3,091 research outputs found
Cut Elimination inside a Deep Inference System for Classical Predicate Logic
Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivation
Normalisation in Deep Inference
Στην διπλωματική αυτή εργασία γίνεται μια
αναλυτική παρουσίαση του λογισμού των δομών,
ενός φορμαλισμού της θεωρίας αποδείξεων
που χρησιμοποιεί deep inference. Αυτό σημαίνει
ότι οι συμπερασματικοί κανόνες εφαρμόζονται
οσοδήποτε βαθιά στην πολυπλοκότητα ενός τύπου.
Έπεται ότι οι αποδείξεις έχουν συμμετρική αντί
για δενδρική μορφή. Εισάγουμε ένα σύστημα
για την κλασική πρωτοβάθμια λογική και το συγκρίνουμε με το αντίστοιχο στον
λογισμό
ακολουθητών. Βλέπουμε πως επιτυγχάνεται τοπικότητα, δηλαδή κάθε λογικός κανόνας
έχει
σταθερή πολυπλοκότητα. Η εργασία τελικά
εστιάζει στους διάφορους ορισμούς της κανονικής
μορφής μιας απόδειξης.In this thesis we present the calculus of structures, a proof-theoretic
formalism
using deep inference. This means that inference rules apply arbitrarily
deep inside formulas. It follows that derivations are now symmetric instead
of tree-shape objects. A system for classical predicate logic is introduced
and compared with the corresponding sequent calculus system. They both
have an admissible Cut rule. However, locality can be obtained with deep
inference, meaning that the effort of applying a rule is always bounded. Then
we investigate what normal forms of deductions have been defined. Besides
cut elimination, we can adopt two other notions of normalisation that allow
cuts inside a derivation, under some constraints. We will try to remark
common things and differences between normalisation in deep and shallow
inference
Deep Inference and Symmetry in Classical Proofs
In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems
De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic
This paper explores the proof theory necessary for recommending an expressive
but decidable first-order system, named MAV1, featuring a de Morgan dual pair
of nominal quantifiers. These nominal quantifiers called `new' and `wen' are
distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers.
The novelty of these nominal quantifiers is they are polarised in the sense
that `new' distributes over positive operators while `wen' distributes over
negative operators. This greater control of bookkeeping enables private names
to be modelled in processes embedded as formulae in MAV1. The technical
challenge is to establish a cut elimination result, from which essential
properties including the transitivity of implication follow. Since the system
is defined using the calculus of structures, a generalisation of the sequent
calculus, novel techniques are employed. The proof relies on an intricately
designed multiset-based measure of the size of a proof, which is used to guide
a normalisation technique called splitting. The presence of equivariance, which
swaps successive quantifiers, induces complex inter-dependencies between
nominal quantifiers, additive conjunction and multiplicative operators in the
proof of splitting. Every rule is justified by an example demonstrating why the
rule is necessary for soundly embedding processes and ensuring that cut
elimination holds.Comment: Submitted for review 18/2/2016; accepted CONCUR 2016; extended
version submitted to journal 27/11/201
Locality for Classical Logic
In this paper we will see deductive systems for classical propositional and
predicate logic in the calculus of structures. Like sequent systems, they have
a cut rule which is admissible. In addition, they enjoy a top-down symmetry and
some normal forms for derivations that are not available in the sequent
calculus. Identity axiom, cut, weakening and also contraction can be reduced to
atomic form. This leads to rules that are local: they do not require the
inspection of expressions of unbounded size
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
Normalisation Control in Deep Inference via Atomic Flows
We introduce `atomic flows': they are graphs obtained from derivations by
tracing atom occurrences and forgetting the logical structure. We study simple
manipulations of atomic flows that correspond to complex reductions on
derivations. This allows us to prove, for propositional logic, a new and very
general normalisation theorem, which contains cut elimination as a special
case. We operate in deep inference, which is more general than other syntactic
paradigms, and where normalisation is more difficult to control. We argue that
atomic flows are a significant technical advance for normalisation theory,
because 1) the technique they support is largely independent of syntax; 2)
indeed, it is largely independent of logical inference rules; 3) they
constitute a powerful geometric formalism, which is more intuitive than syntax
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