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Volume 12, Issue 1
2016
1. Type-checking Liveness for Collaborative Processes with Bounded and Unbounded Recursion
We present the first session typing system guaranteeing request-response liveness properties for possibly non-terminating communicating processes. The types augment the branch and select types of the standard binary session types with a set of required responses, indicating that whenever a particular label is selected, a set of other labels, its responses, must eventually also be selected. We prove that these extended types are strictly more expressive than standard session types. We provide a type system for a process calculus similar to a subset of collaborative BPMN processes with internal (data-based) and external (event-based) branching, message passing, bounded and unbounded looping. We prove that this type system is sound, i.e., it guarantees request-response liveness for dead-lock free processes. We exemplify the use of the calculus and type system on a concrete example of an infinite state system.
2. Proof equivalence in MLL is PSPACE-complete
MLL proof equivalence is the problem of deciding whether two proofs in multiplicative linear logic are related by a series of inference permutations. It is also known as the word problem for star-autonomous categories. Previous work has shown the problem to be equivalent to a rewiring problem on proof nets, which are not canonical for full MLL due to the presence of the two units. Drawing from recent work on reconfiguration problems, in this paper it is shown that MLL proof equivalence is PSPACE-complete, using a reduction from Nondeterministic Constraint Logic. An important consequence of the result is that the existence of a satisfactory notion of proof nets for MLL with units is ruled out (under current complexity assumptions). The PSPACE-hardness result extends to equivalence of normal forms in MELL without units, where the weakening rule for the exponentials induces a similar rewiring problem.
3. Weak topologies for Linear Logic
We construct a denotational model of linear logic, whose objects are all the locally convex and separated topological vector spaces endowed with their weak topology. The negation is interpreted as the dual, linear proofs are interpreted as continuous linear functions, and non-linear proofs as sequences of monomials. We do not complete our constructions by a double-orthogonality operation. This yields an interpretation of the polarity of the connectives in terms of topology.
4. (Leftmost-Outermost) Beta Reduction is Invariant, Indeed
Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is lambda-calculus a reasonable machine? Is there a way to measure the computational complexity of a lambda-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of lambda-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating lambda-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the lambda-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in […]
5. Separating Regular Languages with First-Order Logic
Given two languages, a separator is a third language that contains the first one and is disjoint from the second one. We investigate the following decision problem: given two regular input languages of finite words, decide whether there exists a first-order definable separator. We prove that in order to answer this question, sufficient information can be extracted from semigroups recognizing the input languages, using a fixpoint computation. This yields an EXPTIME algorithm for checking first-order separability. Moreover, the correctness proof of this algorithm yields a stronger result, namely a description of a possible separator. Finally, we generalize this technique to answer the same question for regular languages of infinite words.
6. Simulation Problems Over One-Counter Nets
One-counter nets (OCN) are finite automata equipped with a counter that can store non-negative integer values, and that cannot be tested for zero. Equivalently, these are exactly 1-dimensional vector addition systems with states. We show that both strong and weak simulation preorder on OCN are PSPACE-complete.
7. History-Register Automata
Programs with dynamic allocation are able to create and use an unbounded number of fresh resources, such as references, objects, files, etc. We propose History-Register Automata (HRA), a new automata-theoretic formalism for modelling such programs. HRAs extend the expressiveness of previous approaches and bring us to the limits of decidability for reachability checks. The distinctive feature of our machines is their use of unbounded memory sets (histories) where input symbols can be selectively stored and compared with symbols to follow. In addition, stored symbols can be consumed or deleted by reset. We show that the combination of consumption and reset capabilities renders the automata powerful enough to imitate counter machines, and yields closure under all regular operations apart from complementation. We moreover examine weaker notions of HRAs which strike different balances between expressiveness and effectiveness.
8. A Sorted Semantic Framework for Applied Process Calculi
Applied process calculi include advanced programming constructs such as type systems, communication with pattern matching, encryption primitives, concurrent constraints, nondeterminism, process creation, and dynamic connection topologies. Several such formalisms, e.g. the applied pi calculus, are extensions of the the pi-calculus; a growing number is geared towards particular applications or computational paradigms. Our goal is a unified framework to represent different process calculi and notions of computation. To this end, we extend our previous work on psi-calculi with novel abstract patterns and pattern matching, and add sorts to the data term language, giving sufficient criteria for subject reduction to hold. Our framework can directly represent several existing process calculi; the resulting transition systems are isomorphic to the originals up to strong bisimulation. We also demonstrate different notions of computation on data terms, including cryptographic primitives and a lambda-calculus with erratic choice. Finally, we prove standard congruence and structural properties of bisimulation; the proof has been machine-checked using Nominal Isabelle in the case of a single name sort.
9. Order-Invariant Types and Their Applications
Our goal is to show that the standard model-theoretic concept of types can be applied in the study of order-invariant properties, i.e., properties definable in a logic in the presence of an auxiliary order relation, but not actually dependent on that order relation. This is somewhat surprising since order-invariant properties are more of a combinatorial rather than a logical object. We provide two applications of this notion. One is a proof, from the basic principles, of a theorem by Courcelle stating that over trees, order-invariant MSO properties are expressible in MSO with counting quantifiers. The other is an analog of the Feferman-Vaught theorem for order-invariant properties.