Presenting Distributive Laws

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages

Pi-Calculus in Logical Form

Abramsky’s logical formulation of domain theory is extended to encompass the domain theoretic model for picalculus processes of Stark and of Fiore, Moggi and Sangiorgi. This is done by defining a logical counterpart of categorical constructions including dynamic name allocation and name exponentiation, and showing that they are dual to standard constructs in functor categories. We show that initial algebras of functors defined in terms of these constructs give rise to a logic that is sound, complete, and characterises bisimilarity. The approach is modular, and we apply it to derive a logical formulation of pi-calculus. The resulting logic is a modal calculus with primitives for input, free output and bound output

Automata for context-dependent connectors

Recent approaches to component-based software engineering employ coordinating connectors to compose components into software systems. For maximum flexibility and reuse, such connectors can themselves be composed, resulting in an expressive calculus of connectors whose semantics encompasses complex combinations of synchronisation, mutual exclusion, non-deterministic choice and state-dependent behaviour. A more expressive notion of connector includes also context-dependent behaviour, namely, whenever the choices the connector can take change non-monotonically as the context, given by the pending activity on its ports, changes. Context dependency can express notions of priority and inhibition. Capturing context-dependent behaviour in formal models is non-trivial, as it is unclear how to propagate context in- formation through composition. In this paper we present an intuitive automata-based formal model of context-dependent connectors, and argue that it is superior to previous attempts at such a model for the coordination language Reo

Simulating Quantum Circuits by Model Counting

Quantum circuit compilation comprises many computationally hard reasoning tasks that nonetheless lie inside #$\mathbf{P}$ and its decision counterpart in $\mathbf{PP}$. The classical simulation of general quantum circuits is a core example. We show for the first time that a strong simulation of universal quantum circuits can be efficiently tackled through weighted model counting by providing a linear encoding of Clifford+T circuits. To achieve this, we exploit the stabilizer formalism by Knill, Gottesmann, and Aaronson and the fact that stabilizer states form a basis for density operators. With an open-source simulator implementation, we demonstrate empirically that model counting often outperforms state-of-the-art simulation techniques based on the ZX calculus and decision diagrams. Our work paves the way to apply the existing array of powerful classical reasoning tools to realize efficient quantum circuit compilation; one of the obstacles on the road towards quantum supremacy

Towards an infinitary logic of domains : Abramsky logic for transition systems

We give a new characterization of sober spaces in terms of their completely distributive lattice of saturated sets. This characterization is used to extend Abramsky's results about a domain logic for transition systems. The Lindenbaum algebra generated by the Abramsky finitary logic is a distributive lattice dual to an SFP-domain obtained as a solution of a recursive domain equation. We prove that the Lindenbaum algebra generated by the infinitary logic is a completely distributive lattice dual to the same SFP-domain. As a consequence soundness and completeness of the infinitary logic is obtained for a class of transition systems that is computational interesting

Interacting via the Heap in the Presence of Recursion

Almost all modern imperative programming languages include operations for dynamically manipulating the heap, for example by allocating and deallocating objects, and by updating reference fields. In the presence of recursive procedures and local variables the interactions of a program with the heap can become rather complex, as an unbounded number of objects can be allocated either on the call stack using local variables, or, anonymously, on the heap using reference fields. As such a static analysis is, in general, undecidable. In this paper we study the verification of recursive programs with unbounded allocation of objects, in a simple imperative language for heap manipulation. We present an improved semantics for this language, using an abstraction that is precise. For any program with a bounded visible heap, meaning that the number of objects reachable from variables at any point of execution is bounded, this abstraction is a finitary representation of its behaviour, even though an unbounded number of objects can appear in the state. As a consequence, for such programs model checking is decidable. Finally we introduce a specification language for temporal properties of the heap, and discuss model checking these properties against heap-manipulating programs.Comment: In Proceedings ICE 2012, arXiv:1212.345

Equivalence Checking of Quantum Circuits by Model Counting

Verifying equivalence between two quantum circuits is a hard problem, that is nonetheless crucial in compiling and optimizing quantum algorithms for real-world devices. This paper gives a Turing reduction of the (universal) quantum circuits equivalence problem to weighted model counting (WMC). Our starting point is a folklore theorem showing that equivalence checking of quantum circuits can be done in the so-called Pauli-basis. We combine this insight with a WMC encoding of quantum circuit simulation, which we extend with support for the Toffoli gate. Finally, we prove that the weights computed by the model counter indeed realize the reduction. With an open-source implementation, we demonstrate that this novel approach can outperform a state-of-the-art equivalence-checking tool based on ZX calculus and decision diagrams