Forbidden Strings Research Articles

Improved Descriptional Complexity Results for Simple Semi-Conditional Grammars

A simple semi-conditional (SSC) grammar is a form of regulated rewriting system where the derivations are controlled either by a permitting string alone or by a forbidden string alone and this condition is specified in the rule. The maximum length i (j, resp.) of the permitting (forbidden, resp.) strings serves as a measure of descriptional complexity known as the degree of such grammars. In addition to the degree, the numbers of nonterminals and of conditional rules are also counted into the descriptional complexity measures of these grammars. We improve on some previously obtained results on the computational completeness of SSC grammars by minimizing the number of nonterminals and / or the number of conditional rules for a given degree (i, j). More specifically we prove, using a refined analysis of a normal form for type-0 grammars due to Geffert, that every recursively enumerable language is generated by an SSC grammar of (i) degree (2, 1) with eight conditional rules and nine nonterminals, (ii) degree (3, 1) with seven conditional rules and seven nonterminals (iii) degree (4, 1) with six conditional rules and seven nonterminals and (iv) degree (4, 1) with eight conditional rules and six nonterminals.

On the computational completeness of matrix simple semi-conditional grammars

In matrix grammars, context-free rules have to be applied in a certain order. In simple semi-conditional (SSC) grammars, the derivations are controlled either by a permitting string or by a forbidden string associated to each rule. In SSC grammars, the maximum length i of permitting strings and the maximum length j of forbidden strings, the numbers of conditional rules and of nonterminals serve as measures of descriptional complexity and the pair (i,j) is called the degree of such SSC grammars. Matrix grammars with appearance checking with three nonterminals are computationally complete; however, the matrix length is unbounded. Matrix SSC grammars (MSSC) put matrix grammar control on SSC rules. In this paper, we show that MSSC grammars with degrees (2,1), (2,0) and (3,0) are computationally complete, restricting several other descriptional complexity measures. With our constructions, we even bound the matrix length for MSSC grammars.

Computational completeness of simple semi-conditional insertion–deletion systems of degree (2,1)

Insertion–deletion (or ins–del for short) systems are simple models of bio-inspired computing. They are well studied in formal language theory, especially regarding their computational completeness. This concerns the question if all recursively enumerable languages can be generated. This ultimately addresses the question if one can build general-purpose computers rooted in this formalism. The descriptional complexity of an ins–del system is typically measured by its size, a 6-dimensional tuple of non-negative integers $$(e,e',e'';d,d',d'')$$ where e is the maximum length of the insertion string, $$e'$$ (and $$e''$$ ) is the maximum length of the left (and right) context used for insertion; the last three parameters $$d,d',d''$$ are similarly understood for deletion rules. Computational completeness for ins–del systems can even be achieved with rule size (1, 1, 1; 1, 1, 1) but with no rule size strictly smaller than this. This fact has motivated to study ins–del systems in combination with regulation mechanisms. In this context, the six-tuple explained above is called the ID size of a system. Several regulations like graph-control, matrix and semi-conditional have been imposed on ins–del systems. Typically, the computational completeness results are obtained as trade-offs, reducing the ID size, say, to (1, 1, 0; 1, 1, 0) at the expense of increasing other measures of descriptional complexity. In this paper, we study simple semi-conditional ins–del systems, where an ins–del rule can be applied only in the presence or absence of substrings of the derivation string. This brings along two further natural parameters to measure descriptional complexity, namely, the maximum permitting string length p and the maximum forbidden string length f, usually summarized as the degree $$d=(p,f)$$ . We show that simple semi-conditional ins–del systems of degree (2, 1) and with ID sizes $$(1+e,e',e'';1+d,d',d'')$$ are computationally complete for any $$e,e',e'',d,d',d''\in \{0,1\}$$ , with $$e+e'+e''=1$$ and $$d+d'+d''=1$$ . The obtained results complement and improve on the existing results known from the literature. To prove our results, we also show a new normal form for type-0 grammars that appears to be interesting in its own right.

Forbidden traces and forbidden subtraces

The concepts of forbidden strings and forbidden subsequences are generalized to traces. This paper presents algorithms for constructing sets of minimal forbidden traces and minimal forbidden subtraces for a given trace.

Safe Diagnosability of Stochastic Discrete Event Systems

Recently, safe diagnosability of discrete event systems (DESs) was investigated by Paoli and Lafortune, which was viewed as the first necessary step of fault-tolerant supervision. In this paper, we consider the problem of safe diagnosability in the framework of stochastic discrete event systems (SDESs). We define the notion of safe diagnosability for stochastic automata, in which fault detection occurs before any given forbidden string in the failed mode of system is executed. The relationship between diagnosability and safe diagnosability for SDESs is analyzed. In particular, a necessary and sufficient condition for safe diagnosability of SDESs is presented by constructing the recognizer of illegal language and the safe diagnoser. Some examples are described to illustrate the results.

Safe diagnosability for fault-tolerant supervision of discrete-event systems

The problem of achieving fault-tolerant supervision of discrete-event systems is considered from the viewpoint of safe and timely diagnosis of unobservable faults. To this end, the new property of safe diagnosability is introduced and studied. Standard definitions of diagnosability of discrete-event systems deal with the problem of detecting the occurrence of unobservable fault events using model-based inferencing from observed sequences of events. In safe diagnosability, it is required in addition that fault detection occur prior to the execution of a given set of forbidden strings in the failed mode of operation of the system. For instance, this constraint could be required to prevent local faults from developing into failures that could cause safety hazards. If the system is safe diagnosable, reconfiguration actions could be forced upon the detection of faults prior to the execution of unsafe behaviour, thus achieving the objective of fault-tolerant supervision. Necessary and sufficient conditions for safe diagnosability are derived. In addition, the problem of explicitly considering safe diagnosability in controller design, termed “active safe diagnosis problem”, is formulated and solved. A brief discussion of safe diagnosability for timed models of discrete-event systems is also provided.

A combinatorial approach to finding the capacity of the discrete noiseless channel

Shannon defined the capacity of the discrete noiseless channel (DNC) and considered a finite-state model from which the capacity can be calculated. Alternatively, the DNC may be (and often is) represented by a finite list of forbidden strings. In this correspondence, we demonstrate the application of combinatorial techniques to finding the Shannon capacity of the DNC directly from the forbidden list. In our derivations, the case of noninteger symbol durations, as introduced by Csiszar, is considered.