Fabio Zanasi | University College London (original) (raw)

conference papers by Fabio Zanasi

Logic in Computer Science (LICS)

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal cat... more String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory. An important role in many such approaches is played by equa-tional theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation for this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hy-pergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure. We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids.

Mathematical Foundations of Program Semantics (MFPS'16)

This paper establishes a link between Bayesian inference (learning) and predicate and state trans... more This paper establishes a link between Bayesian inference (learning) and predicate and state transformer operations from programming semantics and logic. Specifically, a very general definition of backward inference is given via first applying a predicate transformer and then conditioning. Analogously, forward inference involves first conditioning and then applying a state transformer. These definitions are illustrated in many examples in discrete and continuous probability theory and also in quantum theory.

Mathematical Foundations of Program Semantics (MFPS'16)

Recent work by the author with Bonchi and Soboci´nski shows how PROPs of linear relations can be ... more Recent work by the author with Bonchi and Soboci´nski shows how PROPs of linear relations can be presented by generator and equations through a so-called " cube construction " , based on letting very simple structures interact according to PROP operations of sum, fibered sum and composition via a distributive law. This paper shows how the same construction can be used in a cartesian setting to obtain presentations by generator and equations for the PROP of equivalence relations and of partial equivalence relations.

Coalgebraic Methods in Computer Science (CMCS), workshop of ETAPS'16

PROPs and Lawvere categories are related notions adapted to the study of algebraic structures bor... more PROPs and Lawvere categories are related notions adapted to the study of algebraic structures borne by an object in a category, but whereas PROPs are symmetric monoidal, Lawvere categories are cartesian. This paper formulates the connection between the two notions using Lack's technique for composing PROPs via distributive laws. We show Lawvere categories can be seen as resulting from a distributive law of two PROPs — one expressing the algebraic structure in linear form and the other expressing the ability of copying and discarding variables.

POPL 2015, 2015

Network theory uses the string diagrammatic language of monoidal categories to study graphical st... more Network theory uses the string diagrammatic language of monoidal categories to study graphical structures formally, eschewing specialised translations into intermediate formalisms. Recently, there has been a concerted research focus on developing a network theoretic approach to signal flow graphs, which are classical structures in control theory, signal processing and a cornerstone in the study of feedback. In this approach, signal flow graphs are given a relational denotational semantics in terms of formal power series.

Thus far, the operational behaviour of such signal flow graphs has only been discussed at an intuitive level. In this paper we equip them with a structural operational semantics. As is typically the case, the purely operational picture is too concrete -- two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised -- rewritten, using the graphical theory, into an executable form where the operational behavior and the denotation coincides.

CONCUR '14, Sep 2014

"We introduce IH, a sound and complete graphical theory of vector subspaces over the fi eld of p... more "We introduce IH, a sound and complete graphical theory of
vector subspaces over the fi eld of polynomial fractions, with relational composition. The theory is constructed in modular fashion, using Lack's approach to composing PROPs with distributive laws.
We then view string diagrams of IH as generalised stream circuits by using a formal Laurent series semantics. We characterize the subtheory where circuits adhere to the classical notion of signal
ow graphs, and illustrate the use of the graphical calculus on several examples."

CSL-LICS 2014, 2014

We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is eq... more We prove that the bisimulation-invariant fragment of weak monadic second-order logic (WMSO) is equivalent to the fragment of the modal µ-calculus where the application of the least fixpoint operator µp.ϕ is restricted to formulas ϕ that are continuous in p. Our proof is automata-theoretic in nature; in particular, we introduce a class of automata characterizing the expressive power of WMSO over tree models of arbitrary branching degree. The transition map of these automata is defined in terms of a logic FOE ∞ 1 that is the extension of firstorder logic with a generalized quantifier ∃ ∞ , where ∃ ∞ x.ϕ means that there are infinitely many objects satisfying ϕ. An important part of our work consists of a model-theoretic analysis of FOE ∞ 1 .

CMCS'14, 2014

We propose an abstract framework for modeling state-based systems with internal behavior as e.g. ... more We propose an abstract framework for modeling state-based systems with internal behavior as e.g. given by silent or ǫ-transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems.

FoSSaCS 2014, 2014

"Frobenius Algebras and Bialgebras are different ways in which monoids and comonoids can interact... more "Frobenius Algebras and Bialgebras are different ways in which monoids and comonoids can interact as part of the same theory. Such theories are important in many fields: e.g. quantum computation, compositional algebras of Petri nets, network algebra and component-based programming.
In this paper we study an important sub-theory of Coecke and Duncan's ZX-calculus, related to strongly-complementary observables, where two Frobenius algebras interact. We characterize its free model as a category of \Z_2-vector subspaces. Moreover, we use the framework of PROPs to exhibit the modular structure of its algebra via a universal construction involving span and cospans categories of \Z_2-matrices and distributive laws between PROPs. This approach shows the Frobenius structure results from the interaction of Bialgebras."

