Extended addressing machines (EAMs) have been introduced to represent higher-order sequential computations. Previously, we have shown that they are capable of simulating -- via an easy encoding -- the operational semantics of PCF, extended with explicit substitutions. In this paper we prove that the simulation is actually an equivalence: a PCF program terminates in a numeral exactly when the corresponding EAM terminates in the same numeral. It follows that the model of PCF obtained by quotienting typable EAMs by a suitable logical relation is adequate. From a definability result stating that every EAM in the model can be transformed into a PCF program with the same observational behavior, we conclude that the model is fully abstract for PCF. arXiv admin note: text overlap with arXiv:2212.11147

Turing machines and register machines have been used for decades in theoretical computer science as abstract models of computation. Also the $\lambda$-calculus has played a central role in this domain as it allows to focus on the notion of functional computation, based on the substitution mechanism, while abstracting away from implementation details. The present article starts from the observation that the equivalence between these formalisms is based on the Church-Turing Thesis rather than an actual encoding of $\lambda$-terms into Turing (or register) machines. The reason is that these machines are not well-suited for modelling $\lambda$-calculus programs. We study a class of abstract machines that we call "addressing machine" since they are only able to manipulate memory addresses of other machines. The operations performed by these machines are very elementary: load an address in a register, apply a machine to another one via their addresses, and call the address of another machine. We endow addressing machines with an operational semantics based on leftmost reduction and study their behaviour. The set of addresses of these machines can be easily turned into a combinatory algebra. In order to obtain a model of the full untyped $\lambda$-calculus, we need to introduce a rule that bares similarities with the $\omega$-rule and the rule $\zeta_\beta$ from combinatory logic.

The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees.

A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.

Healthcare organizations are increasingly pushed to improve the quality of care service taking into account the increasing complexity in patient treatment and the continuous reduction of available resources. The adoption of Business Process Management (BPM) practices is thus becoming a key enabler for the improvement of healthcare processes (HPs). Accordingly, methods and tools are required to address behavioral and performance aspects from the early phases of the process lifecycle in order to improve the quality of healthcare, reduce costly reworks and increase the effectiveness of BPM approaches. This paper specifically addresses the specification and analysis phases of the process lifecycle and introduces a model-driven method for healthcare process simulation. The proposed method is based on a model transformation approach that takes as input the process specification in BPMN, appropriately extended to include the performance properties of the process, and yields as output the corresponding process simulation code, ready to be executed. In order to illustrate the method and its effectiveness, the paper describes an example application to a process dealing with the hip fracture for elderly patients.

In this paper we present SyBeL (System Behaviour modelling Language), an XML based formalism for software system modelling. In particular, SyBeL focuses on the description of the system behaviour in order to capture its functional requirements and has been designed to fulfill some of the most trendy software engineering issues. The use of the underlying XML language makes the artifacts generated by SyBeL immediately available to further automatic manipulation (e. g., to automatically generate test cases) without the need of intermediate models, as usually done in semi-formal approaches. Moreover, we are experimenting SyBeL on a variety of practical case studies.

The λ-theory ℋ is obtained from β-conversion by identifying all closed unsolvable terms (or, equivalently, terms without head normal form). The range problem for the theory ℋ asks whether a closed term has always (up to equality in ℋ) either an infinite range or a singleton range (that is, it is a constant function). Here we give a solution to a natural version of this problem, giving a positive answer for the theory ℋ restricted to Combinatory Logic. The method of proof applies also to the Lazy λ-Calculus.

In this paper we present SMDP (Scenario Model Development Process), an XML-based methodology for the description and manipulation of scenarios that are used to formalize and reuse software requirements. SMDP is an iterative and incremental process that supports scenario evolution during the requirements engineering process. The formalization of scenarios through the underlying XML-based language of SMDP makes them immediately available to further automatic manipulation (e.g., to automatically generate test cases) without the need for intermediate models, as it is usually done in semi-formal approaches. Thanks to the implementation of a software assistant environment for SMDP, the methodology is currently being experimented on a variety of case studies, in particular web applications.

H is the theory extending β-conversion by identifying all closed unsolvables. Hω is the closure of this theory under the ω-rule (and β-conversion). A long-standing conjecture of H. Barendregt states that the provable equations of Hω form Π11-complete set. Here we prove that conjecture.

We give a many-one reduction of the set of true /spl Pi//sub 2//sup 0/ sentences to the set of consequences of the lambda calculus with the omega rule. This solves in the affirmative a well known problem of H. Barendregt. The technique of proof has interest in itself and can be extended to prove that the theory which identifies all unsolvable terms together with the omega rule is H/sub 1//sup 1/-complete which solves another long-standing conjecture of H. Barendregt.

Scenarios are a new way of representing knowledge that has been attracting a lot of attention from practitioners and researchers. In this paper we present the SDML formalism, an XML definition language to support scenario-based requirements engineering. The definition of scenarios through SDML enables to exploit the emerging XML technologies in order to offer powerful ways to create, maintain, distribute and use scenarios. Moreover, we are experimenting the SDML language on a variety of practical case studies.

The main barrier to automatic verification of concurrent systems is the huge amount of memory required to complete the verification task (state explosion). In this paper we present a probabilistic algorithm for automatic verification via model checking. Our algorithm trades space with time. In particular, when memory is full because of state explosion our algorithm does not give up verification. Instead it just proceeds at a lower speed and its results will only hold with some arbitrarily small error probability. Our preliminary experimental results show that by using our probabilistic algorithm we can typically save more than 30% of RAM with an average time penalty of about 100% w.r.t. a deterministic state space exploration with enough memory to complete the verification task. This is better than giving up the verification task because of lack of memory.

Terms finitely representing infinite sequences of finite first-order terms have received attention by several authors. In this paper, we consider the class of recurrent terms proposed by H. Chen and J. Hsiang, and we extend it to allow infinite terms. This extension helps in clarifying the relationships between matching and unification over the class of terms we consider, that we call iterative terms. In fact, it holds that if a term s matches a term t by a substitution Γ, then the limit of iterations of the matching Γ, if it exists, is a most general unifier of s and t. A crucial feature of iterative terms is the notion of maximally-folded normal form that allows for a comprehensive treatment of both finite and infinite iterative terms. In this setting, infinite terms can be simply characterized as limits of sequences of finite terms. For finite terms we positively settle an open problem of H. Chen and J. Hsiang on the number of most general unifiers for a pair of terms.