arXiv subjects

Max Kanovich

Publications and source records attributed to Max Kanovich.

17 recordsLinked to original sources

arXiv2024

Technical Report: Time-Bounded Resilience

Most research on formal system design has focused on optimizing various measures of efficiency. However, insufficient attention has been given to the design of systems optimizing resilience, the ability of systems to adapt to unexpected changes or adversarial disruptions. In our prior work, we formalized the intuitive notion of resilience as a property of cyber-physical systems by using a multiset rewriting language with explicit time. In the present paper, we study the computational complexity of a formalization of time-bounded resilience problems for the class of $\eta$-simple progressing planning scenarios, where, intuitively, it is simple to check that a system configuration is critical, and only a finite number of actions can be carried out in a bounded time period. We show that, in the time-bounded model with $n$ (potentially adversarially chosen) updates, the corresponding time-bounded resilience problem for this class of systems is complete for the $\Sigma^P_{2n+1}$ class of the polynomial hierarchy, PH. To support the formal models and complexity results, we perform automated experiments for time-bounded verification using the rewriting logic tool Maude.

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arXiv2023

Explorations in Subexponential Non-associative Non-commutative Linear Logic

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, exhibiting a classical one-sided multi-succedent classical analogue of our intuitionistic system, following the exponential-free calculi of Buszkowski, and de Groote, Lamarche. A large fragment of the intuitionistic calculus is shown to embed faithfully into the classical fragment.

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arXiv2021

On the Complexity of Verification of Time-Sensitive Distributed Systems: Technical Report

This paper develops a Multiset Rewriting language with explicit time for the specification and analysis of Time-Sensitive Distributed Systems (TSDS). Goals are often specified using explicit time constraints. A good trace is an infinite trace in which the goals are satisfied perpetually despite possible interference from the environment. In our previous work (FORMATS 2016), we discussed two desirable properties of TSDSes, realizability (there exists a good trace) and survivability (where, in addition, all admissible traces are good). Here we consider two additional properties, recoverability (all compliant traces do not reach points-of-no-return) and reliability (the system can always continue functioning using a good trace). Following (FORMATS 2016), we focus on a class of systems called Progressing Timed Systems (PTS), where intuitively only a finite number of actions can be carried out in a bounded time period. We prove that for this class of systems the properties of recoverability and reliability coincide and are PSPACE-complete. Moreover, if we impose a bound on time (as in bounded model-checking), we show that for PTS the reliability property is in the $Π_2^p$ class of the polynomial hierarchy, a subclass of PSPACE. We also show that the bounded survivability is both NP-hard and coNP-hard.

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arXiv2020

The Multiplicative-Additive Lambek Calculus with Subexponential and Bracket Modalities

We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill's calculi, and focus on their fragments including multiplicative (Lambek) connectives, additive conjunction and disjunction, brackets and bracket modalities, and the ! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut (cut elimination theorem). We also prove algorithmic undecidability for both calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages.

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arXiv2020

Language Models for Some Extensions of the Lambek Calculus

We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this difference keeps valid for systems with permutation and/or weakening structural rules, that is, intuitionistic linear and affine logics and affine multiplicative-additive Lambek calculus. For the extension of the Lambek with the unit constant, we present a calculus which reflects natural algebraic properties of the empty word. We do not claim completeness for this calculus, but we prove undecidability for the whole range of systems extending this minimal calculus and sound w.r.t. language models. As a corollary, we show that in the language with the unit there exissts a sequent that is true if all variables are interpreted by regular language, but not true in language models in general.

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arXiv2019

Reconciling Lambek's restriction, cut-elimination, and substitution in the presence of exponential modalities

The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called "Lambek's restriction," that is, the antecedent of any provable sequent should be non-empty. In this paper we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek's restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek's restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek's restriction and cut elimination and another system with Lambek's restriction and substitution. For both calculi we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek's restriction is folklore and known to be undecidable.

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arXiv2018

Compliance in Real Time Multiset Rewriting Models

The notion of compliance in Multiset Rewriting Models (MSR) has been introduced for untimed models and for models with discrete time. In this paper we revisit the notion of compliance and adapt it to fit with additional nondeterminism specific for dense time domains. Existing MSR with dense time are extended with critical configurations and non-critical traces, that is, traces involving no critical configurations. Complexity of related {\em non-critical reachability problem} is investigated. Although this problem is undecidable in general, we prove that for balanced MSR with dense time the non-critical reachability problem is PSPACE-complete.

