2,781 research outputs found
Logic of Non-Monotonic Interactive Proofs (Formal Theory of Temporary Knowledge Transfer)
We propose a monotonic logic of internalised non-monotonic or instant
interactive proofs (LiiP) and reconstruct an existing monotonic logic of
internalised monotonic or persistent interactive proofs (LiP) as a minimal
conservative extension of LiiP. Instant interactive proofs effect a fragile
epistemic impact in their intended communities of peer reviewers that consists
in the impermanent induction of the knowledge of their proof goal by means of
the knowledge of the proof with the interpreting reviewer: If my peer reviewer
knew my proof then she would at least then (in that instant) know that its
proof goal is true. Their impact is fragile and their induction of knowledge
impermanent in the sense of being the case possibly only at the instant of
learning the proof. This accounts for the important possibility of
internalising proofs of statements whose truth value can vary, which, as
opposed to invariant statements, cannot have persistent proofs. So instant
interactive proofs effect a temporary transfer of certain propositional
knowledge (knowable ephemeral facts) via the transmission of certain individual
knowledge (knowable non-monotonic proofs) in distributed systems of multiple
interacting agents.Comment: continuation of arXiv:1201.3667 ; published extended abstract:
DOI:10.1007/978-3-642-36039-8_16 ; related to arXiv:1208.591
Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)
We produce a decidable classical normal modal logic of internalised
negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP)
from an existing logical counterpart of non-monotonic or instant interactive
proofs (LiiP). LDiiP internalises agent-centric proof theories that are
negation-complete (maximal) and consistent (and hence strictly weaker than, for
example, Peano Arithmetic) and enjoy the disjunction property (like
Intuitionistic Logic). In other words, internalised proof theories are
ultrafilters and all internalised proof goals are definite in the sense of
being either provable or disprovable to an agent by means of disjunctive
internalised proofs (thus also called epistemic deciders). Still, LDiiP itself
is classical (monotonic, non-constructive), negation-incomplete, and does not
have the disjunction property. The price to pay for the negation completeness
of our interactive proofs is their non-monotonicity and non-communality (for
singleton agent communities only). As a normal modal logic, LDiiP enjoys a
standard Kripke-semantics, which we justify by invoking the Axiom of Choice on
LiiP's and then construct in terms of a concrete oracle-computable function.
LDiiP's agent-centric internalised notion of proof can also be viewed as a
negation-complete disjunctive explicit refinement of standard KD45-belief, and
yields a disjunctive but negation-incomplete explicit refinement of
S4-provability.Comment: Expanded Introduction. Added Footnote 4. Corrected Corollary 3 and 4.
Continuation of arXiv:1208.184
Logic of Intuitionistic Interactive Proofs (Formal Theory of Perfect Knowledge Transfer)
We produce a decidable super-intuitionistic normal modal logic of
internalised intuitionistic (and thus disjunctive and monotonic) interactive
proofs (LIiP) from an existing classical counterpart of classical monotonic
non-disjunctive interactive proofs (LiP). Intuitionistic interactive proofs
effect a durable epistemic impact in the possibly adversarial communication
medium CM (which is imagined as a distinguished agent), and only in that, that
consists in the permanent induction of the perfect and thus disjunctive
knowledge of their proof goal by means of CM's knowledge of the proof: If CM
knew my proof then CM would persistently and also disjunctively know that my
proof goal is true. So intuitionistic interactive proofs effect a lasting
transfer of disjunctive propositional knowledge (disjunctively knowable facts)
in the communication medium of multi-agent distributed systems via the
transmission of certain individual knowledge (knowable intuitionistic proofs).
Our (necessarily) CM-centred notion of proof is also a disjunctive explicit
refinement of KD45-belief, and yields also such a refinement of standard
S5-knowledge. Monotonicity but not communality is a commonality of LiP, LIiP,
and their internalised notions of proof. As a side-effect, we offer a short
internalised proof of the Disjunction Property of Intuitionistic Logic
(originally proved by Goedel).Comment: continuation of arXiv:1201.3667; extended start of Section 1 and 2.1;
extended paragraph after Fact 1; dropped the N-rule as primitive and proved
it derivable; other, non-intuitionistic family members: arXiv:1208.1842,
arXiv:1208.591
Knowledge Spaces and the Completeness of Learning Strategies
We propose a theory of learning aimed to formalize some ideas underlying
Coquand's game semantics and Krivine's realizability of classical logic. We
introduce a notion of knowledge state together with a new topology, capturing
finite positive and negative information that guides a learning strategy. We
use a leading example to illustrate how non-constructive proofs lead to
continuous and effective learning strategies over knowledge spaces, and prove
that our learning semantics is sound and complete w.r.t. classical truth, as it
is the case for Coquand's and Krivine's approaches
Parametric Constructive Kripke-Semantics for Standard Multi-Agent Belief and Knowledge (Knowledge As Unbiased Belief)
We propose parametric constructive Kripke-semantics for multi-agent
KD45-belief and S5-knowledge in terms of elementary set-theoretic constructions
of two basic functional building blocks, namely bias (or viewpoint) and
visibility, functioning also as the parameters of the doxastic and epistemic
accessibility relation. The doxastic accessibility relates two possible worlds
whenever the application of the composition of bias with visibility to the
first world is equal to the application of visibility to the second world. The
epistemic accessibility is the transitive closure of the union of our doxastic
accessibility and its converse. Therefrom, accessibility relations for common
and distributed belief and knowledge can be constructed in a standard way. As a
result, we obtain a general definition of knowledge in terms of belief that
enables us to view S5-knowledge as accurate (unbiased and thus true)
KD45-belief, negation-complete belief and knowledge as exact KD45-belief and
S5-knowledge, respectively, and perfect S5-knowledge as precise (exact and
accurate) KD45-belief, and all this generically for arbitrary functions of bias
and visibility. Our results can be seen as a semantic complement to previous
foundational results by Halpern et al. about the (un)definability and
(non-)reducibility of knowledge in terms of and to belief, respectively
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
RPP: Automatic Proof of Relational Properties by Self-Composition
Self-composition provides a powerful theoretical approach to prove relational
properties, i.e. properties relating several program executions, that has been
applied to compare two runs of one or similar programs (in secure dataflow
properties, code transformations, etc.). This tool demo paper presents RPP, an
original implementation of self-composition for specification and verification
of relational properties in C programs in the FRAMA-C platform. We consider a
very general notion of relational properties invoking any finite number of
function calls of possibly dissimilar functions with possible nested calls. The
new tool allows the user to specify a relational property, to prove it in a
completely automatic way using classic deductive verification, and to use it as
a hypothesis in the proof of other properties that may rely on it
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