58 research outputs found
Sub-classical Boolean Bunched Logics and the Meaning of Par
We investigate intermediate logics between the bunched logics Boolean BI and Classical BI, obtained by combining classical propositional logic with various flavours of Hyland and De Paiva\u27s full intuitionistic linear logic. Thus, in addition to the usual multiplicative conjunction (with its adjoint implication and unit), our logics also feature a multiplicative disjunction (with its adjoint co-implication and unit). The multiplicatives behave "sub-classically", in that disjunction and conjunction are related by a weak distribution principle, rather than by De Morgan equivalence.
We formulate a Kripke semantics, covering all our sub-classical bunched logics, in which the multiplicatives are naturally read in terms of resource operations. Our main theoretical result is that validity according to this semantics coincides with provability in a corresponding Hilbert-style proof system.
Our logical investigation sheds considerable new light on how one can understand the multiplicative disjunction, better known as linear logic\u27s "par", in terms of resource operations. In particular, and in contrast to the earlier Classical BI, the models of our logics include the heap-like memory models of separation logic, in which disjunction can be interpreted as a property of intersection operations over heaps
Sequent calculus proof systems for inductive definitions
Inductive definitions are the most natural means by which to represent many families of structures
occurring in mathematics and computer science, and their corresponding induction / recursion
principles provide the fundamental proof techniques by which to reason about such
families. This thesis studies formal proof systems for inductive definitions, as needed, e.g., for
inductive proof support in automated theorem proving tools. The systems are formulated as
sequent calculi for classical first-order logic extended with a framework for (mutual) inductive
definitions.
The default approach to reasoning with inductive definitions is to formulate the induction
principles of the inductively defined relations as suitable inference rules or axioms, which are
incorporated into the reasoning framework of choice. Our first system LKID adopts this direct
approach to inductive proof, with the induction rules formulated as rules for introducing atomic
formulas involving inductively defined predicates on the left of sequents. We show this system
to be sound and cut-free complete with respect to a natural class of Henkin models. As a
corollary, we obtain cut-admissibility for LKID.
The well-known method of infinite descent `a la Fermat, which exploits the fact that there are
no infinite descending chains of elements of well-ordered sets, provides an alternative approach
to reasoning with inductively defined relations. Our second proof system LKIDw formalises
this approach. In this system, the left-introduction rules for formulas involving inductively
defined predicates are not induction rules but simple case distinction rules, and an infinitary,
global soundness condition on proof trees ā formulated in terms of ātracesā on infinite paths
in the tree ā is required to ensure soundness. This condition essentially ensures that, for
every infinite branch in the proof, there is an inductive definition that is unfolded infinitely
often along the branch. By an infinite descent argument based upon the well-foundedness of
inductive definitions, the infinite branches of the proof can thus be disregarded, whence the
remaining portion of proof is well-founded and hence sound. We show this system to be cutfree
complete with respect to standard models, and again infer the admissibility of cut.
The infinitary system LKIDw is unsuitable for formal reasoning. However, it has a natural
restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs.
This restricted ācyclicā proof system, CLKIDw, is suitable for formal reasoning since proofs
have finite representations and the soundness condition on proofs is thus decidable.
We show how the formulation of our systems LKIDw and CLKIDw can be generalised to
obtain soundness conditions for a general class of infinite proof systems and their corresponding
cyclic restrictions. We provide machinery for manipulating and analysing the structure of
proofs in these essentially arbitrary cyclic systems, based primarily on viewing them as generating
regular infinite trees, and we show that any proof can be converted into an equivalent
proof with a restricted cycle structure. For proofs in this ācycle normal formā, a finitary, localised soundness condition exists that is strictly stronger than the general, infinitary soundness
condition, but provides more explicit information about the proof.
Finally, returning to the specific setting of our systems for inductive definitions, we show
that any LKID proof can be transformed into a CLKIDw proof (that, in fact, satisfies the finitary
soundness condition). We conjecture that the two systems are in fact equivalent, i.e. that proof
by induction is equivalent to regular proof by infinite descent
On Model Structures Relating to Spectral Sequences
In [CELW19] Cirici, Egas Santander, Livernet and Whitehouse define model structures on filtered chain complexes and bicomplexes whose weak equivalences are the -weak equivalences, i.e. isomorphisms on the -pages of the associated spectral sequences. In this thesis we study and generalise these model structures. These generalisations and for fixed such are indexed by subsets of containing in the former case and and in the latter and are finitely cofibrantly generated.
