695 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
Tool support for reasoning in display calculi
We present a tool for reasoning in and about propositional sequent calculi.
One aim is to support reasoning in calculi that contain a hundred rules or
more, so that even relatively small pen and paper derivations become tedious
and error prone. As an example, we implement the display calculus D.EAK of
dynamic epistemic logic. Second, we provide embeddings of the calculus in the
theorem prover Isabelle for formalising proofs about D.EAK. As a case study we
show that the solution of the muddy children puzzle is derivable for any number
of muddy children. Third, there is a set of meta-tools, that allows us to adapt
the tool for a wide variety of user defined calculi
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
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
To Teach Modal Logic: An Opinionated Survey
I aim to promote an alternative agenda for teaching modal logic chiefly
inspired by the relationships between modal logic and philosophy. The guiding
idea for this proposal is a reappraisal of the interest of modal logic in
philosophy, which do not stem mainly from mathematical issues, but which is
motivated by central problems of philosophy and language. I will point out some
themes to start elaborating a guide for a more comprehensive approach to teach
modal logic, and consider the contributions of dual-process theories in
cognitive science, in order to explore a pedagogical framework for the proposed
point of view.Comment: Proceedings of the Fourth International Conference on Tools for
Teaching Logic (TTL2015), Rennes, France, June 9-12, 2015. Editors: M.
Antonia Huertas, Jo\~ao Marcos, Mar\'ia Manzano, Sophie Pinchinat,
Fran\c{c}ois Schwarzentrube
Hybrid Branching-Time Logics
Hybrid branching-time logics are introduced as extensions of CTL-like logics
with state variables and the downarrow-binder. Following recent work in the
linear framework, only logics with a single variable are considered. The
expressive power and the complexity of satisfiability of the resulting logics
is investigated.
As main result, the satisfiability problem for the hybrid versions of several
branching-time logics is proved to be 2EXPTIME-complete. These branching-time
logics range from strict fragments of CTL to extensions of CTL that can talk
about the past and express fairness-properties. The complexity gap relative to
CTL is explained by a corresponding succinctness result.
To prove the upper bound, the automata-theoretic approach to branching-time
logics is extended to hybrid logics, showing that non-emptiness of alternating
one-pebble Buchi tree automata is 2EXPTIME-complete.Comment: An extended abstract of this paper was presented at the International
Workshop on Hybrid Logics (HyLo 2007
Relation-Changing Logics as Fragments of Hybrid Logics
Relation-changing modal logics are extensions of the basic modal logic that
allow changes to the accessibility relation of a model during the evaluation of
a formula. In particular, they are equipped with dynamic modalities that are
able to delete, add, and swap edges in the model, both locally and globally. We
provide translations from these logics to hybrid logic along with an
implementation. In general, these logics are undecidable, but we use our
translations to identify decidable fragments. We also compare the expressive
power of relation-changing modal logics with hybrid logics.Comment: In Proceedings GandALF 2016, arXiv:1609.0364
- …