Proof Polynomials: A Unified Semantics for Modality and lambda-terms (original) (raw)
Unified Semantics for Modality and lambda-terms via Proof Polynomials
Algebras, Diagrams and Decisions in Language, Logic and Computation, 2002
It is shown that the modal logic S4, simple-calculus and modal-calculus admit arealization in a very simple propositional logical system LP, which has an exact provabilitysemantics. In LP both modality and-terms become objects of the same nature, namely, proof polynomials. The provability interpretation of modal-terms presented here maybe regarded as a system-independent generalization of the Curry-Howard isomorphism ofproofs and-terms. 1 IntroductionThe Logic of Proofs (LP, see Section 2) is a system in the propositional ...
Proof Polynomials vs. lambda-terms
1998
Abstract: The Logic of Proofs provides a basic framework for the formalization of reasoning about proofs. It incorporates proof terms into the propositional language, using labeled logical operators" t:" with the intended reading of t: F being" T is a proof of F". In the current paper we demonstrate how the typed lambda calculus and the modal lambda-calculus can be realized in the Logic of Proofs.
Polynomial Ring Calculus for Modal Logics: A New Semantics and Proof Method for Modalities
The Review of Symbolic Logic, 2011
A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.
Proofnets and Context Semantics for the Additives
Computer Science Logic, 2002
We provide a context semantics for Multiplicative-Additive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cut-elimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lévy, who provided a "geometry of optimal λ-reduction" (context semantics) for λ-calculus and Multiplicative-Exponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a read-back procedure that computes a cut-free proof from the semantics, which is closely related to full abstraction theorems. Peut-être que la logique se trompe. -Yves Lafont Linear Logic appeared in 1987 and turned out quickly to be an interesting tool to model programming languages, specifically reasoning that is sensitive to the notion of consumable resources. Indeed the multiplicative fragment of Linear Logic (⊗, ) allows linear products (pairing and unpairing), implementing functions: a context pairs a continuation and an argument, a function unpairs and connects the two. The additive fragment (⊕, &) allows linear sums (injection and case dispatch), implementing features of processes in the style of CSP or CCS . The crucial difference between these two components is the way they take care with consumption of resources. The exponential fragment implements sharing of resources: arguments, control contexts. We can then implement, for example, graph reduction technology for λ-calculus with control operators (call/cc, abort, jumps), and related mechanical proof systems for classical logic-taking care of the sharing and copying, implicit in these calculi . Linear Logic was initially a sequent calculus, but this sequentialized structure was too strong and the design of a nice cut-elimination procedure was complicated. Therefore, proofnets were introduced as a more flexible representation of proofs . Further, Geometry of Interaction (GoI) developed the idea that the reduction of proofs can be seen as a local interaction process . Its intensional features provide a mediating Purgatory between the Heaven of denotational semantics, and the Hell of operational semantics. GoI was simplified in the "geometry of optimal λ-reduction" by Gonthier, Abadi and Lévy in the context of the MELL fragment. They reduced Hilbert spaces to simple data-structures, known as context semantics, and developed a proofnet technology which implemented the context semantics locally. Reduction on proofnets preserves the semantics, and Lamping's algorithm for optimal reduction of λ-terms [14] is a method of graph reduction. They further indicated how to read back any part of the Böhm tree (normal form) of a λ-term from its context semantics. Can this program be carried out for full Linear Logic? In this paper we extend these result to the MALL fragment (multiplicatives and additives): this may be a step towards a satisfactory proofnet syntax for full Linear Logic with a good characterization of proofs. The MALL fragment is quite problematic since it does not have a nice cutelimination procedure-unlike MLL, which enjoys a straightforward one. Part of the work will be to understand and improve the reduction procedure for MALL. The main contributions of this paper are to provide an integrated development of (1) a context semantics for the MALL fragment; (2) a proofnet technology allowing normalization of MALL proofs, using the ideas of multidimensional boxes and generalized garbage; and (3) a read-back procedure that inputs a valid context semantics and outputs a normalized proofnet. In Section 1, we recall some basic definitions on MALL, proofnets and Linear Logic. We introduce in Section ?? a form of context semantics for MALL proofs, and we derive from it a bus-notation based proofnet syntax in Section 3. Then, we envisage the main problems that come from the reduction on the additives in Section 4. The first difficulty is that the additive cut-elimination is not really linear, since an additive reduction step erases a whole part of the proof (this is also a problem for the locality of the reduction we would like to achieve). The second problem also stems from the additives: the way one should reduce a cut involving auxiliary formulas of two &-links is unclear. The solution to this problem will come from an extension of the MALL syntax. After this adaptation, we will be able to get a much more satisfactory cut-elimination procedure. In Section 5, we show how a normalized proof can be read-back from the context semantics of a proof, and we see how this result can be related to a form of full completeness. In Section 6, we compare our approach with other works.
The λ Calculus and the Unity of Structural Proof Theory
Theory of Computing Systems / Mathematical Systems Theory, 2009
In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.
The Coq Proof Assistant A Tutorial
Rapport Technique
C OQ is a Proof Assistant for a Logical Framework known as the Calculus of Induc- tive Constructions. It allows the interactive construction of formal proofs, and also the manipulation of functional programs consistently with ...
“Proof Frameworks—A Way to Get Started” (with John Selden and Ahmed Benkhalti), Tennessee Technological University Technical Report No. 2016-1., 2016
Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and prepare for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on writing proofs. This is the technique of writing proof frameworks, based on the logical structure of the statement of the theorem and associated definitions. Often there is both a first-level and a second-level proof framework.