On Symmetric Circuits and Fixed-Point Logics

We study properties of relational structures such as graphs that are decided by families of Boolean circuits. Circuits that decide such properties are necessarily invariant to permutations of the elements of the input structures. We focus on families of circuits that are symmetric, i.e., circuits whose invariance is witnessed by automorphisms of the circuit induced by the permutation of the input structure. We show that the expressive power of such families is closely tied to definability in logic. In particular, we show that the queries defined on structures by uniform families of symmetric Boolean circuits with majority gates are exactly those definable in fixed-point logic with counting. This shows that inexpressibility results in the latter logic lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201

A logic with temporally accessible iteration

Deficiency in expressive power of the first-order logic has led to developing its numerous extensions by fixed point operators, such as Least Fixed-Point (LFP), inflationary fixed-point (IFP), partial fixed-point (PFP), etc. These logics have been extensively studied in finite model theory, database theory, descriptive complexity. In this paper we introduce unifying framework, the logic with iteration operator, in which iteration steps may be accessed by temporal logic formulae. We show that proposed logic FO+TAI subsumes all mentioned fixed point extensions as well as many other fixed point logics as natural fragments. On the other hand we show that over finite structures FO+TAI is no more expressive than FO+PFP. Further we show that adding the same machinery to the logic of monotone inductions (FO+LFP) does not increase its expressive power either

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

Connectionist modal logic: Representing modalities in neural networks

AbstractModal logics are amongst the most successful applied logical systems. Neural networks were proved to be effective learning systems. In this paper, we propose to combine the strengths of modal logics and neural networks by introducing Connectionist Modal Logics (CML). CML belongs to the domain of neural-symbolic integration, which concerns the application of problem-specific symbolic knowledge within the neurocomputing paradigm. In CML, one may represent, reason or learn modal logics using a neural network. This is achieved by a Modalities Algorithm that translates modal logic programs into neural network ensembles. We show that the translation is sound, i.e. the network ensemble computes a fixed-point meaning of the original modal program, acting as a distributed computational model for modal logic. We also show that the fixed-point computation terminates whenever the modal program is well-behaved. Finally, we validate CML as a computational model for integrated knowledge representation and learning by applying it to a well-known testbed for distributed knowledge representation. This paves the way for a range of applications on integrated knowledge representation and learning, from practical reasoning to evolving multi-agent systems

On Symmetric Circuits and Fixed-Point Logics

We study properties of relational structures, such as graphs, that are decided by families of Boolean circuits. Circuits that decide such properties are necessarily invariant to permutations of the elements of the input structures. We focus on families of circuits that are symmetric, i.e., circuits whose invariance is witnessed by automorphisms of the circuit induced by the permutation of the input structure. We show that the expressive power of such families is closely tied to definability in logic. In particular, we show that the queries defined on structures by uniform families of symmetric Boolean circuits with majority gates are exactly those definable in fixed-point logic with counting. This shows that inexpressibility results in the latter logic lead to lower bounds against polynomial-size families of symmetric circuits.This research was supported by EPSRC grant EP/H026835