2,327 research outputs found
A Uniform Substitution Calculus for Differential Dynamic Logic
This paper introduces a new proof calculus for differential dynamic logic
(dL) that is entirely based on uniform substitution, a proof rule that
substitutes a formula for a predicate symbol everywhere. Uniform substitutions
make it possible to rely on axioms rather than axiom schemata, substantially
simplifying implementations. Instead of nontrivial schema variables and
soundness-critical side conditions on the occurrence patterns of variables, the
resulting calculus adopts only a finite number of ordinary dL formulas as
axioms. The static semantics of differential dynamic logic is captured
exclusively in uniform substitutions and bound variable renamings as opposed to
being spread in delicate ways across the prover implementation. In addition to
sound uniform substitutions, this paper introduces differential forms for
differential dynamic logic that make it possible to internalize differential
invariants, differential substitutions, and derivations as first-class axioms
in dL
Uniform Substitution for Differential Game Logic
This paper presents a uniform substitution calculus for differential game
logic (dGL). Church's uniform substitutions substitute a term or formula for a
function or predicate symbol everywhere. After generalizing them to
differential game logic and allowing for the substitution of hybrid games for
game symbols, uniform substitutions make it possible to only use axioms instead
of axiom schemata, thereby substantially simplifying implementations. Instead
of subtle schema variables and soundness-critical side conditions on the
occurrence patterns of logical variables to restrict infinitely many axiom
schema instances to sound ones, the resulting axiomatization adopts only a
finite number of ordinary dGL formulas as axioms, which uniform substitutions
instantiate soundly. This paper proves soundness and completeness of uniform
substitutions for the monotone modal logic dGL. The resulting axiomatization
admits a straightforward modular implementation of dGL in theorem provers
On the Syntax of Logic and Set Theory
We introduce an extension of the propositional calculus to include abstracts
of predicates and quantifiers, employing a single rule along with a novel
comprehension schema and a principle of extensionality, which are substituted
for the Bernays postulates for quantifiers and the comprehension schemata of ZF
and other set theories. We prove that it is consistent in any finite Boolean
subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti,
and others, and discuss the relationship of the system to other set theoretic
systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order
quantification and abstraction, and then very briefly discuss the application
of one of these methods to areas of mathematics outside of logic.Comment: 34 pages, accepted, to appear in the Review of Symbolic Logi
A New Arithmetically Incomplete First- Order Extension of Gl All Theorems of Which Have Cut Free Proofs
Reference [12] introduced a novel formula to formula translation tool (“formulators”) that enables syntactic metatheoretical investigations of first-order modal logics, bypassing a need to convert them first into Gentzen style logics in order to rely on cut elimination and the subformula property. In fact, the formulator tool, as was already demonstrated in loc. cit., is applicable even to the metatheoretical study of logics such as QGL, where cut elimination is (provably, [2]) unavailable. This paper applies the formulator approach to show the independence of the axiom schema _A ! _8xA of the logics M3 and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics (compare with the more complex proofs in [2, 8]).This research was partially supported by NSERC grant No. 8250
Abstraction in situation calculus action theories
We develop a general framework for agent abstraction based on the situation calculus and the ConGolog agent programming language. We assume that we have a high-level specification and a low-level specification of the agent, both repre- sented as basic action theories. A refinement mapping specifies how each high-level action is implemented by a low- level ConGolog program and how each high-level fluent can be translated into a low-level formula. We define a notion of sound abstraction between such action theories in terms of the existence of a suitable bisimulation between their respective models. Sound abstractions have many useful properties that ensure that we can reason about the agent’s actions (e.g., executability, projection, and planning) at the abstract level, and refine and concretely execute them at the low level. We also characterize the notion of complete abstraction where all actions (including exogenous ones) that the high level thinks can happen can in fact occur at the low level
Abstraction in situation calculus action theories
We develop a general framework for agent abstraction based on the situation calculus and the ConGolog agent programming language. We assume that we have a high-level specification and a low-level specification of the agent, both repre- sented as basic action theories. A refinement mapping specifies how each high-level action is implemented by a low- level ConGolog program and how each high-level fluent can be translated into a low-level formula. We define a notion of sound abstraction between such action theories in terms of the existence of a suitable bisimulation between their respective models. Sound abstractions have many useful properties that ensure that we can reason about the agent’s actions (e.g., executability, projection, and planning) at the abstract level, and refine and concretely execute them at the low level. We also characterize the notion of complete abstraction where all actions (including exogenous ones) that the high level thinks can happen can in fact occur at the low level
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is
accessible to a broad mathematical audience. Gentzen's cut elimination theorem
is not as well known as it deserves to be, and it is tied to a lot of
interesting mathematical structure. In particular we try to indicate some
dynamical and combinatorial aspects of cut elimination, as well as its
connections to complexity theory. We discuss two concrete examples where one
can see the structure of short proofs with cuts, one concerning feasible
numbers and the other concerning "bounded mean oscillation" from real analysis
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