# Volume 12, Issue 4

2016

### 1. Extending Unification in $\mathcal{EL}$ to Disunification: The Case of Dismatching and Local Disunification

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic $\mathcal{EL}$ to disunification since negative constraints can be used to avoid unwanted unifiers. While decidability of the solvability of general $\mathcal{EL}$-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we consider only solutions that are constructed from terms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of disunification problems.

### 2. Model Checking Spatial Logics for Closure Spaces

Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of programs; however, properties of space are typically not taken into account explicitly. We present a topology-based approach to formal verification of spatial properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. We lift the topological definitions to the more general setting of closure spaces, also encompassing discrete, graph-based structures. We extend the framework with a spatial surrounded operator, a propagation operator and with some collective operators. The latter are interpreted over arbitrary sets […]

### 3. Some theorems on passing from local to global presence of properties of functions

When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to all of them belong to this class. The collections with the formulated property are said to be strongly join permitting for the given class (the notion of join permitting collection is defined in the same way, but without the words "a subset of"). Three theorems concerning certain instances of the problem are proved. A necessary and sufficient condition for being strongly join permitting is given for the case when, for some $n$, the class consists of the potentially partial recursive functions of $n$ variables, and the collection consists of sets of $n$-tuples of natural numbers. The second theorem gives a sufficient condition for the case when the class consists of the continuous partial functions between two given topological spaces, and […]

### 4. Borel-piecewise continuous reducibility for uniformization problems

We study a fine hierarchy of Borel-piecewise continuous functions, especially, between closed-piecewise continuity and $G_\delta$-piecewise continuity. Our aim is to understand how a priority argument in computability theory is connected to the notion of $G_\delta$-piecewise continuity, and then we utilize this connection to obtain separation results on subclasses of $G_\delta$-piecewise continuous reductions for uniformization problems on set-valued functions with compact graphs. This method is also applicable for separating various non-constructive principles in the Weihrauch lattice.

### 5. Discovering, quantifying, and displaying attacks

In the design of software and cyber-physical systems, security is often perceived as a qualitative need, but can only be attained quantitatively. Especially when distributed components are involved, it is hard to predict and confront all possible attacks. A main challenge in the development of complex systems is therefore to discover attacks, quantify them to comprehend their likelihood, and communicate them to non-experts for facilitating the decision process. To address this three-sided challenge we propose a protection analysis over the Quality Calculus that (i) computes all the sets of data required by an attacker to reach a given location in a system, (ii) determines the cheapest set of such attacks for a given notion of cost, and (iii) derives an attack tree that displays the attacks graphically. The protection analysis is first developed in a qualitative setting, and then extended to quantitative settings following an approach applicable to a great many contexts. The […]

### 6. Automata for Specifying and Orchestrating Service Contracts

An approach to the formal description of service contracts is presented in terms of automata. We focus on the basic property of guaranteeing that in the multi-party composition of principals each of them gets his requests satisfied, so that the overall composition reaches its goal. Depending on whether requests are satisfied synchronously or asynchronously, we construct an orchestrator that at static time either yields composed services enjoying the required properties or detects the principals responsible for possible violations. To do that in the asynchronous case we resort to Linear Programming techniques. We also relate our automata with two logically based methods for specifying contracts.

### 7. Honesty by Typing

We propose a type system for a calculus of contracting processes. Processes can establish sessions by stipulating contracts, and then can interact either by keeping the promises made, or not. Type safety guarantees that a typeable process is honest - that is, it abides by the contracts it has stipulated in all possible contexts, even in presence of dishonest adversaries. Type inference is decidable, and it allows to safely approximate the honesty of processes using either synchronous or asynchronous communication.

### 8. On the locality of arb-invariant first-order formulas with modulo counting quantifiers

We study Gaifman locality and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MOD_p, for short) with arbitrary numerical predicates. We require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas. This paper gives a detailed picture of locality and non-locality properties of arb-invariant FO+MOD_p. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MOD_p is neither Hanf nor Gaifman local with respect to a sublinear locality radius. However, in case that p is an odd prime power, it is weakly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime powers p, arb-invariant FO+MOD_p is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MOD_p formulas […]

### 9. On linear rewriting systems for Boolean logic and some applications to proof theory

Linear rules have played an increasing role in structural proof theory in recent years. It has been observed that the set of all sound linear inference rules in Boolean logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left- and right-)linear rewrite rule. In this paper we study properties of systems consisting only of linear inferences. Our main result is that the length of any 'nontrivial' derivation in such a system is bound by a polynomial. As a consequence there is no polynomial-time decidable sound and complete system of linear inferences, unless coNP=NP. We draw tools and concepts from term rewriting, Boolean function theory and graph theory in order to access some required intermediate results. At the same time we make several connections between these areas that, to our knowledge, have not yet been presented and constitute a rich theoretical framework for reasoning about linear TRSs for Boolean logic.

### 10. Coalgebraic trace semantics via forgetful logics

We use modal logic as a framework for coalgebraic trace semantics, and show the flexibility of the approach with concrete examples such as the language semantics of weighted, alternating and tree automata, and the trace semantics of generative probabilistic systems. We provide a sufficient condition under which a logical semantics coincides with the trace semantics obtained via a given determinization construction. Finally, we consider a condition that guarantees the existence of a canonical determinization procedure that is correct with respect to a given logical semantics. That procedure is closely related to Brzozowski's minimization algorithm.

### 11. On the Mints Hierarchy in First-Order Intuitionistic Logic

We stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the $\Delta_2$ level is undecidable and that $\Sigma_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $\Sigma_1$ is complete for co-Nexptime.

### 12. Compositional bisimulation metric reasoning with Probabilistic Process Calculi

We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of non-expansiveness) captures the essential nature of compositional reasoning and allows now also to reason compositionally about recursive processes. We characterize the distance between probabilistic processes composed by standard process algebra operators. Combining these results, we demonstrate how compositional reasoning about systems specified by continuous process algebra operators allows for metric assume-guarantee like performance validation.

### 13. Encoding Monomorphic and Polymorphic Types

Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to untyped first-order logic. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of "cover". The new encodings are implemented in Isabelle/HOL as part of the Sledgehammer tool. We include informal proofs of soundness and correctness, and have formalised the monomorphic part of this work in Isabelle/HOL. Our evaluation finds the new encodings vastly superior to previous schemes.