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arXiv2018

On the Complexity of Pointer Arithmetic in Separation Logic (an extended version)

We investigate the complexity consequences of adding pointer arithmetic to separation logic. Specifically, we study extensions of the points-to fragment of symbolic-heap separation logic with various forms of Presburger arithmetic constraints. Most significantly, we find that, even in the minimal case when we allow only conjunctions of simple "difference constraints" (x'\leq x+k) where k is an integer, polynomial-time decidability is already impossible: satisfiability becomes NP-complete, while quantifier-free entailment becomes coNP-complete and quantified entailment becomes P2-complete (P2 is the second class in the polynomial-time hierarchy) In fact we prove that the upper bound is the same, P2, even for the full pointer arithmetic but with a fixed pointer offset, where we allow any Boolean combinations of the elementary formulas (x'=x+k0), (x'\leq x+k0), and (x'<x+k0), and, in addition to the points-to formulas, we allow spatial formulas of the arrays the length of which is bounded by k0 and lists which length is bounded by k0, etc, where k0 is a fixed integer. However, if we allow a significantly more expressive form of pointer arithmetic - namely arbitrary Boolean combinations of elementary formulas over arbitrary pointer sums - then the complexity increase is relatively modest for satisfiability and quantifier-free entailment: they are still NP-complete and coNP-complete respectively, and the complexity appears to increase drastically for quantified entailments.

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arXiv2017

A polynomial time algorithm for the Lambek calculus with brackets of bounded order

Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determ- ining provability of bounded depth formulas in the Lambek calculus with empty antecedents allowed. Pentus' algorithm is based on tabularisation of proof nets. Lambek calculus with brackets is a conservative extension of Lambek calculus with bracket modalities, suitable for the modeling of syntactical domains. In this paper we give an algorithm for provability the Lambek calculus with brackets allowing empty antecedents. Our algorithm runs in polynomial time when both the formula depth and the bracket nesting depth are bounded. It combines a Pentus-style tabularisation of proof nets with an automata-theoretic treatment of bracketing.

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arXiv2017

Time, Computational Complexity, and Probability in the Analysis of Distance-Bounding Protocols

Many security protocols rely on the assumptions on the physical properties in which its protocol sessions will be carried out. For instance, Distance Bounding Protocols take into account the round trip time of messages and the transmission velocity to infer an upper bound of the distance between two agents. We classify such security protocols as Cyber-Physical. Time plays a key role in design and analysis of many of these protocols. This paper investigates the foundational differences and the impacts on the analysis when using models with discrete time and models with dense time. We show that there are attacks that can be found by models using dense time, but not when using discrete time. We illustrate this with a novel attack that can be carried out on most Distance Bounding Protocols. In this attack, one exploits the execution delay of instructions during one clock cycle to convince a verifier that he is in a location different from his actual position. We additionally present a probabilistic analysis of this novel attack. As a formal model for representing and analyzing Cyber-Physical properties, we propose a Multiset Rewriting model with dense time suitable for specifying cyber-physical security protocols. We introduce Circle-Configurations and show that they can be used to symbolically solve the reachability problem for our model, and show that for the important class of balanced theories the reachability problem is PSPACE-complete. We also show how our model can be implemented using the computational rewriting tool Maude, the machinery that automatically searches for such attacks.

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arXiv2017

Subexponentials in non-commutative linear logic

Linear logical frameworks with subexponentials have been used for the specification of among other systems, proof systems, concurrent programming languages and linear authorization logics. In these frameworks, subexponentials can be configured to allow or not for the application of the contraction and weakening rules while the exchange rule can always be applied. This means that formulae in such frameworks can only be organized as sets and multisets of formulae not being possible to organize formulae as lists of formulae. This paper investigates the proof theory of linear logic proof systems in the non-commutative variant. These systems can disallow the application of exchange rule on some subexponentials. We investigate conditions for when cut-elimination is admissible in the presence of non-commutative subexponentials, investigating the interaction of the exchange rule with local and non-local contraction rules. We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.

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arXiv2017

Simulating Linear Logic in 1-Only Linear Logic

Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. It turned out that full propositional Linear Logic is undecidable (Lincoln, Mitchell, Scedrov, and Shankar) and, hence, it is more expressive than (modalized) classical or intuitionistic logic. In this paper we focus on the study of the simplest fragments of Linear Logic, such as the one-literal and constant-only fragments (the latter contains no literals at all). Here we demonstrate that all these extremely simple fragments of Linear Logic (one-literal, $\bot$-only, and even unit-only) are exactly of the same expressive power as the corresponding full versions. We present also a complete computational interpretation (in terms of acyclic programs with stack) for bottom-free Intuitionistic Linear Logic. Based on this interpretation, we prove the fairness of our encodings and establish the foregoing complexity results.

