40,757 research outputs found
Nominal Abstraction
Recursive relational specifications are commonly used to describe the
computational structure of formal systems. Recent research in proof theory has
identified two features that facilitate direct, logic-based reasoning about
such descriptions: the interpretation of atomic judgments through recursive
definitions and an encoding of binding constructs via generic judgments.
However, logics encompassing these two features do not currently allow for the
definition of relations that embody dynamic aspects related to binding, a
capability needed in many reasoning tasks. We propose a new relation between
terms called nominal abstraction as a means for overcoming this deficiency. We
incorporate nominal abstraction into a rich logic also including definitions,
generic quantification, induction, and co-induction that we then prove to be
consistent. We present examples to show that this logic can provide elegant
treatments of binding contexts that appear in many proofs, such as those
establishing properties of typing calculi and of arbitrarily cascading
substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio
Nominal Abstraction
International audienceRecursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and co-induction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments
Automatically Deriving Schematic Theorems for Dynamic Contexts
International audienceHypothetical judgments go hand-in-hand with higher-order abstract syntax for meta-theoretic reasoning. Such judgments have two kinds of assumptions: those that are statically known from the specification, and the dynamic assumptions that result from building derivations out of the specification clauses. These dynamic assumptions often have a simple regular structure of repetitions of blocks of related assumptions, with each block generally involving one or several variables and their properties, that are added to the context in a single backchaining step. Reflecting on this regular structure can let us derive a number of structural properties about the elements of the context. We present an extension of the Abella theorem prover, which is based on a simply typed intuitionistic reasoning logic supporting (co-)inductive definitions and generic quantification. Dynamic contexts are repre-sented in Abella using lists of formulas for the assumptions and quantifier nesting for the variables, together with an inductively defined context relation that specifies their structure. We add a new mechanism for defining particular kinds of regular context relations, called schemas, and tacticals to derive theorems from these schemas as needed. Importantly, our extension leaves the trusted kernel of Abella unchanged. We show that these tacticals can eliminate many commonly encountered kinds of administrative lemmas that would otherwise have to be proven manually, which is a common source of complaints from Abella users
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Thinking intuitively: the rich (and at times illogical) world of concepts
Intuitive knowledge of the world involves knowing what kinds of things have which properties. We express it in generalities such as “ducks lay eggs”. It contrasts with extensional knowledge about actual individuals in the world, which we express in quantified statements such as “All US Presidents are male”. Reasoning based on this intuitive knowledge, while highly fluent and plausible may in fact lead us into logical fallacy. Several lines of research point to our conceptual memory as the source of this logical failure. We represent concepts with prototypical properties, judging likelihood and argument strength on the basis of similarity between ideas. Evidence that our minds represent the world in this intuitive way can be seen in a range of phenomena, including how people interpret logical connectives applied to everyday concepts, studies of creativity and emergence in conceptual combination, and demonstrations of the logically inconsistent beliefs that people express in their everyday language
Relating Nominal and Higher-order Abstract Syntax Specifications
Nominal abstract syntax and higher-order abstract syntax provide a means for
describing binding structure which is higher-level than traditional techniques.
These approaches have spawned two different communities which have developed
along similar lines but with subtle differences that make them difficult to
relate. The nominal abstract syntax community has devices like names,
freshness, name-abstractions with variable capture, and the new-quantifier,
whereas the higher-order abstract syntax community has devices like
lambda-binders, lambda-conversion, raising, and the nabla-quantifier. This
paper aims to unify these communities and provide a concrete correspondence
between their different devices. In particular, we develop a
semantics-preserving translation from alpha-Prolog, a nominal abstract syntax
based logic programming language, to G-, a higher-order abstract syntax based
logic programming language. We also discuss higher-order judgments, a common
and powerful tool for specifications with higher-order abstract syntax, and we
show how these can be incorporated into G-. This establishes G- as a language
with the power of higher-order abstract syntax, the fine-grained variable
control of nominal specifications, and the desirable properties of higher-order
judgments.Comment: To appear in PPDP 201
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Generics as reflecting conceptual knowledge
Generics are proposed to reflect the content of the conceptual system, whose prototype structure and vague boundaries make an unreliable basis for traditional treatments of truth and logic. Examples from the psychological literature are used to illustrate the relation between generics, similarity‐based reasoning and concepts
Conceptual thinking in Hegel’s Science of logic
Filozofia analityczna po logicyzmie Fregego i atomizmie logicznym Russella odziedziczyła szereg założeń związanych z istnieniem rodzajowej dziedziny bytów indywidualnych, których tożsamość i elementarne określenia już mamy zdefiniowane. Te „indywidua” istnieją tylko w idealnych „światach możliwych” i nie są niczym innym jak zbiorami posiadającymi strukturę bądź czystymi zbiorami matematycznymi. W przeciwieństwie do takich czysto abstrakcyjnych modeli, Hegel analizuje rolę pojęciowych rozróżnień i odpowiednich brakujących inferencji w rzeczywistym świecie. Tutaj wszystkie obiekty są przestrzennie i czasowo skończone. Nawet jeśli rzeczywiste rzeczy poruszają się zgodnie z pewnymi formami, są tylko momentami w całościowym procesie. Wszelako, formy te nie są przedmiotami bezpośredniej, empirycznej obserwacji, lecz zakładają udane i powtarzalne działania i akty mowy. W rezultacie żadna semantyka odnoszącej się do świata referencji nie może obyć się bez kategorii Heglowskich, które wykraczają daleko poza narzędzia opartej wyłącznie na relacjach logiki matematycznej
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