Polynomial agorithms for BR-nets and for a fragment of Girard's Linear Logic (original) (raw)

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Abstract

In this paper we consider some classes of nets with bounded types of distributed resources (BR-nets). The succsesful execution problem for BR-nets is equivalent to the provability problem for the logical calculus based on Horn fragment of Girard's Linear Logic . We show that, in spite of NP-completeness of the problem for all BR-nets, it is in PTIME for some interesting subclasses of BR-nets.

Polynomially graded logic I. A graded version of system T

[1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science, 1989

We investigate a logical framework for programming languages which treats requirements on computational resources as part of the formal program specification. Resource bounds are explicit in the syntax of all programs. In a programming language based on this approach, compliance of a program with imposed resource bounds would be assured by verifying the syntactic correctness wing a compiler with a static type checking feature. The principal innovation of this paper is the introduction of systems of logical inference, called polynomially graded logics. These logics make resource bounds part of every proposition and eve y deduction. The sample calculus presented in this paper is a restriction of GodePs system T to polynomial time resources. We prove that the numerical jhnctions representable in this calculus are exactly the PTIME functions.

Hybrid Logics and NP Graph Properties

Lecture Notes in Computer Science, 2011

We show that for each property of graphs G in NP there is a sequence φ 1 , φ 2 ,. .. of formulas of the full hybrid logic which are satisfied exactly by the frames in G. Moreover, the size of φn is bounded by a polynomial. We also show that the same holds for each graph property in the polynomial hierarchy. These results lead to the definition of syntactically defined fragments of hybrid logic whose model checking problem is complete for each degree in the polynomial hierarchy.

The Complexity of Generalized Satisfiability for Linear Temporal Logic

Electronic Colloquium on Computational Complexity, 2006

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.

Polynomially solvable satisfiability problems

Information Processing Letters, 1988

We address the well-known satisfiability problem (SAT). i.e., the problem of checking whether a given propositional formula is satisfiable. Although the general satisfiability problem is NP-complete, some particular cases of SAT are known to be easy. The most important of those cases is HORN-SAT, i.e., the satisfiability problem in the case of Horn clauses: actually, an instance of HORN-SAT can be solved in linear time. Yamasaki and Doshita (1983) have introduced a new subclass of SAT, S,, which is polynomially solvable and which strictly includes HORN-SAT. Here we introduce a family of subclasses of

A Modal View on Resource-Bounded Propositional Logics

Studia Logica

Classical propositional logic plays a prominent role in industrial applications, and yet the complexity of this logic is presumed to be non-feasible. Tractable systems such as depth-bounded boolean logics approximate classical logic and can be seen as a model for resource-bounded agents whose reasoning style is nonetheless classical. In this paper we first study a hierarchy of tractable logics that is not defined by depth. Then we extend it into a modal logic where modalities make explicit the assumptions discharged in propositional proofs, thereby expressing blueprints for proofs. A natural deduction system is provided that permits to reason about and manage such proof blueprints.

On the computational complexity of satisfiability in propositional logics of programs

Theoretical Computer Science, 1982

The satisfiability problems of propositional algorithmic logic and propositional dynamic logic are shown to be complete in the classes of languages accepted in polynomial space by the Ijeterministic and alternating Turing machines respectively. Explicit upper and lower bounds on. 1:he space complexity ase calculated. Exponential lower bounds on the space co;Bplexity of tht: !;atisfiability problems of rthese logics extended by adding a certain program connective are proved.

A Uniform Proof-Theoretic Investigation of Linear Logic Programming

Journal of Logic and Computation, 1994

In this paper we consider the problem of identifying logic programming languages for linear logic. Our analysis builds on a notion of goal-directed provability, characterized by the so-called uniform proofs, previously introduced for minimal and intuitionistic logic. A class of uniform proofs in linear logic is identi ed by an analysis of the permutability of inferences in the linear sequent calculus. We show that this class of proofs is complete (for logical consequence) for a certain (quite large) fragment of linear logic, which thus forms a logic programming language. We obtain a notion of resolution proof, in which only one left rule, of clause-directed resolution, is required. We also consider a translation, resembling those of Girard, of the hereditary Harrop fragment of intuitionistic logic into our framework. We show that goal-directed provability is preserved under this translation. the right rule for that connective. Moreover, it is easy to show that the strategy of constructing proofs (from root to leaves, regarding the rules as reduction operators 1 ) by applying a right rule wherever one is applicable is complete. This property arises from the permutability properties of the rules of the intuitionistic sequent calculus. For example, it should be clear that an intuitionistic subproof of the form

On the system CL12 of computability logic

Logical Methods in Computer Science, 2015

Computability logic (CoL) is a long-term project for redeveloping logic on the basis of a constructive game semantics, with games seen as abstract models of interactive computational problems. Among the fragments of CoL successfully axiomatized so far is CL12-a conservative extension of classical first-order logic, whose language augments that of classical logic with the so called choice ("constructive") sorts of quantifiers and connectives. This system has already found fruitful applications as a logical basis for constructive and complexity-bound versions of Peano arithmetic, such as arithmetics for polynomial time computability, polynomial space computability, and beyond. The present paper introduces a third, indispensable complexity measure for interactive computations termed amplitude complexity, and establishes the adequacy (soundness/completeness) of CL12 and the associated Logical Consequence mechanism with respect to (simultaneously) A amplitude, S space and T time computability under certain minimal conditions on the triples (A, S, T) of function classes. This result very substantially broadens the potential application areas of CL12, even when time and/or space complexity is the only concern. It would be sufficient to point out that, for instance, now CL12 can be reliably used as a logical basis of systems for logarithmic space or exponential time computabilitiessomething that the earlier-known crude adequacy results for CL12 were too weak to allow us to do. This paper is self-contained, and targets readers with no prior familiarity with the subject.