CALCO 2013, 2013

A series of recent papers introduces a coalgebraic semantics for logic programming, where the beh... more A series of recent papers introduces a coalgebraic semantics for logic programming, where the behavior of a goal is represented by a parallel model of computation called coinductive tree. This semantics fails to be compositional, in the sense that the coalgebra formalizing such behavior does not commute with the substitutions that may apply to a goal. We suggest that this is an instance of a more general phenomenon, occurring in the setting of interactive systems (in particular, nominal process calculi), when one tries to model their semantics with coalgebrae on presheaves. In those cases, compositionality can be obtained through saturation. We apply the same approach to logic programming: the resulting semantics is compositional and enjoys an elegant formulation in terms of coalgebrae on presheaves and their right Kan extensions.

LICS 2013, 2013

We provide a characterization theorem, in the style of van Benthem and Janin-Walukiewicz, for the... more We provide a characterization theorem, in the style of van Benthem and Janin-Walukiewicz, for the alternation-free fragment of the modal mu-calculus. For this purpose we introduce a variant of standard monadic second-order logic (MSO), which we call well-founded monadic second-order logic (WFMSO). When interpreted in a tree model, the second-order quantifiers of WFMSO range over subsets of well-founded subtrees. The first main result of the paper states that the expressive power of WFMSO over trees exactly corresponds to that of weak MSO-automata. Using this automata-theoretic characterization, we then show that, over the class of all transition structures, the bisimulation-invariant fragment of WFMSO is the alternation-free fragment of the modal mu-calculus. As a corollary, we find that the logics WFMSO and WMSO (weak monadic second-order logic, where second-order quantification concerns finite subsets), are incomparable in expressive power.

journal papers by Fabio Zanasi

Theoretical Computer Science, 2015

We propose an abstract framework for modeling state-based systems with internal behaviour as e.g.... more We propose an abstract framework for modeling state-based systems with internal behaviour as e.g. given by silent or epsilon-transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics
to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-
based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems and non-deterministic transducers.

Journal of Pure and Applied Algebra, 2014

We introduce the symmetric monoidal theory of Interacting Hopf Algebras for a principal ideal dom... more We introduce the symmetric monoidal theory of Interacting Hopf Algebras for a principal ideal domain R and we show that it characterizes the PROP of subspaces over the field of fractions of R.

Bialgebrae provide an abstract framework encompassing the semantics of different kinds of computa... more Bialgebrae provide an abstract framework encompassing the semantics of different kinds of computational models. In this paper we propose a bialgebraic approach to the semantics of logic programming. Our methodology is to study logic programs as reactive systems and exploit abstract techniques developed in that setting. First we use saturation to model the operational semantics of logic programs as coalgebrae on presheaves. Then, we make explicit the underlying algebraic structure by using bialgebrae on presheaves. The resulting semantics turns out to be compositional with respect to conjunction and term substitution. Also, it encodes a parallel model of computation, whose soundness is guaranteed by a built-in notion of synchronisation between different threads.

We introduce a graphical syntax for signal flow diagrams based on the language of symmetric monoi... more We introduce a graphical syntax for signal flow diagrams based on the language of symmetric monoidal categories. Using universal categorical constructions, we provide a stream semantics and a sound complete axiomatisation.
A certain class of diagrams captures the orthodox notion of signal flow graph used in control theory; we show that any diagram of our syntax can be realised, via rewriting in the equational theory, as a signal flow graph.

Books by Fabio Zanasi

Scientists in diverse fields use diagrammatic formalisms to reason about various kinds of network... more Scientists in diverse fields use diagrammatic formalisms to reason about various kinds of networks, or compound systems. Examples include electrical circuits, signal flow graphs, Penrose and Feynman diagrams, Bayesian networks, Petri nets, Kahn process networks, proof nets, UML specifications, amongst many others.
Graphical languages provide a convenient abstraction of some underlying mathematical formalism, which gives meaning to diagrams. For instance, signal flow graphs, foundational structures in control theory, are traditionally translated into systems of linear equations. This is typical: diagrammatic languages are used as an interface for more traditional mathematics, but rarely studied per se.
Recent trends in computer science analyse diagrams as first-class objects using formal methods from programming language semantics. In these approaches, diagrams are generated as the arrows of a PROP --- a special kind of monoidal category --- by a two-dimensional syntax and equations. The domain of interpretation of diagrams is also formalised as a PROP, and the (compositional) semantics is expressed as a functor preserving the PROP structure.
The first main contribution of this thesis is the characterisation of SV, the PROP of linear subspaces over a field k. This is an important domain of interpretation for diagrams appearing in diverse research areas, like the signal flow graphs mentioned above. We present by generators and equations the PROP IH of string diagrams whose free model is SV. The name IH stands for interacting Hopf algebras: indeed, the equations ofIH arise by distributive laws between Hopf algebras, which we obtain using S. Lack's technique for composing PROPs.
The significance of the result is two-fold. On the one hand, it offers a canonical string diagrammatic syntax for linear algebra: linear maps, kernels, subspaces and the standard linear algebraic transformations are all faithfully represented in the graphical language. On the other hand, the equations of IH describe familiar algebraic structures --- Hopf algebras and Frobenius algebras --- which are at the heart of graphical formalisms as seemingly diverse as quantum circuits, signal flow graphs, simple electrical circuits and Petri nets. Our characterisation enlightens the provenance of these axioms and reveals their linear algebraic nature.
Our second main contribution is an application of IH to the semantics of signal processing circuits. We develop a formal theory of signal flow graphs, featuring a string diagrammatic syntax for circuits, a structural operational semantics and a denotational semantics. We prove soundness and completeness of the equations ofIH for denotational equivalence. Also, we study the full abstraction question: it turns out that the purely operational picture is too concrete --- two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised --- rewritten, using the equations of IH, into an executable form where the operational behavior and the denotation coincide. This realisability theorem --- which is the culmination of our developments --- suggests a reflection about the role of causality in the semantics of signal flow graphs and, more generally, of computing devices.

