From truth to computability II

Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are defined as a certain sort games between a machine and its environment, with logical operators standing for operations on such games. Within the ambitious program of finding axiomatizations for incrementally rich fragments of this semantically introduced logic, the earlier article "From truth to computability I" proved soundness and completeness for system CL3, whose language has the so called parallel connectives (including negation), choice connectives, choice quantifiers, and blind quantifiers. The present paper extends that result to the significantly more expressive system CL4 with the same collection of logical operators. What makes CL4 expressive is the presence of two sorts of atoms in its language: elementary atoms, representing elementary computational problems (i.e. predicates, i.e. problems of zero degree of interactivity), and general atoms, representing arbitrary computational problems. CL4 conservatively extends CL3, with the latter being nothing but the general-atom-free fragment of the former. Removing the blind (classical) group of quantifiers from the language of CL4 is shown to yield a decidable logic despite the fact that the latter is still first-order. A comprehensive online source on computability logic can be found at http://www.cis.upenn.edu/~giorgi/cl.htm

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

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