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arXiv2017

Undecidability of the Lambek calculus with subexponential and bracket modalities

The Lambek calculus is a well-known logical formalism for modelling natural language syntax. The original calculus covered a substantial number of intricate natural language phenomena, but only those restricted to the context-free setting. In order to address more subtle linguistic issues, the Lambek calculus has been extended in various ways. In particular, Morrill and Valentin (2015) introduce an extension with so-called exponential and bracket modalities. Their extension is based on a non-standard contraction rule for the exponential that interacts with the bracket structure in an intricate way. The standard contraction rule is not admissible in this calculus. In this paper we prove undecidability of the derivability problem in their calculus. We also investigate restricted decidable fragments considered by Morrill and Valentin and we show that these fragments belong to the NP class.

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arXiv2016

Biabduction (and Related Problems) in Array Separation Logic

We investigate array separation logic (ASL), a variant of symbolic-heap separation logic in which the data structures are either pointers or arrays, i.e., contiguous blocks of allocated memory. This logic provides a language for compositional memory safety proofs of imperative array programs. We focus on the biabduction problem for this logic, which has been established as the key to automatic specification inference at the industrial scale. We present an NP decision procedure for biabduction in ASL that produces solutions of reasonable quality, and we also show that the problem of finding a consistent solution is NP-hard. Along the way, we study satisfiability and entailment in our logic, giving decision procedures and complexity bounds for both problems. We show satisfiability to be NP-complete, and entailment to be decidable with high complexity. The somewhat surprising fact that biabduction is much simpler than entailment is explained by the fact that, as we show, the element of choice over biabduction solutions enables us to dramatically reduce the search space.

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arXiv2016

Undecidability of the Lambek calculus with a relevant modality

Morrill and Valentin in the paper "Computational coverage of TLG: Nonlinearity" considered an extension of the Lambek calculus enriched by a so-called "exponential" modality. This modality behaves in the "relevant" style, that is, it allows contraction and permutation, but not weakening. Morrill and Valentin stated an open problem whether this system is decidable. Here we show its undecidability. Our result remains valid if we consider the fragment where all division operations have one direction. We also show that the derivability problem in a restricted case, where the modality can be applied only to variables (primitive types), is decidable and belongs to the NP class.

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arXiv2016

Timed Multiset Rewriting and the Verification of Time-Sensitive Distributed Systems

Time-Sensitive Distributed Systems (TSDS), such as applications using autonomous drones, achieve goals under possible environment interference (\eg, winds). Moreover, goals are often specified using explicit time constraints which must be satisfied by the system \emph{perpetually}. For example, drones carrying out the surveillance of some area must always have \emph{recent pictures}, \ie, at most $M$ time units old, of some strategic locations. This paper proposes a Multiset Rewriting language with explicit time for specifying and analysing TSDSes. We introduce two properties, \emph{realizability} (some trace is good) and \emph{survivability} (where, in addition, all admissible traces are good). A good trace is an infinite trace in which goals are perpetually satisfied. We propose a class of systems called \emph{progressive timed systems} (PTS), where intuitively only a finite number of actions can be carried out in a bounded time period. We prove that for this class of systems both the realizability and the survivability problems are PSPACE-complete. Furthermore, if we impose a bound on time (as in bounded model-checking), we show that for PTS, realizability becomes NP-complete, while survivability is in the $\Delta_2^p$ class of the polynomial hierarchy. Finally, we demonstrate that the rewriting logic system Maude can be used to automate time bounded verification of PTS.

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arXiv2015

Horn Linear Logic and Minsky Machines

Here we give a detailed proof for the crucial point in our Minsky machine simulation - that any linear logic derivation for a specific Horn sequent can be transformed into a Minsky computation leading from an initial configuration to the halting configuration. Among other things, the presentation advantage of the 3-step program is that the non-trivial tricky points are distributed between the independent parts each of which we justify following its own intrinsic methodology (to say nothing of the induction used in the opposite directions): (1) From LL to HLL - we use purely proof-theoretic arguments. (2) From HLL to Horn programs - we translate trees (HLL derivations) into another trees (Horn programs)of the same shape, almost. (3) From Horn programs to Minsky computations - we use purely computational arguments.

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