Wied Pakusa

Choiceless Polynomial Time on Structures with Small Abelian Colour Classes

Choiceless 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 (FPC) but to date it is unknown whether it expresses all polynomial-time properties of finite structures. We study the CPT-definability of the isomorphism problem for relational structures of bounded colour class size q (for short, q-bounded structures). Our main result gives a positive answer, and even CPT-definable canonisation procedures, for classes of q-bounded structures with small Abelian groups on the colour classes. Such classes of q-bounded structures with Abelian colours naturally arise in many contexts. For instance, 2-bounded structures have Abelian colours which shows that CPT captures Ptime on 2-bounded structures. In particular, this shows that the isomorphism problem of multipedes is definable in CPT, an open question posed by Blass, Gurevich, and Shelah.

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields.

Characterising Choiceless Polynomial Time with First-Order Interpretations

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).

Descriptive complexity of linear equation systems and applications to propositional proof complexity

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”.

Definability of summation problems for Abelian groups and semigroups

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.

Defining Winning Strategies in Fixed-Point Logic

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-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, reachability, eventual safety (Co-Buchi) and recurrent reachability (Buchi), 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.

Model-Theoretic Properties of ω-Automatic Structures

We investigate structural properties of ω-automatic presentations of infinite structures in order to sharpen our methods to determine whether a given structure is ω-automatic. We apply these methods to show that several classes of structures such as pairing functions and infinite integral domains do not have an ω-automatic model.