Symmetric Computation | Gregory Wilsenach

Lecture 1

Thu, 29 Jul 2021 00:00:00 +0000

We resolve computational problems (e.g. kidney matching and train scheduling) by first modelling them in terms of abstract data structures (e.g graphs). We note that the solutions to these problems respect these abstractions in the sense of being invariants of the these structures. We informally discuss the notion of a symmetric model of computation, i.e. algorithms that avoid breaking abstraction and so ensure isomorphism-invariant computation. As a case study we discuss the graph matching problem (which underlies kidney matching) and examine which of the algorithms used to solve this problem respect abstraction, and are thus symmetric in our sense, and which rely on details extrinsic to the graph (e.g. an arbitrary order on nodes). We note in particular that while it’s often easy to find some symmetric algorithm for a decidable problem, the interesting question is whether there is an efficient symmetric algorithm.

In this course we will discuss a number of different models of computation and how symmetry arrises in these various contexts, including in: logic, relational machines, circuits, and linear programming. The details of how these different models relate to one another will be discussed in later lectures. We begin by looking at how symmetric algorithms may be generated from high-level specifications, and in particular database query languages and logics. We discuss how these questions first arose in database theory and how formulas of well-known logics, e.g. first-order and second-order logic, and their extensions generate symmetric algorithms. We discuss how this topic was formalised in descriptive complexity, the subject concerned with understanding the relationship between high-level specification languages and conventional complexity classes (e.g. P and NP). In particular, we discuss Fagin’s theorem and the question of a logic for P.

We finally introduce symmetric circuits, a natural notion of a symmetric algorithm in the context of circuit complexity. We see how this notion captures what it means for an algorithm to respect abstraction and present a series of results establishing a tight connection between the computational power of symmetric circuits and logics of central importance in descriptive complexity.

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.

Lecture 2

Thu, 29 Jul 2021 00:00:00 +0000

In this lecture we discuss descriptive complexity in some more detail. This subject is concerned with understanding the relationship between logics, whose formulas generate symmetric algorithms, and complexity classes conventionally defined in terms models with no such symmetry condition (e.g. Turing machines). We discuss the foundational result in this field, Fagin’s theorem, which establishes that those problems in NP are precisely those that can be defined in existential second-order logic. This leads naturally to the central open question in the field: Is there a similar characterisation for P?

We introduce various logics that have arisen naturally in the field, often motivated by this central question, including fixed-point logic (FP) . We discuss the Immerman-Vardi theorem, which establishes that FP characterises P over ordered structures, and how understanding the definability of order is central to understanding the relationship between logic and computation.

We discuss finite-variable fragments of logics and game-based characterisations of these logics, as well how these may be used to establish inexpressibility results for fixed-point logic and its extensions. First, we discuss finite-variable fragments of first-order logic and show that their expressive power may be characterised using a pebble game. We note that each formula of FP may be translated to a family of first-order formulas with a constant bound on the number of variables used in each formula. We show how this relationship enables the use of pebble games for establishing inexpressibility results for FP. We discussion an FP with a counting operator, a logic we call fixed-point logic with counting (FPC). This logic is often said to express almost all problems that are obviously in P and is of central importance in descriptive complexity. More formally, FPC has been shown to capture P over all proper minor closed classes of graphs. We note, however, that FPC provably doesn’t capture P.

We discuss counting quantifiers and extensions of finite-variable fragments with these quantifiers. We recall the equivalence of these finite-variable counting logics and the Weisfeiler-Lehman algorithms for distinguishing graphs. We also discuss various other characterisations of these logics from a number of different fields and two (equivalent) bijection-game characterisations of their expressive power. In the same way as for FP, FPC formulas may be translated to families first-order formulas with counting quantifiers with a constant bound on the number of variables. This allows us to use the bijection-game characterisation to prove inexpressibility results for FPC.

We lastly discuss a family of graph constructions named for Cai, Fürer, and Immerman, called the CFI constructions. We observe, but do not yet prove, that these graphs may be used to construct a query in P not inexpressible in FPC. The proof of this result, which uses bijection games, is discussed in the next lecture.

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.

Lecture 3

Thu, 29 Jul 2021 00:00:00 +0000

In this lecture we continue our discussion of CFI constructions and complete the proof that these graphs give a query separating FPC from P. We also discuss the related notion of counting width.

We discuss relational machines formally and note that polynomial-time relational machines have the same expressive power as FPC. We turn to circuit complexity and discuss invariant circuits, i.e. those that take as input graphs and compute queries invariant under isomorphism. We introduce symmetric circuits, i.e. those where each permutation on the universe of the input graph extend to automorphisms of the circuit (which ensures invariance). We recount results due to Anderson and Dawar establishing an equivalence in expressive power between P-uniform families of symmetric circuits (with threshold gates) and FPC.

In the final part of this lecture we discuss some of the tools used to establish this circuit-based characterisation of FPC. In particular, we discuss what it means for a gate in a circuit to have a support and sketch the proof of a key result establishing bounds on the size of a support of a gate in terms of the size of its orbit. We sketch a translation from symmetric circuits to formulas. We also return to the bijection games used to characterise counting logics and show that they can be played directly on circuits, and so establish a direct game-based characterisation (i.e. one that does not go through the equivalence with counting logics) of the expressive power of symmetric circuits. This result establishes a close connection between the sizes of the supports of the gates in a circuit and the counting width of the query computed, a crucial result used for establishing lower bounds.

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.

Lecture 4

Thu, 29 Jul 2021 00:00:00 +0000

We have so far seen that polynomial-time relational machines, P-uniform families of symmetric circuits, and FPC all have the same expressive power. We also have a means of characterising the expressive power of these models via bijection games. This suggests a robust theory of symmetric computation. In this lecture we discuss a natural symmetric model of computation that arose for independent reasons in linear programming.

We describe what it means for a system of linear equations to decide a language (roughly, it defines a polytope whose projection includes all members in the language and excludes those not in the language). We discuss a recent result establishing a close connection between polynomial-size families of symmetric linear programs, bounded-variable polynomial-size families of formulas of counting logics, and polynomial-size families of symmetric circuits (with threshold gates). In order to sketch a proof of this result we first discuss a theorem establishing that linear program feasibility is in FPC, which itself is proved by showing that the ellipsoid method can be implemented in the logic. We also discuss how this can be used to prove that the perfect matching problem is definable in FPC. We finally return to the characterisation of symmetric linear programs and sketch the proof. We note that we use similar tools in our analysis of symmetric linear programs as for the characterisation of symmetric circuits (e.g. supports).

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.

Lecture 5

Thu, 29 Jul 2021 00:00:00 +0000

In this lecture we discuss a few applications of the results on symmetric computation presented in this course. First, we discuss an application to proving unconditional lower bounds for certain algorithmic methods used to solve constraint satisfaction problems. Second, we discuss an application to hardness of approximation. These hardness results establish that certain NP-hard problems cannot be approximated beyond some threshold of accuracy in polynomial-time so long as P is not equal to NP. We discuss recent work showing that for certain NP-hard problems inapproximability in FPC can be established without reference to any condition on P and NP.

We finally discuss other notions of symmetric computation which, like FPC, can express only problems in P but can also express problems known to inexpressible in FPC. In particular, we discuss choiceless polynomial time and fixed-point logic with rank.

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.