We show each of these model structures is a left (and right) proper, cellular and stable model category. We construct a left adjoint to the product totalisation functor and show, by means of Greenlees and Shipleyās cellularization principle, that it is a Quillen equivalence for suitable indexing sets . As a consequence all the model categories considered thus far have equivalent homotopy categories induced via a zig-zag of Quillen equivalences given by compositions of the -product totalisation, identity-identity and shift-dĆ©calage adjunctions. The model structures with -weak equivalences are shown to have no left Bousfield localisation to a model structure with -weak equivalences. We also derive existence of various bounded variants of the model structures .
We then focus on the model structures on filtered chain complexes, give a classification of their cofibrant objects and cofibrations with a boundedness restriction on their filtrations and show the satisfy the unit and pushout-product axioms thereby giving monoidal model categories. Furthermore the satisfy the monoid axiom of Schwede and Shipley yielding model structures on modules and algebras enhancing the homotopy theory of Halperin and TanrƩ on filtered differential graded algebras to a model category structure
Model checking for symbolic-heap separation logic with inductive predicates
We investigate the model checking problem for symbolic-heap separation logic with user-defined inductive predicates, i.e., the problem of checking that a given stack-heap memory state satisfies a given formula in this language, as arises e.g. in software testing or runtime verification.
First, we show that the problem is decidable; specifically, we present a bottom-up fixed point algorithm that decides the problem and runs in exponential time in the size of the problem instance.
Second, we show that, while model checking for the full language is EXPTIME-complete, the problem becomes NP-complete or PTIME-solvable when we impose natural syntactic restrictions on the schemata defining the inductive predicates. We additionally present NP and PTIME algorithms for these restricted fragments.
Finally, we report on the experimental performance of our procedures on a variety of specifications extracted from programs, exercising multiple combinations of syntactic restrictions
Realizability in Cyclic Proof:Extracting Ordering Information for Infinite Descent
In program veri_cation, measures for proving the termination of programs are typically constructed using (notions of size for) the data manipulated by the program. Such data are often described by means of logical formulas. For example, the cyclic proof technique makes use of semantic approximations of inductively de_ned predicates to construct Fermat-style in_nite descent arguments. However, logical formulas must often incorporate explicit size information (e.g. a list length parameter) in order to support inter-procedural analysis. In this paper, we show that information relating the sizes of inductively de_ned data can be automatically extracted from cyclic proofs of logical entailments.We characterise this information in terms of a graph-theoretic condition on proofs, and show that this condition can be encoded as a containment between weighted automata. We also show that under certain conditions this containment falls within known decidability results. Our results can be viewed as a form of realizability for cyclic proof theory
Complete Sequent Calculi for Induction and Infinite Descent
This paper formalises and compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system, LKID, supports traditional proof by induction, with induction rules formulated as rules for introducing inductively defined predicates on the left of sequents. We show LKID to be cut-free complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system, LKID Ļ, uses infinite (non-well-founded) proofs to represent arguments by infinite descent. In this system, the left-introduction rules for inductively defined predicates are simple case-split rules, and an infinitary, global condition on proof trees is required in order to ensure soundness. We show LKID Ļ to be cut-free complete with respect to standard models, and again infer the eliminability of cut. The infinitary system LKID Ļ is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs, which is so suited. We demonstrate that this restricted ācyclic ā proof system, CLKID Ļ, subsumes LKID, and conjecture that CLKID Ļ and LKID are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
Classical BI: Its Semantics and Proof Theory
We present Classical BI (CBI), a new addition to the family of bunched logics
which originates in O'Hearn and Pym's logic of bunched implications BI. CBI
differs from existing bunched logics in that its multiplicative connectives
behave classically rather than intuitionistically (including in particular a
multiplicative version of classical negation). At the semantic level,
CBI-formulas have the normal bunched logic reading as declarative statements
about resources, but its resource models necessarily feature more structure
than those for other bunched logics; principally, they satisfy the requirement
that every resource has a unique dual. At the proof-theoretic level, a very
natural formalism for CBI is provided by a display calculus \`a la Belnap,
which can be seen as a generalisation of the bunched sequent calculus for BI.
In this paper we formulate the aforementioned model theory and proof theory for
CBI, and prove some fundamental results about the logic, most notably
completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure
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