Inductive Definition and Domain Theoretic Properties of Fully Abstract

A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF + "parallel conditional function"), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent non-deterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given level-by-level in the finite type hierarchy. To prove full abstraction and non-dcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF^+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are non-omega-complete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omega-completeness), "naturally" continuous (with respect to existing directed "pointwise", or "natural" lubs) and also "naturally" omega-algebraic and "naturally" bounded complete -- appropriate generalisation of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page

Non uniform (hyper/multi)coherence spaces

In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, vertices represent results of computations and the edge relation witnesses the ability of being assembled into a same piece of data or a same (strongly) stable function, at arrow types. In (hyper)coherence semantics, the argument of a (strongly) stable functional is always a (strongly) stable function. As a consequence, comparatively to the relational semantics, where there is no edge relation, some vertices are missing. Recovering these vertices is essential for the purpose of reconstructing proofs/terms from their interpretations. It shall also be useful for the comparison with other semantics, like game semantics. In [BE01], Bucciarelli and Ehrhard introduced a so called non uniform coherence space semantics where no vertex is missing. By constructing the co-free exponential we set a new version of this last semantics, together with non uniform versions of hypercoherences and multicoherences, a new semantics where an edge is a finite multiset. Thanks to the co-free construction, these non uniform semantics are deterministic in the sense that the intersection of a clique and of an anti-clique contains at most one vertex, a result of interaction, and extensionally collapse onto the corresponding uniform semantics.Comment: 32 page

Extensional Collapse Situations I: non-termination and unrecoverable errors

We consider a simple model of higher order, functional computation over the booleans. Then, we enrich the model in order to encompass non-termination and unrecoverable errors, taken separately or jointly. We show that the models so defined form a lattice when ordered by the extensional collapse situation relation, introduced in order to compare models with respect to the amount of "intensional information" that they provide on computation. The proofs are carried out by exhibiting suitable applied {\lambda}-calculi, and by exploiting the fundamental lemma of logical relations

On Linear Information Systems

Scott's information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a new-Seely category), with a "set-theoretic" interpretation of exponentials that recovers Scott continuous functions via the co-Kleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic

The Small-Is-Very-Small Principle

The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable cut, then it shows that the property has a very small witness: i.e. a witness below a given standard number. We draw various consequences from the central result. For example (in rough formulations): (i) Every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative w.r.t. formulas of complexity $\leq n$. (ii) Every sequential model has, for any $n$, an extension that is elementary for formulas of complexity $\leq n$, in which the intersection of all definable cuts is the natural numbers. (iii) We have reflection for $\Sigma^0_2$-sentences with sufficiently small witness in any consistent restricted theory $U$. (iv) Suppose $U$ is recursively enumerable and sequential. Suppose further that every recursively enumerable and sequential $V$ that locally inteprets $U$, globally interprets $U$. Then, $U$ is mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measure depth of quantifier alternations

Pairs, sets and sequences in first-order theories

Asuransi sebagai aktivitas bisnis diharuskan memenuhi prinsip-prinsip hukum asuransi. Salah satu prinsip yang harus dipegang teguh adalah principle of  utmost good faith, di samping prinsip yang lain. Prinsip ini berbunyi bahwa seorang tertanggung wajib memberi informasi secara jujur terhadap apa yang dipertanggungkan kepada penanggung. Dalam bisnis Islam, kejujuran merupakan prinsip yang harus dijunjung tinggi. Secara hukum, prinsip ini diatur dalam KUH Dagang. Persoalannya adalah apakah prinsip ini dianggap cukup dari sudut pandang hukum perjanjian syariah. Secara sekilas bahwa prinsip iktikad baik sempurna ini telah memenuhi asas perjanjian syariah, namun demikian tidak memiliki kriteria maksimal kejujuran. Ketiadaan kejujuran dalam bisnis asuransi akan berdampak pada batalnya perjanjian asuransi karena ada unsur cacat kehendak (‘uyub ar-ridla). Insurance as a business activity must fulfill principles of insurance law. One of the principles that must be hold on is the principle of  utmost good faith. The principle says that an endured person must honestly give information of  what should be given responsibility to the guarantor. In Islamic business, honesty is a principle that should be respected. From point of  view of  law, the principle is settled in commerce law. The problem is that whether the principle is represenative enough if it is viewed from law of  syariah agreement. At glance, the principle has fulfilled the basic of syariah agreement, however, it does not have maximum criteria of  honesty. Unavailability of honesty in insurance business will give effect of  invalidate of  insurance agreement, for there is a deformity of desire (‘uyub ar-ridla).</p