"In this thesis we study the expressive power of variants of monadic second-order logic (MSO) on ... more "In this thesis we study the expressive power of variants of monadic second-order logic (MSO) on infinite trees by means of automata. In particular we are interested in weak MSO and well-founded MSO, where the second-order quantifiers range respectively over finite sets and over subsets of well-founded trees. On finitely branching trees, weak and well-founded MSO have the same expressive power and are both strictly weaker than MSO. The associated class of automata (called weak MSO-automata) is a restriction of the class characterizing MSO-expressivity.
We show that, on trees with arbitrary branching degree, weak MSO-automata characterize the expressive power of well-founded MSO, which turns out to be incomparable with weak MSO. Indeed, in this generalized setting, weak MSO gives an account of properties of the ‘horizontal dimension’ of trees, which cannot be described by means of MSO or well-founded MSO formulae.
In analogy with the result of Janin and Walukiewicz for MSO and the modal μ-calculus, this raises the issue of which modal logic captures the bisimulation-invariant fragment of well-founded MSO and weak MSO. We show that the alternation-free fragment of the modal μ-calculus and the bisimulation-invariant fragment of well-founded MSO have the same expressive power on trees of arbitrary branching degree. We motivate the conjecture that weak MSO modulo bisimulation collapses inside MSO and well-founded MSO."

Talks by Fabio Zanasi

Talk at the PhD defense (5 October 2015) Abstract of the PhD thesis: Scientists in diverse fiel... more Talk at the PhD defense (5 October 2015)

Abstract of the PhD thesis:

Scientists in diverse fields use diagrammatic formalisms to reason about various kinds of networks, or compound systems. Examples include electrical circuits, signal flow graphs, Penrose and Feynman diagrams, Bayesian networks, Petri nets, Kahn process networks, proof nets, UML specifications, amongst many others.
Graphical languages provide a convenient abstraction of some underlying mathematical formalism, which gives meaning to diagrams. For instance, signal flow graphs, foundational structures in control theory, are traditionally translated into systems of linear equations. This is typical: diagrammatic languages are used as an interface for more traditional mathematics, but rarely studied per se.
Recent trends in computer science analyse diagrams as first-class objects using formal methods from programming language semantics. In these approaches, diagrams are generated as the arrows of a PROP --- a special kind of monoidal category --- by a two-dimensional syntax and equations. The domain of interpretation of diagrams is also formalised as a PROP, and the (compositional) semantics is expressed as a functor preserving the PROP structure.
The first main contribution of this thesis is the characterisation of SV, the PROP of linear subspaces over a field k. This is an important domain of interpretation for diagrams appearing in diverse research areas, like the signal flow graphs mentioned above. We present by generators and equations the PROP IH of string diagrams whose free model is SV. The name IH stands for interacting Hopf algebras: indeed, the equations ofIH arise by distributive laws between Hopf algebras, which we obtain using S. Lack's technique for composing PROPs.
The significance of the result is two-fold. On the one hand, it offers a canonical string diagrammatic syntax for linear algebra: linear maps, kernels, subspaces and the standard linear algebraic transformations are all faithfully represented in the graphical language. On the other hand, the equations of IH describe familiar algebraic structures --- Hopf algebras and Frobenius algebras --- which are at the heart of graphical formalisms as seemingly diverse as quantum circuits, signal flow graphs, simple electrical circuits and Petri nets. Our characterisation enlightens the provenance of these axioms and reveals their linear algebraic nature.
Our second main contribution is an application of IH to the semantics of signal processing circuits. We develop a formal theory of signal flow graphs, featuring a string diagrammatic syntax for circuits, a structural operational semantics and a denotational semantics. We prove soundness and completeness of the equations ofIH for denotational equivalence. Also, we study the full abstraction question: it turns out that the purely operational picture is too concrete --- two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised --- rewritten, using the equations of IH, into an executable form where the operational behavior and the denotation coincide. This realisability theorem --- which is the culmination of our developments --- suggests a reflection about the role of causality in the semantics of signal flow graphs and, more generally, of computing devices.