16 research outputs found
A semantical approach to equilibria and rationality
Game theoretic equilibria are mathematical expressions of rationality.
Rational agents are used to model not only humans and their software
representatives, but also organisms, populations, species and genes,
interacting with each other and with the environment. Rational behaviors are
achieved not only through conscious reasoning, but also through spontaneous
stabilization at equilibrium points.
Formal theories of rationality are usually guided by informal intuitions,
which are acquired by observing some concrete economic, biological, or network
processes. Treating such processes as instances of computation, we reconstruct
and refine some basic notions of equilibrium and rationality from the some
basic structures of computation.
It is, of course, well known that equilibria arise as fixed points; the point
is that semantics of computation of fixed points seems to be providing novel
methods, algebraic and coalgebraic, for reasoning about them.Comment: 18 pages; Proceedings of CALCO 200
Diagrammatic Semantics for Digital Circuits
We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications
Comonadic Notions of Computation
AbstractWe argue that symmetric (semi)monoidal comonads provide a means to structure context-dependent notions of computation such as notions of dataflow computation (computation on streams) and of tree relabelling as in attribute evaluation. We propose a generic semantics for extensions of simply typed lambda calculus with context-dependent operations analogous to the Moggi-style semantics for effectful languages based on strong monads. This continues the work in the early 90s by Brookes, Geva and Van Stone on the use of computational comonads in intensional semantics
Shades of Iteration: from Elgot to Kleene
Notions of iteration range from the arguably most general Elgot iteration to
a very specific Kleene iteration. The fundamental nature of Elgot iteration has
been extensively explored by Bloom and Esik in the form of iteration theories,
while Kleene iteration became extremely popular as an integral part of
(untyped) formalisms, such as automata theory, regular expressions and Kleene
algebra. Here, we establish a formal connection between Elgot iteration and
Kleene iteration in the form of Elgot monads and Kleene monads, respectively.
We also introduce a novel class of while-monads, which like Kleene monads admit
a relatively simple description in algebraic terms. Like Elgot monads,
while-monads cover a large variety of models that meaningfully support
while-loops, but may fail the Kleene algebra laws, or even fail to support a
Kleen iteration operator altogether.Comment: Extended version of the accepted one for "Recent Trends in Algebraic
Development Techniques - 26th IFIP WG 1.3 International Workshop, WADT 2022
Shadows and traces in bicategories
Traces in symmetric monoidal categories are well-known and have many
applications; for instance, their functoriality directly implies the Lefschetz
fixed point theorem. However, for some applications, such as generalizations of
the Lefschetz theorem, one needs "noncommutative" traces, such as the
Hattori-Stallings trace for modules over noncommutative rings. In this paper we
study a generalization of the symmetric monoidal trace which applies to
noncommutative situations; its context is a bicategory equipped with an extra
structure called a "shadow." In particular, we prove its functoriality and
2-functoriality, which are essential to its applications in fixed-point theory.
Throughout we make use of an appropriate "cylindrical" type of string diagram,
which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical
string diagrams; v3: final version, to appear in JHR
Span(Graph): a Canonical Feedback Algebra of Open Transition Systems
We show that Span(Graph)*, an algebra for open transition systems introduced
by Katis, Sabadini and Walters, satisfies a universal property. By itself, this
is a justification of the canonicity of this model of concurrency. However, the
universal property is itself of interest, being a formal demonstration of the
relationship between feedback and state. Indeed, feedback categories, also
originally proposed by Katis, Sabadini and Walters, are a weakening of traced
monoidal categories, with various applications in computer science. A state
bootstrapping technique, which has appeared in several different contexts,
yields free such categories. We show that Span(Graph)* arises in this way,
being the free feedback category over Span(Set). Given that the latter can be
seen as an algebra of predicates, the algebra of open transition systems thus
arises - roughly speaking - as the result of bootstrapping state to that
algebra. Finally, we generalize feedback categories endowing state spaces with
extra structure: this extends the framework from mere transition systems to
automata with initial and final states.Comment: 48 pages, 33 figures, journal versio