Abstraction. As stated in the introduction, the strength of model learning is that it can produce simple models of complex systems. This, of course, depends on the application of an appropriate abstraction. In the above description of model learning, such an abstraction is hidden in functions beh and xxxX . While in practice, xxxX usually is the semantics associated with the class of models that is inferred by some learning algorithm, the function beh abstracts the actual observable behavior of a program to the level of this semantics. Angluin’s MAT framework, e.g., has been implemented for Xxxxx machine models over finite sets of inputs and outputs [46], where xxxX is a mapping from sequences of inputs to outputs. On the other hand, learning Xxxxx machine models of realistic software components requires a test harness which translates the abstract sequences of inputs to concrete seqeuences of method invocations on the component interface, and abstracts concrete return values of invocations to abstract outputs. The choice of a class of models requires the existence of a learning algo- rithm for this class of models as well as the definition of a function beh that abstracts concrete program executions to traces in the semantics of this class of models. Defining such an appropriate abstraction beh oftentimes is not trivial as it is required to be deterministic and determines the aspects of a component’s behavior that becomes observable. The extension of learning algorithms to richer classes of models is an effort that has two positive impacts in this scenario: On the one hand, using more expressive classes of models can help representing more interesting aspects of a component’s behavior in a model. On the other hand, using more expres- sive models can mitigate the laborious and often error-prone burden of defining appropriate functions beh. This has led to multiple lines of works that extend Angluin’s MAT framework to richer classes of models — most notably classes that can describe control-flow as well as data-flow or timing information. Extensions require finding right- congruences for more expressive classes of automata. One principal challenge that all these works face is that in a black-box setting, models can only be learned from observable behavior. Inferring complex causal relations like data manipu- lations or timed behavior quickly requires many queries and often has principle
Abstraction. → ∈ → The foundations of abstraction have been formalized by Xxxxxx & Cousot [CC77] in the abstract interpretation framework. In principle, the semantics of a pro- gram is specified using lattices. Lattices A and C are then used to specify state in the concrete and abstract domains, respectively, and importantly these lattices are connected by an abstraction function α : A C and a concretization function β : C A. For c C and a correct abstraction function α, the value α(c) then describes c in the sense that it contains c, and possibly more values. This form of imprecision preserves soundness, but may lead to false positive (or spurious) warnings. Often, abstract systems are sufficient to prove interesting system properties. However, if this is not the case, the abstraction has to be refined into a more precise representation of the concrete system semantics, an approach that has widely been automated using techniques such as counterexample guided abstraction refinement [GS97]. However, of course abstract interpretation techniques have widely been ap- plied to the verification of hybrid systems [Hen96]. For example, Xxxxxxx- xxxxxxxxx et al. [SDI08] have combined symbolic model checking with states encoded on top of template polyhedra, that is, conjunctions of linear in- i=0 equalities Σn ci · vi ≤ k where the ci are fixed a priori. However, such works target an entirely different setting than our work since it is entirely based on abstracting formally specified hybrid automata, whereas we focus on continuous-time models that may not necessarily have a formal semantics (the outputs may, for example, be computed using a controller that is directly connected to the system). Further, the scalability of complex abstractions such as template polyhedra in a network of components is uncertain. As stated by Xxxxxxxxxxxxxxxx et al. [SDI08, Sect. 1], “hybrid systems veri- fication is a challenge even for small systems”, which of course applies to networks of hybrid systems.
Abstraction form instances: A class can be de xxx only if there are in- stances in the relevant universe possessing all properties of the type that de nes the class.
Abstraction. For Xxxx, in order to get to grips with the dynamic manner in which a multitude of social difference comes to articulate with the capital form, and in doing so extend beyond the capital-labour-land economic relation, requires a method of analysis that appreciates the contingent and active role of socio-spatial difference as it articulates with capital. Xxxx critiques Hegelian idealism, to argue for a contingent and critical approach to deconstructing abstract categories, he argues that: “The method of rising from the abstract to the concrete is only the way in which thought appropriates the concrete, reproduces it as concrete in the mind” (Xxxx 1993, 101). For Xxxx even the simplest “category” of social life, one thought to be in its absolute essence, is in fact composed of multiple, contingent realities and thereby always already an abstraction; “it can never exist other than as an abstract, one sided relation within an already given concrete living whole”. Using the example of “the category of labour” Xxxx sets out his dialectical method as taking seriously the relation between historical contingency and abstract category. He argues the example of labour: “Shows strikingly how even the most abstract categories, despite their validity - precisely because of their abstractness - for all epochs, are nevertheless, in the specific character of this abstraction, themselves likewise a product of historic relations, and possess their full validity only for and within these relations” (Xxxx 1993, 105). Xxxx’x method of dialectically moving from the abstract to the concrete was drawn upon by Xxxxxx in his development of a dialectical schema in Justice, Nature and the Geography of Difference (1996). In this work Xxxxxx conceptualises a dialectical approach that focuses on: “Elements or ‘things’ [as] constituted out of flows, processes and relations…[these] things and systems which many researchers treat as irreducible and therefore unproblematic are seen in dialectical thought as internally contradictory by virtue of the multiple processes that constitute them...If all things are heterogeneous by virtue of the complex processes…which constitute them, then the only way we can understand the…attributes of ‘things’ is by understanding the processes and relationship which they internalise” (ibid, 49). Echoing Xxxx, Xxxxxx asserts an understanding of totalities, such as the “city”, as composed dialectically through multiple contingent relational processes. This conceptu...
Abstraction. Our world is filled with abstract symbols. Words are abstractions for things and ideas. Letters are abstractions for sounds, numbers for quantity. Three can mean three apples or three wars. Yet young children live in a concrete world. Mother is one person who feels and cares for the infant. Dog is grandpa's longhaired Eng- lish sheepdog. Playing is the process a child uses to slowly learn to move from a reliance of the concrete object in all his thinking to manipulating abstract con- cepts in his mind. Thus the doll at the center initially represents the boy's baby sister, later is any baby that needs to be comforted, and, finally, a teacher might say, "Is that your baby?," as the boy just rocks his empty arms. Blocks are used to create cities; a banana is a telephone; a Lego block is an apple. Eventually the child can talk about cities, telephones, and apples without these symbols. Social skills and moral rules When young children play in the dra- matic play area at child care, they usually select roles very different from who they are. They might choose to be a parent or teacher and direct the activity. Or, a more aggressive child might choose a sub- servient role. In this way the children learn what it’s like to be the other per- son in a real life situation. Because play lacks the pressure of product, evalua- tion, and time frames, children are free to try out new roles. They also learn to adjust their own roles for the good of the play activity. You can't have two children directing the activity. They learn to repress their indi- vidual wishes for the good of the group. Many play activities, especially ones that require children to cooperate – ball games, swings, skipping – teach chil- dren how to work together, how to take turns, and how to reciprocate. Social skill development is closely related to moral development. One pre- requisite for moral development is the ability for a child to put himself in the other child's shoes. Another is for the child to realize he needs to follow some basic rules for the good of the group if he wishes to continue to play. Thus play helps children through the complex process of moral development. Control of the environment A playing child selects toys and play materials to create an interesting activ- ity. As he gets bored, he will either select new materials or use the original ones in novel ways. Because a child does not like being bored, and because play is self-motivated, the child will continually select and man...
Abstraction. If M is a λ-term, and x is a variable, then (λx. M ) is a λ-term.
Abstraction. If xσ is a typed variable and Mτ is a typed λ-term, then (λxσ . Mτ )σ→τ is a typed λ-term.
