power to quantify over a linear order. Indeed the trivial summation algorithm cannot be expressed in any polynomial-time logic or, in fact, in any computational model which works on abstract mathematical structures in an isomorphism-invariant way without violating polynomial resource bounds. The surprising difficulty, in terms of logical definability, of this basic mathematical problem is the reason why Ben Rossman asked, more than ten years ago, whether it can be expressed in the logic Choiceless Polynomial Time with counting (CPT). Note that, to date, CPT is one of the most powerful known candidates for a logic that might be capable of defining every polynomial-time property of finite structures. In this paper we clarify the status of the definability for the summation problem for Abelian groups and semigroups in important polynomial-time logics. In our first main result we show that the problem can be defined in fixed-point logic with counting (FPC). Since FPC is contained in CPT this settles Rossman's question. Our proof is based on a dynamic programming approach and heavily uses the counting mechanism of FPC. In our second main result we give a matching lower bound and show that the use of counting operators cannot be avoided: the summation problem, even over Abelian groups, cannot be defined in pure fixed-point logic without counting. Our proof is based on a probabilistic argument.

Choiceless Polynomial Time (CPT) is one of the most promising candidates in the search for a logic capturing Ptime. The question whether there is a logic that expresses exactly the polynomial-time computable properties of finite structures, which has been open for more than 30 years, is one of the most important and challenging problems in finite model theory. The strength of Choiceless Polynomial Time is its ability to perform isomorphism-invariant computations over structures, using hereditarily finite sets as data structures. But, as it preserves symmetries, it is choiceless in the sense that it cannot select an arbitrary element of a set - an operation which is crucial for many classical algorithms. CPT can define many interesting Ptime queries, including (the original version of) the Cai-Fürer-Immerman (CFI) query. The CFI query is particularly interesting because it separates fixed-point logic with counting from Ptime, and has since remained the main benchmark for the expressibility of logics within Ptime. The CFI construction associates with each connected graph a set of CFI-graphs that can be partitioned into exactly two isomorphism classes called odd and even CFI-graphs. The problem is to decide, given a CFI-graph, whether it is odd or even. In the original version, the underlying graphs are linearly ordered, and for this case, Dawar, Richerby and Rossman proved that the CFI query is CPT-definable. However, the CFI query over general graphs remains one of the few known examples for which CPT-definability is open. Our first contribution generalises the result by Dawar, Richerby and Rossman to the variant of the CFI query where the underlying graphs have colour classes of logarithmic size, instead of colour class size one. Secondly, we consider the CFI query over graph classes where the maximal degree is linear in the size of the graphs. For these classes, we establish CPT-definability using only sets of small, constant rank, which is known to be impossible for the general case. In our CFI-recognising procedures we strongly make use of the ability of CPT to create sets, rather than tuples only, and we further prove that, if CPT worked over tuples instead, no such procedure would be definable. We introduce a notion of "sequence-like objects" based on the structure of the graphs' symmetry groups, and we show that no CPT-program which only uses sequence-like objects can decide the CFI query over complete graphs, which have linear maximal degree. From a broader perspective, this generalises a result by Blass, Gurevich, and van den Bussche about the power of isomorphism-invariant machine models (for polynomial time) to a setting with counting.

We study definability questions for positional winning strategies in infinite games on graphs. The quest for efficient algorithmic constructions of winning regions and winning strategies in infinite games, in particular parity games, is of importance in many branches of logic and computer science. A closely related, yet different, facet of this problem concerns the definability of winning regions and winning strategies in logical systems such as monadic second-order logic, least fixed-point logic LFP, the modal μ-calculus and some of its fragments. While a number of results concerning definability issues for winning regions have been established, so far almost nothing has been known concerning the definability of winning strategies. We make the notion of logical definability of positional winning strategies precise and study systematically the possibility of translations between definitions of winning regions and definitions of winning strategies. We present explicit LFP-definitions for winning strategies in games with relatively simple objectives, such as safety, reach ability, eventual safety (Co-Büchi) and recurrent reach ability (Büchi), and then prove, based on the Stage Comparison Theorem, that winning strategies for any class of parity games with a bounded number of priorities are LFP-definable. For parity games with an unbounded number of priorities, LFP-definitions of winning strategies are provably impossible on arbitrary (finite and infinite) game graphs. On finite game graphs however, this definability problem turns out to be equivalent to the fundamental open question about the algorithmic complexity of parity games. Indeed, based on a general argument about LFP-translations we prove that LFP definable winning strategies on the class of all finite parity games exist if, and only if, parity games can be solved in polynomial time, despite the fact that LFP is, in general, strictly weaker than polynomial time.

Choice less Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time. It is a strict extension of fixed-point logic with counting, but to date the question is open whether it expresses all polynomial-time properties of finite structures. We present here alternative characterisations of Choice less Polynomial Time (with and without counting) based on iterated first-order interpretations. The fundamental mechanism of Choice less Polynomial Time is the manipulation of hereditarily finite sets over the input structure by means of set-theoretic operations and comprehension terms. While this is very convenient and powerful for the design of abstract computations on structures, it makes the analysis of the expressive power of CPT rather difficult. We aim to reduce this functional framework operating on higher-order objects to an approach that evaluates formulae on less complex objects. We propose a more model-theoretic formalism, called polynomial-time interpretation logic (PIL), that replaces the machinery of hereditarily finite sets and comprehension terms by traditional first-order interpretations, and handles counting by Härtig quantifiers. In our framework, computations on finite structures are captured by iterations of interpretations, and a run is a sequence of states, each of which is a finite structure of a fixed vocabulary. Our main result is that PIL has precisely the same expressive power as Choice less Polynomial Time. We also analyse the structure of PIL and show that many of the logical formalisms or database languages that have been proposed in the quest for a logic for polynomial time reappear as fragments of PIL, obtained by restricting interpretations in a natural way (e.g. By omitting congruences or using only one-dimensional interpretations).