Proof Relevant Corecursive Resolution

Resolution lies at the foundation of both logic programming and type class context reduction in functional languages. Terminating derivations by resolution have well-defined inductive meaning, whereas some non-terminating derivations can be understood coinductively. Cycle detection is a popular method to capture a small subset of such derivations. We show that in fact cycle detection is a restricted form of coinductive proof, in which the atomic formula forming the cycle plays the role of coinductive hypothesis. This paper introduces a heuristic method for obtaining richer coinductive hypotheses in the form of Horn formulas. Our approach subsumes cycle detection and gives coinductive meaning to a larger class of derivations. For this purpose we extend resolution with Horn formula resolvents and corecursive evidence generation. We illustrate our method on non-terminating type class resolution problems.Comment: 23 pages, with appendices in FLOPS 201

Correlation of the rate of Type Ia supernovae with the parent galaxy properties: Light and shadows

The identification of the progenitors of Type Ia Supernovae (SNIa) is extremely important in several astrophysical contexts, ranging from stellar evolution in close binary systems to evaluating cosmological parameters. Determining the distribution of the delay times (DTD) of SNIa progenitors can shed light on their nature. In this paper we investigate on the diagnostic capabilities on the DTD of the correlation between the SNIa rate and the parent galaxy properties by examining its systematics with the various parameters at play: simple stellar population models, the adopted description for the star formation history in galaxies, and the way in which the masses of the galaxies are evaluated. We compute models for the correlations of the SNIa rate with the parent galaxy color and specific star formation rate for a variety of input ingredients, and for a few astrophysically motivated DTD laws. The models are compared to the results of three independent observational surveys. We find that the scaling of the SNIa rate with the properties of the parent galaxy is sensitive to all input ingredients mentioned above. This is a severe limitation on the possibility to discriminate alternative DTDs. In addition, current surveys show some discrepancies for the rate measured in the reddest and bluest galaxies, likely due to limited statistics and inhomogeneity of the observations. For galaxies with intermediate colors the rates are in agreement, leading to a robust determination of the productivity of SNIa from stellar populations of $\simeq$ 0.8 events per 1000 \msun. Large stastistics of SNIa events along with accurate measurements of the star formation history in the galaxies are required to derive firm constraints on the DTD. LSST will achieve these results by providing the homogeneous, unbiased and vast database on both SNIa and galaxies.Comment: Astronomy and Astrophysics in press. Includes one more figure in the appendix. Notice the slight change of titl

Adequacy of compositional translations for observational semantics

We investigate methods and tools for analysing translations between programming languages with respect to observational semantics. The behaviour of programs is observed in terms of may- and must-convergence in arbitrary contexts, and adequacy of translations, i.e., the reflection of program equivalence, is taken to be the fundamental correctness condition. For compositional translations we propose a notion of convergence equivalence as a means for proving adequacy. This technique avoids explicit reasoning about contexts, and is able to deal with the subtle role of typing in implementations of language extension

Correctness of an STM Haskell implementation

A concurrent implementation of software transactional memory in Concurrent Haskell using a call-by-need functional language with processes and futures is given. The description of the small-step operational semantics is precise and explicit, and employs an early abort of conflicting transactions. A proof of correctness of the implementation is given for a contextual semantics with may- and should-convergence. This implies that our implementation is a correct evaluator for an abstract specification equipped with a big-step semantics

Dynamic Congruence vs. Progressing Bisimulation for CCS

Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g. \alpha.\tau.\beta.nil and \alpha.\beta.nil are woc but \tau.\beta.nil and \beta.nil are not. This fact prevent us from characterizing CCS semantics (when \tau is considered invisible) as a final algebra, since the semantic function would induce an equivalence over the agents that is both a congruence and a bisimulation. In the paper we introduce a new behavioural equivalence for CCS agents, which is the coarsest among those bisimulations which are also congruences. We call it Dynamic Observational Congruence because it expresses a natural notion of equivalence for concurrent systems required to simulate each other in the presence of dynamic, i.e. run time, (re)configurations. We provide an algebraic characterization of Dynamic Congruence in terms of a universal property of finality. Furthermore we introduce Progressing Bisimulation, which forces processes to simulate each other performing explicit steps. We provide an algebraic characterization of it in terms of finality, two logical characterizations via modal logic in the style of HML and a complete axiomatization for finite agents (consisting of the axioms for Strong Observational Congruence and of two of the three Milner's $\tau$-laws). Finally, we prove that Dynamic Congruence and Progressing Bisimulation coincide for CCS agents

Cosmic voids in modified gravity scenarios

Modified gravity (MG) theories aim to reproduce the observed acceleration of the Universe by reducing the dark sector while simultaneously recovering General Relativity (GR) within dense environments. Void studies appear to be a suitable scenario to search for imprints of alternative gravity models on cosmological scales. Voids cover an interesting range of density scales where screening mechanisms fade out, which reaches from a density contrast $\delta \approx -1$ close to their centers to $\delta \approx 0$ close to their boundaries. We present an analysis of the level of distinction between GR and two modified gravity theories, the Hu-Sawicki $f(R)$ and the symmetron theory. This study relies on the abundance, linear bias, and density profile of voids detected in n-body cosmological simulations. We define voids as connected regions made up of the union of spheres with a {\it \textup{mean}} density given by $\overline\rho_v=0.2\,\overline\rho_m$, but disconnected from any other voids. We find that the height of void walls is considerably affected by the gravitational theory, such that it increases for stronger gravity modifications. Finally, we show that at the level of dark matter n-body simulations, our constraints allow us to distinguish between GR and MG models with $|f_{R0}| > 10^{-6}$ and $z_{SSB} > 1$. Differences of best-fit values for MG parameters that are derived independently from multiple void probes may indicate an incorrect MG model. This serves as an important consistency check.Comment: 15 pages, 12 figure

Fitting density models to observational data - The local Schmidt law in molecular clouds

We consider the general problem of fitting a parametric density model to discrete observations, taken to follow a non-homogeneous Poisson point process. This class of models is very common, and can be used to describe many astrophysical processes, including the distribution of protostars in molecular clouds. We give the expression for the likelihood of a given spatial density distribution of protostars and apply it to infer the most probable dependence of the protostellar surface density on the gas surface density. Finally, we apply this general technique to model the distribution of protostars in the Orion molecular cloud and robustly derive the local star formation scaling (Schmidt) law for a molecular cloud. We find that in this cloud the protostellar surface density, $\Sigma_\mathrm{YSO}$, is directly proportional to the square gas column density, here expressed as infrared extinction in the $K$-band, $A_K$: more precisely, $\Sigma_\mathrm{YSO} = (1.65 \pm 0.19) A_K^{(2.03 \pm 0.15)}$ stars pc$^{-2}$.Comment: 6 pages, A&A in press. References fixe