Developing New Models And Abstractions For Modern Computing Paradigms

Formalizing Distributed and Parallel Systems

Modern computer systems increasingly rely on distributed and parallel architectures to meet the growing demands for performance and scalability. Formal models provide mathematical abstractions that can precisely capture the semantics and behaviors of these complex systems.

Leveraging Process Calculi to Model Concurrency

Process calculi such as the pi-calculus provide formalisms to describe concurrent systems and communications between concurrent processes. The pi-calculus uses channels, choice, and recursion to model dynamic topologies and mobility found in distributed systems. Process calculi give computer scientists a way to formally reason about concurrency, studyparallel algorithms, and verify the correctness of implementations.

Using Temporal Logics to Specify Correctness

Temporal logics extend propositional and predicate logic to reason about the temporal ordering of events and states in concurrent systems. Logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL) allow the specification of critical correctness properties such as safety, liveness, and fairness. Model checkers can then algorithmically verify that a system satisfies these high-level temporal specifications, enabling formal verification of complex parallel designs.

Encoding Causality in Graph-Based Models

Causal models such as causal graphs, Bayesian networks, and structural equation models capture the causal relationships between different components in a distributed system. These graphical models provide a way to study and reason about the causal effects that propagate across interconnected systems. By making assumptions of modularity and conditional independence explicit in the graph structure, they can facilitate explanatory and diagnostic analyses for parallel and distributed systems.

Abstracting Emerging Hardware Architectures

In addition to distributed computing, computer science is increasingly concerned with novel and exotic computing paradigms made possible by recent advances in hardware architectures. Theoretical abstractions help uncover computational principles behind unconventional architectures to understand their capabilities.

Applying Category Theory to Quantum Computing

Category theory provides an abstract framework to model quantum computational processes involving superposition, entanglement, and measurement. Compact closed categories can elegantly capture information flow in quantum systems, while adjoint functors model reversible dynamics. Category-theoretic tools aid physics-based analysis to uncover connections between quantum algorithms and interpretations of quantum mechanics.

Representing Neuromorphic Systems with Neural Networks

Neuromorphic engineering seeks to emulate neural systems found in nature using very-large-scale integration systems. Neural networks model the adaptive learning capabilities of biological neural networks by representing neuron activations through graphs. The weights on the edges capture the plasticity of real synapses. Neural networks can be analyzed to study how neuromorphic hardware may one day approach human-like classification, perception, and control tasks.

Capturing Stochasticity with Probabilistic Models

Many emerging substrates for computation involve uncertainty, noise, and randomness. Markov chains and Bayesian networks provide common abstractions for representing the stochastic dynamics of such systems. Their graphical structure encodes conditional dependence, while probability distributions over nodes model stochastic transitions. These techniques have unlocked analyses of molecular, chemical, and quantum systems that harness randomness as a computational resource.

Connecting Theory to Practice

In addition to studying computation abstractly, formal models provide a bridge between theoretical computer science and applied programming by providing semantics to capture the meanings and behaviors of real programming languages and systems.

Relating Models to Programming Language Semantics

Formal semantic frameworks allow precise mathematical definitions of the valid behaviors and outcomes invoked by programs. Operational semantics capture computational steps through abstract machines and rewriting systems while denotational semantics associate programs to mathematical meanings expressed in domains. These techniques enable computer scientists to formally reason about properties like type safety, data races, and termination.

Mapping Abstractions to Systems Implementation

Model-driven engineering uses abstractions to automatically synthesize concrete systems from high-level specifications. Models encode architectural constraints and domain concepts that tools can process to produce standard hardware/software implementations. This raises the level of abstraction to facilitate design reuse and accelerate realization of systems guided by the constraints captured in the models.

Bridging the Gap Between Theory and Applied Research

Domain-specific languages (DSLs) embed abstractions and notations tailored to particular problem domains within programming languages. DSLs enable applied scientists to access the power of programming without needing to master general-purpose languages. Furthermore, because DSLs are formally defined, they permit extending applied research with rigorous analysis methods from theoretical computer science.

Expanding the Horizons of Theoretical Computer Science

Lastly, computer science theory itself continues expanding its own horizons by proposing more expressive models of computation not envisioned in classical computer science.

Exploring New Computational Models like Chemical Computing

Chemical computing harnesses theoretical models of chemical kinetics to perform computation through interactions between molecules and chemical reactions networks. Formally modeling the state transitions facilitated by chemical mixtures broadens the notions of algorithms and architectures beyond traditional digital logic and numerical analysis.

Generalizing Concepts to Capture More Classes of Computation

Hypercomputation theory generalizes classical theoretical computer science by proposing models that can overcome limitations like the halting problem. Quantum tupling, for example, uses infinite specification information to go beyond Turing-computability while abstract accelerated machines exploit transfinite ordinal time to compute some infinite loops. These conceptual expansions aim to uniformly capture both standard and non-standard classes of computation.

Pushing the Boundaries of What is Computable

Models at the frontiers explore truly exotic computing mechanisms ranging from relativistic machines that harness causal paradoxes to oracular devices that have access to uncomputable information. Studying these extreme models through the lens of recursion theory reveals deep connections between physics and computation by formalizing capabilities and limitations of unconventional computing devices.