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References (11)

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Linear Logic for Nets with Bounded Resources

Annals of Pure and Applied Logic, 1996

In this paper we introduce a new type of nets with bounded types of distributed resources (BR-nets). Linear Logic to describe the behaviour of BR-nets is defined. It is based on Girard's Linear Logic but captures not only consumption of resources but their presence as well. Theorem of soundness and completeness of the proposed axiomatization is proved and the complexity of the provability problem is established for the general case and some particular ones.

Linear logic with fixed resources

Archangelsky, D.A. and M.A. Taitslin, Linear logic with fixed resources, Annals of Pure and Applied Logic 67 (1994) 3-28. In this paper we continue the study of Girard's Linear Logic and introduce a new Linear Logic with modalities. Our logic describes not only the consumption, but also the presence of resources. We introduce a new semantics and a new calculus for this logic. In contrast to the results of Lincoln et al. [7] and Kanovich [4] about the NP-completeness of the problem of the construction of a proof for a given sequent in the multiplicative fragment of Girard's Linear Logic, we present here a non-exponential algorithm to construct a proof for a given sequent and a given point of a given model in our Linear Logic.

Decision Problems for Propositional Linear Logic

1990

Linear logic, introduced by Girard, is a re nement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the \structural" rules of contraction and weakening; adding a modal operator; and adding ner versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection, and the control structure of logic programs. In addition, there is a direct correspondence between polynomial-time computation and proof normalization in a bounded form of linear logic.

A Fragment of Dependence Logic Capturing Polynomial Time

Logical Methods in Computer Science, 2014

In this paper we study the expressive power of Horn-formulae in dependence logic and show that they can express NP-complete problems. Therefore we define an even smaller fragment D * -Horn and show that over finite successor structures it captures the complexity class P of all sets decidable in polynomial time. Furthermore we study the question which of our results can ge generalized to the case of open formulae of D * -Horn and so-called downwards monotone polynomial time properties of teams.

Polynomial and Exponential Bounded Logic Programs with Function Symbols: Some New Decidable Classes

Journal of Artificial Intelligence Research

A logic program with function symbols is called finitely ground if there is a finite propositional logic program whose stable models are exactly the same as the stable models of this program. Finite groundability is an important property for logic programs with function symbols because it makes feasible to compute such programs' stable models using traditional ASP solvers. In this paper, we introduce new decidable classes of finitely ground programs called poly-bounded and k-EXP-bounded programs, which, to the best of our knowledge, strictly contain all other decidable classes of finitely ground programs discovered so far in the literature. We also study the relevant complexity properties for these classes of programs. We prove that the membership complexities for poly-bounded and k-EXP-bounded programs are EXPTIME-complete and (k+1)-EXPTIME-complete, respectively.

Complexity results for quantified boolean formulae based on complete propositional languages

2006

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.

Decidability and Complexity of Some Finitely-valued Dynamic Logics

Proceedings of the Eighteenth International Conference on Principles of Knowledge Representation and Reasoning, 2021

Propositional Dynamic Logic, PDL, is a well known modal logic formalizing reasoning about complex actions. We study many-valued generalizations of PDL based on relational models where satisfaction of formulas in states and accessibility between states via action execution are both seen as graded notions, evaluated in a finite Łukasiewicz chain. For each n>1, the logic PDŁn is obtained using the n-element Łukasiewicz chain, PDL being equivalent to PDŁ2. These finitely-valued dynamic logics can be applied in formalizing reasoning about actions specified by graded predicates, reasoning about costs of actions, and as a framework for certain graded description logics with transitive closure of roles. Generalizing techniques used in the case of PDL we obtain completeness and decidability results for all PDŁn. A generalization of Pratt's exponential-time algorithm for checking validity of formulas is given and EXPTIME-hardness of each PDŁn validity problem is established by embeddin...

Unique perfect matchings, edge-colored graphs and proof nets for linear logic with Mix

2020

This paper establishes a bridge between linear logic and mainstream graph theory, building on previous work by Retore (2003). We show that the problem of correctness for MLL+Mix proof nets is equivalent to the problem of uniqueness of a perfect matching. By applying matching theory, we obtain new results for MLL+Mix proof nets: a linear-time correctness criterion, a quasi-linear sequentialization algorithm, and a characterization of the sub-polynomial complexity of the correctness problem. We also use graph algorithms to compute the dependency relation of Bagnol et al. (2015) and the kingdom ordering of Bellin (1997), and relate them to the notion of blossom which is central to combinatorial maximum matching algorithms. Alternating cycles in perfect matchings serve as witnesses of non-uniqueness, and in this significantly expanded journal version, we discuss connections with other kinds of constrained cycles known to be equivalent: semicycles in directed graphs, trails avoiding forb...

Introduction to linear logic and ludics, part II

Arxiv preprint cs/0501039, 2005

This paper is the second part of an introduction to linear logic and ludics, both due to Girard. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic (section 1) and to ludics, which has been recently developped in an aim of further ...